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Chapter 2 Shifts in Supply and Demand Influence Price 2.

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1

2 Chapter 2

3 Shifts in Supply and Demand Influence Price 2

4 3 Economists Love Competitive Markets Demand Coal Qd = f (Pc-, Psb+, Pcm-, Y, T+/-, Pol+/-, #buy+) Ceteris Paribus hold constant everything but P & Q P Q D

5 4 World Coal Use by Sector

6 5 Economists Love Competitive Markets Demand Coal Qd = f (Pc, Psb, Pcm, Y, Tech, Policy, #buy) Ceteris Paribus hold constant everything but P & Q P Q D -+ -+/- +

7 6 Supply Suppliers Qs = f(Pc, Pf, Psm, Pby,T, Pcy, #sel) +--+++/- + P Q S

8 Sum Up 7 Where are coal reserves Conversions E 1 in unit 1, (u1) or E 2 in unit 2 (u2) conversion is units of 1 per unit of 2 (u1/u2) Energy Content

9 Sum Up 8 Qualitative create D and S hold all variables but P&Q constant started to look behind supply Q S D P

10 Behind Supply for firm to maximize profits  = P*Q – TC = P*Q – FC – VC(Q) competitive firms take price as given f.o.c.  /  Q = P -  TC/  Q = P -  VC(Q)/  Q = 0 MC ↑ 2.o.c  2  /  Q 2 = -  TC 2 /  Q 2 = -  MC(Q)/  Q<0  MC(Q)/  Q>0 operate where price equals marginal variable cost short run supply equals marginal cost curve 9

11 Typical Competitive Firm Cost Short Run Supply 10 P Q AVC 1 MC 1 P Q AVC 2 MC 2 Market is horizontal sum S i = MC i above AVC Q S D P sr P Q1Q1 Q2Q2 Q 1 +Q 2

12 11 Where They Cross Determines P & Q Supply = Demand P Q S D Model Building Blocks PePe QeQe

13 12 Out of Equilibrium P Q S D QdQd QsQs PLPL Price too low Price too high PHPH

14 13 Shift in D Change in Q s – movement along S P Q S DD' (increase)→ D"(decrease)← PePe QeQe Pe"Pe" Pe'Pe' Qe'Qe'Qe"Qe"

15 14 Shift in S Movements along the D curve P Q S D S' (increase)→ S"(decrease)← PePe QeQe Pe'Pe' Pe"Pe" Qe"Qe" Qe'Qe'

16 15 More than one Change Coal Mine Productivity Per Miner Increases P Q →S' D S QeQe PePe Pe'Pe' Qe'Qe' PQPQ

17 16 1.Chinese Coal Mine Productivity  2. Plus Cheaper Sequestration P Q S D  S' Qe'Qe' Pe'Pe' PePe QeQe Q →D' D S QeQe PePe Pe'Pe' Qe'Qe' P ↓ Q ↑P ↑ Q↑

18 Supply and Demand Building Blocks-Two markets 17 Natural Gas Coal if all market - general equilibrium

19 Two Markets Qdo = a + bPo +cPg + dY Qso = e + fPo + gPG + hCost Cost Exogenous Y Exogenous Qdg = i + jPo + kPg + lY Qsg = m + nPo + oPG + pCost Qdo = Qso Qdg = Qsg 6 endogenous variables, 6 equations 18

20 Supply and Demand Building Blocks Dynamic -Two time periods 19 Time 2 Time 1 n time periods

21 Trade Models- Two Areas in World 20 P Q1Q1 Q2Q2 QwQw S1S1 S1S1 S2S2 D1D1 D2D2 D 1 +D 2 D S 1 +S 2

22 Market Power Seller 21 P Q S D

23 Market Power Buyer 22 P Q S D

24 23 Quantitative Models Chapter 2&3 Buyers Qd = f(Pc, Psb, Pcm, T, Pot, Pcy, #buy) Suppliers Qs = f(Pc, Pf, Psm, Pby,T, Pcy, #sel) Functions – with numbers often start with qualitative model to get intuition

25 24 Quantitative S-D Example Qd =99 - 2Pc + 1Psb - 2Pcm + 0.1Y Qs =30 + 1Pc – 1Pk - 0.2Pl - 0.4Pnr Pc = price of coal Psb = price of substitute to coal (natural gas) =1 Pcm = a complement to coal =10 Y = income = 200 Pk = price of capital = 20 Pl = price of labor = 40 Pnr = price of other natural resources used in production of coal = 10

26 25 Qd = 99 - 2Pcd + Psb - 2Pcm + 0.1Y Qs =30 + 1Pcs – 1Pk - 0.2Pl - 0.4Pnr Qd = 99 - 2Pcd + 1 - 2*10+0.1*200 = 100 -2Pcd Qs = 30 + Pcs – 1*31 - 0.2*30 - 0.4*10 = -11 + 1Pcs

27 26 Model02.xls: Worksheet S&D

28 27 Inverse Demand Qd = = 100 -2Pcd Qs = -11 + 1Pcs Sometimes want price as function of quantity invert Qd = 100 -2Pcd solve demand for Pcd 2Pcd = 100 – Qd → Pcd = 50 – (1/2)Qd invert Qs = -11 + 1Pcs solve supply for Pcs Pcs = 11 + Qs

29 28 Graph and Forecast Pd = 50 – (1/2)Qd Ps = 11 + Qs Forecast P & Q Pd = Ps 50 – (1/2)Q = 11 + Q 50-11 = Q+(1/2)Q 39 = (3/2)Q Q = (2/3)39 = 26 Pd = 50– (1/2)26 = 37 Ps = 11 + 26 = 37 Q P 60 20 40 100 50 D S P = 37 Q = 26

30 29 Is Equilibrium Stable? Price above Equilibrium Pd = 50 – (1/2)Qd Qd=100 – 2Pd Ps = 11 + Qs Qs = -11 + Ps What if P = 40 Qd=100 – 2*40 = 20 Qs = -11 + 40 = 29 Excess quantity supplied P↓ Q P 60 20 100 D S P = 40 Qd = 20 Qs = 29

31 30 Quantitative Need numbers for ceterus paribus values Substitute in to Qd and Qs Q d = f(P d ) is demand Q s = f(P s ) is supply Solve for P and Q Sometimes inverse is easier or more useful Solve for price as a function of quantity P d = f -1( Q d ) is inverse demand We graph the inverses

32 General Equilibrium Model (1) Think about but not to be tested 31 Markets for all products all factors or production consumers buy m final goods: a,b,c,…. at prices p b, p c, p d,…. their demand for final goods: d b, d c, d d,…. consumers own and sell n factors of production: q t, q p, q k, …. at prices p t, p p, p k, … their supply/of n factors: s t, s p, s k,… m + n unknown prices

33 General Equilibrium Model (2) Think about but not to be tested 32 in real world things are priced in money $/liter, etc in simplest G.E. model no money pick a numeraire good its price is one m + n - 1 unknown prices equilibrium in household sector s t p t + s p p p + s k p k + …. = d a + d b p b + d c p c + …. income = expenditure If holds for each household, holds for market

34 General Equilibrium Model (3) Think about but not to be tested 33 producers buy n factors of production demand: d t, d p, d k, producers produce m end use goods s demand: s t, s p, s k, m commodities and n factors there are m+n unknowns quantities m + n - 1 unknown prices total: 2m + 2n - 1 unknowns

35 General Equilibrium Model (4) Think about but not to be tested 34 Consumer Demand for goods (m-1 independent) d a = d a (p t, p p, p k, …, p b, p c, p d,…) Supply of factors (n) s t = s t (p t, p p, p k, …, p b, p c, p d,…) Producers Demand for factors (n) d t = d t (p t, p p, p k, …, p b, p c, p d,…) Supply of goods m s a = s a (p t, p p, p k, …, p b, p c, p d,…)

36 Last Time - Sum Up 35 Qualitative create D and S hold all variables but P&Q constant Q S D P

37 36 Models for Policy What if Government Sets Maximum Price of 30 Shortages Likely to be black market Could to subsidize What would subsidy cost? To get suppliers to produce 40 Need Ps=11+40 =51 Cost (51-30)(40)=840 Q P 60 100 D S P = 30 Qd = 40Qs = 19 Ps = 51

38 37 What happens with following policies? Q P 60 100 D S Pmin Pmax Controls Non-binding

39 38 Demand Price Elasticity Q responsiveness to price P Q D lr P1P1 Q1Q1 D1D1 may change over time Q2Q2 P2P2 Q lr

40 39 Back to 1973 Oil Market P Q D lr D1D1 Q 79 P 79 Q 82 S 73 S 79 OPEC Supply Shocks 73&79

41 40 Elasticity Definition How much quantity responds to price  d = % change quantity % change in price If  d = –0.5 price goes up by 100%, quantity demanded falls by % change quantity = % change in price*  d = 100%*-0.5 = 50%

42 41 Let’s Develop Formal Definition  d = % change quantity % change in price  Q d *100  d = Q d  P d *100 P d Q 2 -Q 1 = Q 1 P 2 - P 1 P 1

43 42 Suppose We Have Price Increase P Q $2.00/g 500  10 6 g/d $3.00/g 400  10 6 g/d  Q d  d = Q d  P d PdPd (400  10 6 g/d – 500  10 6 g/d) 500 10 6 g/d ($3.00 g – $2.00 g) $2.00 /g = -0.20/0.5 = -0.4 (no units)

44 43 Lets Go Back to Lower Price P Q $2.00/g 500  10 6 g/d $3.00/g 400  10 6 g/d Q 2 – Q 1  d = Q 1 P 2 -P 1 P 1 (500 – 400) 400 = (1/4) = (2– 3) -(1/3) 3 = -(1/4)(3/1) = - 3/4 = - 0.75

45 44 Sum Up Computing Arc Elasticities  d = % change quantity % change in price  Q d  d = Q d  P d P d Q 2 -Q 1 = (Q 1+ Q 2 )/2 P 2 - P 1 (P 1+ P 2 )/2

46 45 Sum Up Elasticity = Responsiveness to Price  x = % change quantity % change in X Q could be quantity demanded Q could be quantity supplied X could be Price X could be income X could be the price of a substitute (cross price elasticity) X could be any other variable that influences Q Q likely more responsive in long run than short run

47 46 More Convenient for Elasticity Q s and Q d responsiveness to other variables  x = % change quantity % change in  Q  x = Q =  Q X  X  X Q X Take limit as  X→0  x =  Q X  X Q

48 47 Where do they come from? Estimate whole function market data Q d = f(P d, Y, P s, P c,..., etc. ) ε p =  Q P   P Q Function Forms linear: Q = a – bP ε p = -b(P/Q)

49 48 Linear Function Q = a - bP  p = -b(P/Q) Graph: P = 0 then Q = a -b*0 = a  = -b(0/a) = 0 Q = 0 = a-bP then P = a/b  = -b(a/b)/0 = -  P = (a/b)/2 then Q = a - b(a/b)/2) = a - a/2 = a/2  = -b(a/b)/2/(a/2) = -1 =-1 Q =0 a/b P =-  D a a/2b a/2 |Elastic| > 1 |Unit Elastic| = 1 |Inelastic| (1,0)

50 49 Demand Price Elasticities and Revenues How Does Price Change Revenue TR = PQ = PQ(P)  TR/  P = Q + (  Q/  P)*P =Q(1+ (  Q/  P)*(P/Q)) Sign of  TR/  P = sign (1+ε p )  TR/  P < 0 when (1+ε p )<0 subtract -1 from both sides ε p <-1 (elastic) = Q(1+ε p ) Raising price lowers revenue Lowering price raises revenue

51 50 Demand Price Elasticities and Revenues  TR/  P < 0 when ε p <-1 elastic P and TR opposite direction P  TR  P  TR   TR/  P = 0 when (1+ε p )=0 ε p = -1 unitary elasticity  TR/  P > 0 when (1+ε p )>0 0> ε p > -1 P  TR? P  TR?

52 51 Demand Price Elasticities and Revenues  TR/  P < 0 when ε p <-1 elastic P and TR opposite direction P  TR  P  TR   TR/  P = 0 when (1+ε p )=0 ε p = -1 unitary elasticity  TR/  P > 0 when (1+ε p )>0 0> ε p > -1 P  TR? P  TR?

53 52 Demand Price Elasticities and Revenues  TR/  P < 0 when ε p <-1 elastic P and TR opposite direction P  TR  P  TR   TR/  P = 0 when (1+ε p )=0 ε p = -1 unitary elasticity  TR/  P > 0 when (1+ε p )>0 0> ε p > -1 P  TR? P  TR?

54 53 Elasticities and Revenues Intuition  d = % change quantity % change in price Revenue = P*Q P  TR  Q  TR  so what happens to TR? depends on whether P or Q effect larger elastic -2 1 unitary elastic -1 1 inelastic -1 2

55 54 Elasticities and Revenues Intuition  d = % change quantity % change in price Revenue = P*Q P  TR  Q  TR  so what happens to TR? depends on whether P or Q effect larger elastic -2 1 unitary elastic -1 1 inelastic -1 2

56 55 Where do they come from? Other Function Forms multiplicative: Q = aP -b ε p = -baP -b-1 P/Q = -baP -b /aP -b = -b Linearize ln Q = ln a -blnP Another way to write  lnQ = -b = ε p =  Q P  lnP  P Q

57 56 Be Able to Compute for Other Functional Forms Other functional forms ln(Q) = a – bP + cY Q = a – bln(P) + c lnY ln(Q) = a - blnP + clnP 2 + dlnY Q = a + bP +cY +dPY

58 57 Good for Back of the Envelope Forecasting New Q = Q+  Q = Q(1+  Q/Q)

59 58 Sum Up Price Elasticity  P = % change quantity % change in P Three ways to compute Q 2 -Q 1  p = Q 1+ Q 2 )/2 P 2 - P 1 (P 1+ P 2 )/2 =  Q P  P Q =  lnQ  lnP

60 59 Price elasticity and revenue elastic P↑ → TR ↓ and P ↓ → TR ↑ elastic P↑ → TR ↑ and P ↓ → TR ↓

61 L9 - More on D&S Responsiveness Elasticities 60 P Q D1D1 S1S1 D2D2 S2S2

62 Direct Purchases 61

63 Direct Purchases vs Cradle to Grave 62

64 63 Last Time Quantitative S-D Example Qd =99 - 2Pc + 1Psb - 2Pcm + 0.1Y Qs =30 + 1Pc – 1Pk - 0.2Pl - 0.4Pnr Pc = price of coal Psb = 1 Pcm = 10 Y = 200 Pk = 20 Pl = 40 Pnr = 10 Qd = 99 - 2Pcd + 1 - 2*10+0.1*200 = 100 -2Pcd Qs = 30 + Pcs – 1*31 - 0.2*30 - 0.4*10 = -11 + 1Pcs Invert, Shift, Solve for equilibrium, Price controls

65 64 Last Time Price Elasticity How responsive Q s or Q d is to price flatter is more responsive  P = % change quantity % change in P P Q D lr D1D1 Q S lr S1S1

66 65 Last Time Price Elasticity  P = % change quantity % change in P Three ways to compute Q 2 -Q 1  p = Q 1+ Q 2 )/2 P 2 - P 1 (P 1+ P 2 )/2 =  Q P  P Q =  lnQ  lnP

67 66 Last Time Price elasticity and revenue elastic P↑ → TR ↓ and P ↓ → TR ↑ inelastic P↑ → TR ↑ and P ↓ → TR ↓

68 67 Elasticities to Forecast Oil Price $40 to $60,  p = -0.2, Q = 80 mb/d New Q = 80-8 = (1-0.1)80 = 72 million b/d  Q = 80*(-0.1) = - 8 mb/d

69 P  Q and Q  P 68 Q D P

70 69 Compute Price Increase P = $2.50 E p = -0.08  Q = -0.10 spread over 8 weeks = -0.0125 Q  P =  Q/Q = -0.0125 = 0.208 P ε p -0.08  P = 0.208*2.5 = 0.521 P new = P +  P = $2.50 + $0.521 = $3.021

71 70 Compute Price Increase P = $1.70 E p = -0.08  Q = -0.10 Q  P =  Q/Q = -0.10 = 1.25 P ε p -0.08  P = 1.25*1.70 = 2.125 P new = P +  P = $1.70 + $2.125 = $3.825

72 71 Sum Up Defined arc and point elasticities Uses of Demand Price Elasticity Relationship of Revenue, Price and Elasticity Simple Forecasting 1. ΔQ=ε p ΔP Q P 2. ΔP = εp ΔQ P Q

73 72 L7- More on Elasticities =-1 Q =0 a/b P =-  D a a/2b a/2 |Elastic| > 1 |Unit Elastic| = 1 |Inelastic| (1,0)

74 73 Last Time Defined demand price elasticities Arc Q 2 -Q 1 (Q 1 + Q 2 )/2 P 2 - P 1 (P 1 + P 2 )/2 Point (∂Q/∂P)(P/Q) = ∂lnQ/∂lnP Relationship of Revenue, Price and Elasticity

75 74 Last Time Demand Elasticities and Revenue ε p =  LnQ =  Q P  LnP  P Q Elastic < -1  TR/  P= 1+ε p <0 P  TR  P  TR  Unit Elastic = -1  TR/  P= 1+ε p = 0 P  TR  P  TR  Inelastic (-1,0)  TR/  P=1+ε p >0 P  TR? P  TR ?

76 75 Last Time Simple Forecasting 1. ΔQ=ε p ΔP Q P 2. ΔP = ε p ΔQ P Q

77 76 a. If ε y China = 0.8, Q = 3 billion tons per year (almost half of world's total) income grows at historical rate of about 9% what is  Q china ? Forecast Using Income Elasticity = 0.8*0.09*3 = 0.216 billion tonnes

78 77 Price Change to Offset Coal Growth Let ε y China = 0.8,  Y/Y=0.09 ε p = -0.5 What  P/P do we need to choke off coal growth

79 78 Cross Price Elasticities of Demand  Px = % change quantity % change in another good X price If Q and X are substitutes what is sign  Px ? coal and natural gas for electricity natural gas and electricity for heating If Q and X are complements what is sign  Px ? coal and boiler gasoline and automobile + -

80 79 Cross Price Iooty, Queiroz, and Roppa (2007) Cross price elasticity of ethanol with respect to gasoline substitute or complement? ε pg =+0.6 Q Eth = 100 P g = $2 per gallon Increases to $3  Q/Q = ε Pg (  P/P) = 0.6*(1/2) = 0.3 Q new = Q(1+  Q/Q) = 100(1+0.3) = 130 What might happen to ethanol price? What might happen to sugar price?

81 80 Create Function from Elasticity Q =  P  Y  Can add more variables ε p = -0.80 ε y = 1.40 P =$1.15 Q = 8.00 Y = 5.40  = -0.8  = 1.4 Q =  P  Y  =  (1.15 -0.8 5.4 1.4 )  = Q/(P  Y  ) = 8/(1.15 -0.8 5.4 1.4 ) = 0.84 Q = 0.84P -0.8 Y 1.4

82 81 Elasticities to create demand equations linear (Q = a + bP + cY) around the following values. Price Elasticity ε p = -0.80 Income Elasticity ε y = 1.40 Price per gallon =$1.15 Consumption millions of barrels per day = 8.00 Income in trillions of U.S. dollars = 5.40

83 82 Create Demand from Elasticities Q = a + bP + cY ε p = -0.80 ε y = 1.40  p = (dQ/dP)(P/Q) = b*(P/Q) -0.8 = b(1.15/8) b = -0.8*8/1.15=-5.57  y = (dQ/dY)Y/Q = c*(Y/Q) 1.4=c(5.4/8) c = 1.4*8/5.4 = 2.07 a = Q - bP - cY = 8 - (-5.57)*1.15 - 2.07*5.4 = 3.2 Q = 3.2 - 5.57Pd + 2.07Yd Could add another variable X Need values ε x and X P =$1.15 Q = 8.00 Y = 5.40

84 83 Example: Q 1 = P 1  Q 2 =P 2  Q 2 /Q 1 = (P 2 /P 1 )  if P growing at 2 then P 2 /P 1 = 1.02P 1 /P 1 = 1.02 Q 2 /Q 1 = (1.02)  only need ratio prices or growth rates no units to forecast Q 2 multiply Q 1 by your forecast of Q 2 /Q 1 only works exactly for constant elasticity functions Another Way With Constant Elasticity of Demand

85 84 ^ Q 2 = (0.98P s1 ) b2 (1.01P pl ) b3 (1.02P al ) b4 (1.015Y 1 ) b5 Q 1 P s1 b2 P pl b3 P al b4 Y 1 b5 = 0.98 b2 1.01 b3 1.02 b4 1.015 b5 = 0.98 -0.82657 1.01 0.199 1.02 0.43714 1.015 1.10167 = 1.0447 Q 1 = 111.9 ^ Q 2 = 1.0447*Q 1 = 1.0447*111.9= 116.9 Forecasts with Elasticity-2

86 85 Sum Up: Why are Demand Elasticities important? Why are they important? Forecast P → Q ΔQ/Q = ε p (ΔP/P) Q → P ΔP/P = (ΔQ/Q)/ε p Y → Q ΔQ/Q = ε y (ΔY/Y) P other → Q ΔQ/Q = ε o (ΔP o /P o ) P,Y, etc.  Q ΔQ 1 /Q 1 + ΔQ 2 /Q 2 Policy analysis P to offset Y increase Effect of carbon tax Create demand from elasticities linear and log

87 Chapter 3

88 87 Elasticities so Far 1. Measure of responsiveness 2. Where do they come from? a. compute from market data P Q 6 80 4 100 (Q 2 -Q 1 ) = (Q 1 + Q 2 )/2 P 2 - P 1 (P 1 + P 2 )/2 problem if other variables change beside P

89 88 Elasticities so Far 2. Where do they come from? b. Estimate whole function market data Q d = f(P d, Y, P s, P c,..., etc. ) ε p =  LnQ =  Q P  LnP  P Q Function Forms linear: Q = a – bP ε p = -b(P/Q) multiplicative: Q = aP -b ε p = -b mixed: ln(Q) = a – bP ε p = -bP mixed: Q = a – bln(P) ε p = -b/Q Other

90 89 Demand Elasticities ε x =  LnQ =  Q X  LnX  X Q |Elastic| > 1 |Unit Elastic| = 1 |Inelastic| (1,0) 3. Uses of elasticity price to revenue (P*Q) forecasting policy: price increase to offset income growth PQPQQPQPYQYQP cross  Q

91 90 Price Change to Offset Coal Growth Let ε y China = 0.8,  Y/Y=0.09 ε p = -0.5 What  P/P do we need to choke off coal growth

92 91 Y  Q and Q  Y? Q D P D' PoPo

93 92 Studies of Oil Price on U.S. Macro Economy  GDP Po  Po GDP studies seem to suggest around 0.05 smaller than in 1970s and 1980s asymmetric affect when prices up not down mechanism GDP (K, L, O, etc.) - less oil GDP  P o  more inflation, tighter monetary policy r up, GDP down P o  income transfer to OPEC

94 93 Elasticity Approximation - Linear Q = 20-4P P =3 Q = 8 ε p =-4*3/8 = -1.5 P2=4P2=4 Use elasticitydQ/Q = ε p *dP/P = -1.5*1/3 = -0.5 Q new = Q(1+dQ/Q) = 4 With functionQ new = 20 - 4*4 = 8(1-0.5) = 4

95 94 Elasticity Approximation - Log Q = 10P -1 P = 2 Q = 5 ε p = -1 P 2 = 2.1 Use elasticity: dQ/Q = ε p *dP/P = -1*0.1/2 = -0.05 Q new = Q(1+dQ/Q)= 5*(1-0.05) =4.75 With functionQ new = 10(2.1) -1 = 4.76 Approximation gets worse the larger the price change

96 95 Cross Price Iooty, Queiroz, and Roppa (2007) Cross price elasticity of ethanol with respect to gasoline substitute or complement? ε pg =+0.6 Q Eth = 100 P g = $2 per gallon Increases to $3  Q/Q = ε Pg (  P/P) = 0.6*(1/2) = 0.3 Q new = Q(1+  Q/Q) = 100(1+0.3) = 130 What might happen to ethanol price? What might happen to sugar price?

97 96 Create Function from Elasticity Q =  P  Y  Can add more variables ε p = -0.80 ε y = 1.40 P =$1.15 Q = 8.00 Y = 5.40  = -0.8  = 1.4 Q =  P  Y  =  (1.15 -0.8 5.4 1.4 )  = Q/(P  Y  ) = 8/(1.15 -0.8 5.4 1.4 ) = 0.84 Q = 0.84P -0.8 Y 1.4

98 97 Elasticities to create demand equations linear (Q = a + bP + cY) around the following values. Price Elasticity ε p = -0.80 Income Elasticity ε y = 1.40 Price per gallon =$1.15 Consumption millions of barrels per day = 8.00 Income in trillions of U.S. dollars = 5.40

99 98 Create Demand from Elasticities Q = a + bP + cY ε p = -0.80 ε y = 1.40  p = (dQ/dP)(P/Q) = b*(P/Q) -0.8 = b(1.15/8) b = -0.8*8/1.15=-5.57  y = (dY/dQ)Y/Q = c*(Y/Q) 1.4=c(5.4/8) c = 1.4*8/5.4 = 2.07 a = Q - bP - cY = 8 - (-5.57)*1.15 - 2.07*5.4 = 3.2 Q = 3.2 - 5.57Pd + 2.07Yd Could add another variable X Need values ε x and X P =$1.15 Q = 8.00 Y = 5.40

100 99 Example: Q 1 = P 1  Q 2 =P 2  Q 2 /Q 1 = (P 2 /P 1 )  if P growing at 2 then P 2 /P 1 = 1.02P 1 /P 1 = 1.02 Q 2 /Q 1 = (1.02)  only need ratio prices or growth rates no units to forecast Q 2 multiply Q 1 by your forecast of Q 2 /Q 1 only works exactly for constant elasticity functions Another Way With Constant Elasticity of Demand

101 100 ^ Q 2 = (0.98P s1 ) b2 (1.01P pl ) b3 (1.02P al ) b4 (1.015Y 1 ) b5 Q 1 P s1 b2 P pl b3 P al b4 Y 1 b5 = 0.98 b2 1.01 b3 1.02 b4 1.015 b5 = 0.98 -0.82657 1.01 0.199 1.02 0.43714 1.015 1.10167 = 1.0447 Q 1 = 111.9 ^ Q 2 = 1.0447*Q 1 = 1.0447*111.9= 116.9 Forecasts with Elasticity-2

102 101 Example: Q 1 = P 1  Q 2 =P 2  Q 2 /Q 1 = (P 2 /P 1 )  if P growing at 2 then P 2 /P 1 = 1.02P 1 /P 1 = 1.02 Q 2 /Q 1 = (1.02)  only need ratio prices or growth rates no units to forecast Q 2 multiply Q 1 by your forecast of Q 2 /Q 1 only works exactly for constant elasticity functions Another Way With Constant Elasticity of Demand

103 102 ^ Q 2 = (0.98P s1 ) b2 (1.01P pl ) b3 (1.02P al ) b4 (1.015Y 1 ) b5 Q 1 P s1 b2 P pl b3 P al b4 Y 1 b5 = 0.98 b2 1.01 b3 1.02 b4 1.015 b5 = 0.98 -0.82657 1.01 0.199 1.02 0.43714 1.015 1.10167 = 1.0447 Q 1 = 111.9 ^ Q 2 = 1.0447*Q 1 = 1.0447*111.9= 116.9 Forecasts with Elasticity

104 103 Tax Qualitative tax affect on price, quantity, government revenue or cost incidence social welfare of tax Unit Ps + t = Pd add to supply Ps = Pd-t subtract from demand Ps Pd-t P Q Qe Pe Ps+t Pd

105 104 Tax Supplier What happens to P and Q Ps + t = Pd add to supply Ps+t Pd P Q Qe Pe Ps Qe’ Pd’ Ps’

106 105 Tax Government Revenues Ps+t Pd P Q Qe Pe Ps Qe’ Pd’ Ps’ t

107 106 Coal Ad Valorem Tax 50 % of Price P Q ←(1+t%)Ps Pd Ps QeQe PePe Pd'Pd' Qe'Qe' tax Ps'Ps' Ad Valorem 50% of Ps (1+0.5)Ps = Pd

108 107 Tax Demander Subtract from Pd: What happens to P and Q Ps = Pd -t Pd-t Pd P Q Qe Pe Ps Qe’ Pd’ Ps’

109 108 Coal Ad Valorem Tax 50 % of Buyer Price P Q ←(1-t%)Pd Pd Ps QeQe PePe Pd'Pd' Qe'Qe' tax Ps'Ps' Ad Valorem 50% of Pd (1-0.5)Pd = Ps

110 109 Quantitative Model Tax supplier Q d = 30 -2P d Q s = -3 + P s Solve for equilibrium 30 -2P=-3+P P = 11, Q = 8 Add tax of 6 to supply price Invert demand and supply P d = 15 - 0.5Q d P s = 3 + Q s P d =15 - 0.5Q d = P s +t = 3 + Q s + 6 Solve Q = 4, P d =13, P s = 7 PsPs PdPd P Q4 11 15 30 3 P s +6 8 13 7

111 110 Government Revenues Supplier tax of 6 Q = 4 t = 6 t*Q = 6*4=24 PsPs PdPd P Q4 11 15 30 3 P s +6 6 13 7

112 111 Who Pays the Tax Depends on Shape of Demand and Supply P Q P Q PdPd PdPd PsPs PsPs P s +t QeQe QeQe PePe PePe Ps'Ps' Pd'Pd' Qe'Qe' Perfectly Elastic Supply P s +t Qe'Qe' P s '= Pd'Pd' share the taxconsumer pays

113 112 Incidence of Tax Depends Shape of Supply and Demand (Practice Four Extreme Cases) P Q P Q PsPs P Q PdPd P Q P s +t D S PdPd PsPs PsPs P d -t

114 113 Incidence Depends on Elasticity Inputs:  d = -0.5  s = 1 t = 4.5 dPd = ε s dPs ε d dPd+|dPs| = t dPd-dPs = t dPd=t+dPs But also dPd = ε s /ε d dPs = (1/-0.5)dPs (1/-0.5)dPs-dPs = 4.50 -3dPs=4.50 dPs =-1.50 dPd =4.5 + (-1.5)=3

115 114 Social Welfare Effects: Behind the Supply Curve Perfect competitors take P from market  = PQ – TC(Q) pick Q to maximize f.o.c  /  Q = P –  TC(Q)/  Q = 0 P – MC = 0 P = MC 2.o.c  2  /  Q 2 = –  TC(Q) 2 /  Q 2 < 0 TC(Q) 2 /  Q 2 > 0 MC slopes up - increasing marginal cost

116 115 Social Welfare - Producer Surplus P Q PsPs PdPd QeQe PePe Price Set by Marginal Producer and Consumer Ricardian Rent

117 116 Social Welfare - Consumer Surplus P Q P s =MC QeQe PePe Pd=Marginal Benefit

118 117 Social Welfare Effects: Behind the Supply Curve Perfect competitors take P from market  = PQ – TC(Q) pick Q to maximize f.o.c  /  Q = P –  TC(Q)/  Q = 0 P – MC = 0 P = MC 2.o.c  2  /  Q 2 = –  TC(Q) 2 /  Q 2 < 0 TC(Q) 2 /  Q 2 > 0 MC slopes up - increasing marginal cost

119 118 Social Welfare Q d = 30 -2P d Q s = -3+ P s Invert demand P d = 15 - 0.5Q d P s = 3 + Q s Solve for Equilibrium P = 11, Q = 8 PsPs PdPd P Q 8 11 Consumer Surplus = ½(15-11)*8 =16 15 30 3 Producer Surplus = (1/2)(11-3)*8= 32

120 119 Welfare Cost of a Tax P d = 15 - 0.5Q d P s = 3 + Q s t = 6 Invert demand P s + t = Pd 3 + Q s + 6 = 15 - 0.5Q d 1.5Q = 6 Q = 4 P s = 7 P d = 13 PsPs PdPd P Q P e =11 CS=16 15 30 P d =13 PS=32 P s +6 4 3 P s =7 Tax revenues = 6*4 = 24 DWL = (1/2)(13-7)*(8-4) 8

121 120 Welfare Cost of a Subsidy P d = 15 - 0.5Q d P s = 3 + Q s Q e =8, P e =11 sb= 3.6 P s - sb = Pd 3 + Q s - 3.6 = 15 - 0.5Q d 1.5Q = 15.6 Q = 10.4 P s = 13.4 P d = 9.8 PsPs PdPd P Q 10.4 11 CS=16 15 30 P s =13.4 PS=32 P s -3.6 8 3 P d= 9.8 Subsidy Cost = 3.6*10.4= DWL = (1/2)(10.4-8)*(13.4-9.8)= CS'=?PS'=?

122 121 Calculating the Deadweight Loss in Practice Supply Elasticity (e S )1.2 Demand Elasticity (e D )0.3 Initial Gasoline Price per Gallon$2.40 Initial Gallons Supplied / Demanded 4,500,000 Per-unit Tax$0.32

123 Sum Up - 122 PsPs PdPd P Q P e =11 15 30 P d =13 4 3 P s =7 8

124 123 DWL and Elasticity P Q P Q P s+t PdPd S D PsPs More losses the more elastic are demand and supply

125 124 U.S. Tariff on Brazilian Ethanol (2008) Q P PwPw D us S us P w + tariff Q1Q1 Q2Q2 Q3Q3 Q4Q4 tariff revenues U.S. Ethanol Market

126 125 Brazil Ethanol 2008 Exports of Q 4 -Q 3 Q P D Brazil S Brazil PwPw Q3Q3 Q4Q4

127 126 MR = MC P Q P(Q) MC ATC MR AC m QmQm PmPm MR = MC 2.o.c. Is slope MR< slope of MC? Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm –  o Qm MC dQ m

128 127 MR = MC P Q P(Q) MC ATC MR AC m QmQm PmPm MR = MC 2.o.c. Is slope MR< slope of MC? Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm –  o Qm MC dQ m

129 128 Social Optimum P = MC P Q PdPd MC ATC Q so P so Losses Choices: regulate or government own P = MC collect losses some other way P = ATC P d = MC ATC o

130 Chapter 4

131 130 Incidence of Tax P Qe Pe Q Pd Ps Ps+t Ps Pd  Pd  Ps Q’ t tax revenues

132 131 Incidence of Tax Depends on Demand Shape Two Extreme Cases PP Q Q D D Ps Ps+t Ps Ps+t Qe= Q’ QeQ’ Pe=Pd Ps P=Ps Pd Consumer Pays Producer Pays Perfectly elastic D Perfectly inelastic D

133 132 Incidence of Tax Depends on Supply Shape Two Extreme Cases P Q Ps PdPs+t QeQ’ Pe=Ps Consumer Pays Perfectly elastic S P Q S Pd-t Qe= Q’ Ps Pd Producer Pays Perfectly inelastic S Pe=Pd

134 133 Incidence of Tax – Depends on Elasticity Depends on elasticity  d = -0.5  s = 1 t = 1.5 dP d = ε s = 1 dP s ε d -0.5 PsPs PdPd P s+t P Q Q'QeQe PsPs PdPd PePe

135 134 Incidence of Tax – Depends on Elasticity dP d = ε s = 1 dP s ε d -0.5 (1) dPd = (1/-0.5)dPs = -2dPs dPd-dPs = t (2) dPd-dPs = 1.5 Two equations two unknowns PsPs PdPd P s+t P Q Q'QeQe PsPs PdPd PePe dP d >0 dP s <0

136 135 Solve Two Equations for dP s, dP d (1) dP d = -2dP s (2) dP d -dP s = 1.5 Substitute (1) into (2) -dP s -dP s = 1.5 -3dP s = 1.5 dP s = -0.5 dP d = -2P s = -2(-0.5) = 1 PsPs PdPd P s+t P Q Q'QeQe PsPs PdPd PePe dP d =1 dP s =-0.5

137 136 Social Welfare Q d = 30 -2P d Q s = -3 + P s P = 11, Q = 8 Invert demand P d = 15 - 0.5Q d P s = 3 + Q s PsPs PdPd P Q 8 11 Consumer Surplus = ½(15-11)*8 =16 15 30 3

138 137 Behind the Supply Curve Perfect competitors take P from market  = PQ – TC(Q) pick Q to maximize f.o.c   /  Q = P – TC(Q)/  Q = 0 P – MC = 0 P = MC 2.o.c  2  /  Q 2 = – TC(Q )2 /  Q 2 < 0 TC(Q )2 /  Q 2 > 0 MC slopes up- increasing marginal cost

139 138 S = MC in competitive market PS = MC

140 139 S = MC in competitive market PS = MC PePe Producer Surplus

141 140 Social Welfare Q d = 30 -2P d Q s = -3 + P s P = 12, Q = 6 Invert demand P d = 15 - 0.5Q d P s = 3 + Q s PsPs PdPd P Q 8 11 Consumer Surplus = ½(15-11)*8 =16 15 30 3 Producer Surplus = (1/2)(11-3)*8 = 32

142 141 Social Welfare - Consumer Surplus P Q P s =MC QeQe PePe Pd=Marginal Benefit

143 142 Social Welfare - Producer Surplus P Q PsPs PdPd QeQe PePe Price Set by Marginal Producer and Consumer Hotelling Rent

144 143 Welfare loss from an ad valorem tax P Q PsPs PdPd QeQe PePe P s (1+t%) PsPs PdPd  Q' Qe P d (Q)dQ -  Q' Qe P s (Q)dQ Q'Q'

145 144 Government Revenues P Q PsPs PdPd QeQe PePe P s (1+t%) PsPs PdPd Q'Q'

146 145 Change in Consumer and Producer Surplus P Q PsPs PdPd QeQe PePe P s (1+t%) PsPs PdPd Q'Q'

147 146 Welfare loss from unit subsidy tax P Q PsPs PdPd QeQe PePe P s -sb PsPs PdPd  Q' Qe P s (Q)dQ -  Q' Qe P d (Q)dQ Q'

148 147 What if You Export Your Product? P Q PsPs PdPd QeQe PePe P s (1+t%) PsPs PdPd Q'Q' gain loss

149 148 What if your Demand is Perfectly Elastic P Q PsPs PdPd QeQe PePe P s (1+t%) Ps'Ps' P d '= Q'Q'

150 149 Tariff is a Tax on Imports Small Consumer and Producer Crude Price determined on world markets PwPw P Q S D QsQs QdQd

151 150 Tariff on Crude Imports Add tariff t PwPw P Q S D QsQs QdQd P w +t Qs'Qs'Qd'Qd'

152 151 Tariff on Crude Imports Add tariff t PwPw P Q S D QsQs QdQd P w +t Qs'Qs'Qd'Qd'

153 152 Welfare loss from unit subsidy tax P Q PsPs PdPd QeQe PePe P s -sb Ps'Ps' Pd'Pd' Q'Q' cost to government benefit to producer benefit to consumer DWL loss

154 153 Welfare one wrinkle Price increase/decrease what happens to consumer welfare e.g. price increase – buy less loss in consumer surplus But two effects use less so lose utility but less real income P1P1 P2P2 P Q D

155 154 Look at real income with two goods P 1 X 1 + P 2 X 2 =Y Example 2X 1 + 4X 2 =100 Graph X 1 = 100/2 – (4/2)X 2 X 1 = 50 – 2X 2 Raise P 1 to 4 X 1 = 100/4 – (4/4)X 2 X 1 = 25 – X 2 X1X1 X2X2 50 25 Budget

156 155 Sum Up Tariff from Last Time Q P PwPw D us S us P w + tariff Q1Q1 Q2Q2 Q3Q3 Q4Q4 Loss in consumer surplus from tariff U.S. Ethanol Market

157 156 Sum Up Tariff from Last Time Q P PwPw D us S us P w + tariff Q1Q1 Q2Q2 Q3Q3 Q4Q4 Gain in domestic producer surplus from tariff U.S. Ethanol Market

158 157 Sum Up Tariff from Last Time Q P PwPw D us S us P w + tariff Q1Q1 Q2Q2 Q3Q3 Q4Q4 tariff revenues U.S. Ethanol Market social loss

159 158 Electricity - Decreasing Cost Industry P Q D ATC Natural Monopoly

160 159 MR = MC P Q P(Q) MC ATC MR AC m QmQm PmPm MR = MC 2.o.c. Is slope MR< slope of MC? Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm –  o Qm MC dQ m

161 160 Example (small village) – Monopoly Solution P is US cents per kWh Q is kWh per year demand is P = 75 - 4Q total cost curve in cents is TC = 19Q - 0.25Q 2 AC = TC/Q = 19 – 0.25Q MC =  TC/  Q = 19 –0.50Q MR = 75 - 8Q Q = 7.467 P = 75 – 4(7.467)=45.132 45.132 P 18.75 19 75 MC AC MR 7.467 Monopoly Profits Q D =MC=19 –0.50Q

162 161 Example (small village) – Monopoly Solution P is US cents per kWh = 45.132 Q is kWh per year = 7,467 TC = 19Q - 0.25Q 2  = PQ – TC =45.132*7.467 -19(7.467) + 0.25(7.467) 2 = 209.1 units Units PQ = cents/kWh*kWh = cents TC must be measured in cents What if TC measured in $ TC$*100¢ $ 45.132 P 18.75 19 75 MC AC MR 7.467 Monopoly Profits Q D

163 162 Social Optimum – Maximize Welfare P Q D MC welfare (W) = sum of consumer plus producer surplus CS = area below demand and above price PS =area above marginal cost and below price W =  0 Q P d (X)dX - PQ + PQ -  0 Q MC(X)dX Q1Q1 P1P1

164 163 What is  0 Q MCdQ Q MC ∫ 0 Q MCdQ=TVC ` P

165 164 What is Social Loss with Natural Monopoly P Q P(Q) MC ATC MR QoQo PmPm Decreasing Average Cost = Natural Monopoly Social Loss market failure QmQm PoPo Monopoly MR = MC Optimum P = MC

166 165 Lets Examine the Optimum P Q PdPd MC ATC Q so P so Losses Choices: regulate or government own P = MC collect losses some other way P = ATC P d = MC ATC o

167 166 Example  p i q i < expenses + s(RB) s = 10.5% 0.08*1,966,667 + 0.05*799,999 < (0.02+0.03)*1,966,667 + (0.02+0.01)*799,999 +0.105*750,000 197,333.31 ? 122,333.32 + 78,750= 201,083.32 < 201,083.32 Rates would be approved

168 R7 Examples Discounting (Annual Compounding) B dollars, interest rate r, in t years, annual compounding B=10, r=0.1, t=20, then B/(1+r) t =10/(1+0.1) 20 = $1.486 B=10, r=0.2, t=20, then B/(1+r) t =10/(1+0.2) 20 = $0.261 B=10, r=0.2, t=40, then B/(1+r) t =10/(1+0.2) 40 = 0.007 B=20, r=0.2, t=40, then B/(1+r) t = 20/(1+0.2) 40 = 0.014 B=20, r=0.0, t=40, then B/(1+r) t = ?

169 R8 Compounding More than Once a Year Compounding twice a year (r annual rate) one half year A (1+r/2) after a year A(1+r/2)(1+r/2) after a year and a half A (1+r/2) 3 after t years or 2t half years A (1+r/2) 2t Example A = 20, r = 8%, t = 10 20(1+0.08/2) 2*10 = $43.82 Compare to compounding annually 20(1+0.08) 10 = $43.18

170 R9 Compounding p times a year compounding p times a year A(1+r/p) tp A = 100, t = 50, r = 10% p = 4 13956.39 p = 10 14477.28 p = 365 14831.16 continuous compounding p goes to  = e rt A = 14841.32

171 R11 Discounting with Compounding p Times a Year B dollars in t years at interest rate 10% is worth ? today A(1+r/p) tp =B  A = B/(1+r/p) tp B = 100, t = 50, r = 10% p = 4 0.717 p = 10 0.691 p = 365 0.674 continuous compounding p goes to  B = e rt A  A = B/e rt = Be -rt = 100e -rt A = 100e -0.10*50 = $0.674

172 R14 Value a Stream of Income D 1 dollars at the end of 1 year D 2 at the end of 2 years NPV = D 1 + D 2 (1+r) (1+r) 2 Example D1 = 50, D2 = 51, r = 0.10 NPV = 50 + 51 = $89.256 (1.1) (1.1) 2 Could have changing interest rates NPV = D 1 + D 2 (1+r 1 ) (1+r 2 ) 2

173 R16 NPV of Power Plant Interest rate is 10%, interest is compounded annually power plant costing 200 now year 0 two years to build Stream of income -100 -100 30 65 65 25 65 65 65 -20 What is the NPV or DCF of this power plant? -100 -100 + 30 + 65 + 65 + 25 + 65 + 65 + 65 – 20 (1.1) (1.1) 2 (1.1) 3 (1.1) 4 (1.1) 5 (1.1) 6 (1.1) 7 (1.1) 8 (1.1) 9

174 R20 Internal r (IRR) Invest Equipment costing 100 now year 0 Yields income after 1 and 2 years of 60 59 Flow of income is -100 60 59 NPV of flow of income is -100 + 60 + 59 (1+r) (1+r) 2 solve for the r that makes NPV = 0

175 R21 Internal r (IRR) Solve for r that Makes NPV 0 -100 + 60 + 59 = 0 (1+r) (1+r) 2 Alternatively rearrange 100 = 60 + 59 (1+r) (1+r) 2 Find r that makes price of asset (100) = DCF of income flow Solve: 100(1+r) 2 = 60(1+r) + 59 100(1+2r+r 2 ) = 60 +60r +59 100r 2 +140r - 19 =0

176 R22 Using Quadratic Formula 100r 2 +140r - 19 =0 Quadratic formula ar 2 +br + c =0 -b  (b 2 - 4ac) 0.5 = -140  (140 2 - 4*100*140) 0.5 2a 2*100 = 0.125 = 1.525 Excel alternative - Put stream of income in A1 to A3 -100 60 59 =irr(a1.a3,guess) = irr(a1.a3, 0.05) = 12.5% seems to always take + root

177 R24 Internal r (IRR) Power Plant Power plant costing 200 now year 0 two years to build Stream of income -100 -100 30 65 65 25 65 65 65 -20 What is the NPV or DCF of this power plant? - -100 -100 + 30 + 65 + 65 + 25 + 65 +... – 20 (1+r) (1+r) 2 (1+r) 3 (1+r) 4 (1+r) 5 (1+r) 6 (1+r) 9 solve for the r that makes above sum zero = irr(addresses, guess) = 14.4% to see other excel functions >insert >function

178 177 Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class residential customers (L) C L = 1200 + 20Q L industrial (H) C H = 1000 + 10Q H if produce both C LH = 1500 + 20 Q L + 10Q H but C L + C H = 2200 + 20Q L + 10Q H sub-additive How to allocate 1500? one group not subsidize another don’t drive anyone out of market > C QLQH

179 178 Marginal Cost Pricing for Low Voltage P L = 80 – 2Q L C L = 1200 + 20Q L C LH = 1500 + 20 Q L + 10Q H MC L = 20 P L = MC L 80 – 2Q L = 20 80-20 = 2Q L Q L = 30 P L = 20 Consumer surplus 0.5(80-20)30 = 900 P QLQL 80 40 20 30 PLPL MC L Standalone Fixed 1200

180 179 You Do Marginal Cost Pricing for High Voltage C LH = 1500 + 20 Q L + 10Q H P H = 100 – 3Q H MC H = Q H = P H = Consumer surplus Standalone fixed PHPH QHQH PHPH MC H

181 180 What is Maximum We Should Charge H 1. Charge less than stand alone 2. Charge less than consumer surplus What is maximum we can charge H? P H = 100 – 3Q H C LH = 1500 + 20Q L + 10Q H C H = 1000 + 10Q H Stand fixed cost = Consumer surplus = P QLQL PHPH MC H

182 181 Pricing Across Time - Peak load pricing one simple case – quantity independent of price in other period peak shifting more complicated problem Q P D opk D pk ckck c k +c o

183 182 Peak load pricing P Q Q pk ' Q opk coco c o +c k Social optimum P pk = c k + c o P opk = c o Q pk Q opk ' CS peak CS offpeak

184 183 Numerical Example Peak Load Pricing No peak switching Q pk = 50 - 5P pk Q opk = 8 - 2P opk c k = 3 c o = 2 P pk = c k + c o P opk = c o P pk = 10 - (1/5)Q pk P opk = 4 - (1/2)Q opk

185 184 Solve for Q pk and Q opk Social optimum P pk = 10 - (1/5)Q pk = c k + c o = 3 + 2 = 5 10 - (1/5)Q pk = 5 Q pk = 25 P opk = 4 - (1/2)Q opk = c o = 2 4 - (1/2)Q opk = 2 Q opk = 4 P Q Q pk Q opk C o = 2 C o +C k =5 P pk P opk 4 25

186 185 Often Charge One Price If Charge One Price: P=5 P Q Q pk Q opk c o = 2 c o + c k =5 P=5 Q pk ' Q opk ' Social Loss

187 186 If charge one price: P = 2 P Q Q pk Q opk c o =2 c o +c k =5 P=2 Q pk Q opk Q pk ' Social Loss also not covering capital cost

188 187 If charge one price: P=2.5 See if you can figure out losses P Q Q pk Q opk C o = 2 C o +C k =5 Q pk Q opk Q opk ' Q pk ' P=2.5 Losses in Peak Losses in Off Peak Peak Load Price if Losses Greater than Metering Cost

189 Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class 188 residential customers (L) C L = 1200 + 20Q L industrial (H) C H = 1000 + 10Q H if produce both C LH = 1500 + 20 Q L + 10Q H but C L + C H = 2200 + 20Q L + 10Q H sub-additive > C LH

190 189 Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class residential customers (L) C L = 1200 + 20Q L industrial (H) C H = 1000 + 10Q H if produce both C LH = 1500 + 20 Q L + 10Q H but C L + C H = 2200 + 20Q L + 10Q H sub-additive How to allocate 1500? one group not subsidize another don’t drive anyone out of market > C QLQH

191 You Do Marginal Cost Pricing for High Voltage 190 C H = 1000 + 10Q H C LH = 1500 + 20Q L + 10Q H P H = 100 – 3Q H MC H = Q H = P H = Consumer surplus Standalone fixed PHPH QHQH PHPH MC H

192 You Do Marginal Cost Pricing for High Voltage 191 C H = 1000 + 10Q H C LH = 1500 + 20 Q L + 10Q H P H = 100 – 3Q H MC H = 10 100 – 3Q H =10 Q H = P H = Consumer surplus Standalone fixed PHPH QHQH PHPH MC H 100 33.33 30 10 30 10 0.5(100-10)*30 =1350 1000

193 Pricing Across Time - Peak load pricing 192 one simple case – quantity independent of price in other period peak shifting more complicated problem Q P D opk D pk ckck c k +c o

194 193 Two Curves Shift - Gasoline Market P Q ←S' D S QeQe PePe Pe'Pe' Qe'Qe' Q →D' D S QeQe PePe Pe'Pe' Qe'Qe' P ↑ Q↓P ↑ Q↑ Oil Prices UpIncome Up

195 194 Incidence of Subsidy – on Supply P Q PsPs PdPd QeQe PePe P s -sb Ps'Ps' Pd'Pd' Q'Q'

196 195 Incidence of Subsidy – on Demand P Q PsPs PdPd QeQe PePe P d +sb Ps'Ps' Pd'Pd' Q'Q'

197 196 Pmax P Q S D QsQs QdQd not binding

198 197 MR = MC P Q P(Q) MC ATC MR AC m QmQm PmPm MR = MC 2.o.c. Is slope MR< slope of MC? Monopoly profit = TR – TC = Pm*Qm – AC*Qm

199 198 Marginal Cost Pricing P QL = 100 – 2Q L P QL = 70 – 4Q L P QL = 100 – 2Q L = MC L = 20 100-20 = 2Q L Q L = 40 P QH = 70 – 4Q H = MC L = 30 70-30 = 4Q H Q H = 10 Haven't allocated fixed costs of 1700 PLPL QLQL 100 50 20 40 P QL MC L

200 199 Marginal Cost Pricing P QL = 100 – 2Q L C QLQH = 1700 + 20 Q L + 30Q H C QL = 1400 + 20 QL Consumer surplus 0.5(100-20)40 = 1600 PLPL QLQL 100 50 20 40 P QL MC L

201 200 Pricing Across Time - Peak load pricing one simple case – quantity independent of price in other period peak shifting more complicated problem

202 201 Example (small village): P is US cents per kilowatt hours Q is measured in kilowatt hours per year demand and total cost curve are P = 75 - 4Q TC = 19Q - 0.25Q 2 AC = TC/Q = 19 – 0.25Q  TC/  Q = MC = 19 –0.50Q MR = 75 - 8Q = MC = 19 –0.50Q => Q = 7.467 P = 75 – 4(7.467)=45.132  = PQ – TC=45.132*7.467* -19(7.467) + 0.25(7.467) 2 = 209.1

203 202 Example (small village): P cents per kilowatt hours Q kilowatt hours per year P = 75 - 4Q MR = 75 – 8Q AC = 19 – 0.25Q MC = 19 – 0.50Q Q = 7.467 P = 45.132  = 209.1 units? 45.132 Q P 18.75 19 75 D MC AC MR 7.467 Monopoly Profits

204 203 Social Optimum P Q D MC welfare (W) = sum of consumer plus producer surplus CS = area below demand and above price PS =area above marginal cost and below price W =  0 Q P d (Q)dQ - PQ + PQ -  0 Q MCdQ Q1Q1 P1P1

205 204 Maximize CS =  0 Q P d (Q)dQ -  0 Q P s (Q)dQ maximizing the area between D and MC f.o.c.  W =   0 Q P d (Q)dQ -   0 Q MCdQ = 0  Q  Q  Q = P d (Q) – MC = 0 2.o.c.  2 CS =  P d (Q) –  MC < 0  Q 2  Q  Q  P d (Q)<  MC  Q  Q Slope of inverse demand less than slope of MC

206 205 Last Time Quiz - Cost Curves Sunk costs are part of total costs Q P FC sunk FC=FC sun + FC nosunk TC = FC+VC ATC = ATC/Q MC =  TC/  TC

207 Chapter 5

208 Generating Costs 207 D D'

209 Price Regulation Transportation and Distribution 208 1. Rate of Return (  p i q i < expenses + s(RB) U.S. 2. Price Cap (RPI-X) prices can to up no more than (RPI) rate of inflation - (X) rate of productivity change CPI07= 115 and CPI08 = 123 RPI=(123-115)/115 = 0.07 productivity change some measure of output/input (O/I) (O/I)07=0.21 and (O/I) 08=0.22

210 Price Regulation Transportation and Distribution 209 X = (0.22-0.21)/0.21 = 0.048 RPI-X = 0.07 - 0.048 = 0.022 P07 = $0.10 P08<(1+0.022)*0.10 = $0.102 popular in UK 3. Light Handed New Zealand 4. Yardstick Scandinavia

211 Wholesale Market Q 1 =19 210 One sided bidding hour ahead, day ahead get bids – put in order Q P one-sided Q1Q1 P1P1

212 Wholesale Market System Marginal Price = SMP 211 Two sided bidding again get bids – put in order Q P Q1Q1 P1P1 $0.07

213 SMP = System Marginal Price + Capacity Charge 212 Total capacity charge loss of load probability (LOLP) times value of the lost load (VOLL) Example: 5% probability of a 10 kWh short fall. loss of output from a 10 kWh shortfall ~ $15 LOLP*VOLL = (0.05*$15)/ = $0.75 Dividing this over all kilowatts consumed (100 kWh) CC = $0.75/100 = $0.0075 Power Pool Price = PPP PPP = SMP + CC = 0.07 + 0.0075 = $0.0775

214 How to allocate power at capacity Role of price signals Gaming the system 213 P Q S D opk D pk Q opk

215 214 Last time: SMP = System Marginal Price + Capacity Charge Total capacity charge loss of load probability (LOLP) times value of the lost load (VOLL) Example: 5% probability of a 10 kWh short fall. loss of output from a 10 kWh shortfall ~ $15 LOLP*VOLL = (0.05*$15)/ = $0.75 Dividing this over all kilowatts consumed (100 kWh) CC = $0.75/100 = $0.0075 PPP = SMP + CC = 0.07 + 0.0075 = $0.0775

216 215 1990 - 1999 Demand up Supply down P Q D D' S S' PePe QeQe Q s' Q d' Imports

217 Chapter 6

218 217 Typical Competitive Firm Cost Short Run Supply P Q AVC 1 MC 1 P Q AVC 2 MC 2 Market is horizontal sum S i = MC i above AVC Q S D P sr P Q1Q1 Q2Q2 Q 1 +Q 2

219 218 Last Time Reviewed- Long Run Supply With Entry and Exit D  sr MC i = S sr D3D3 D1D1 D2D2 P Q S lr Increasing Cost Industry

220 219 Long Run Supply With Entry and Exit D  sr MC i = S sr D3D3 D1D1 D2D2 P Q S lr Increasing Cost Industry

221 220 Long Run Supply With Entry and Exit D  sr MC i = S sr D3D3 D1D1 D2D2 P Q S lr Constant Cost Industry

222 221 Long Run Supply With Entry and Exit D  sr MC i = S sr D3D3 D1D1 D2D2 P Q S lr Constant Cost Industry

223 222 Inelastic Supply and Demand S P D Q D P1P1 S'' S' S'' S' S P Q

224 223 Multiplant Monopoly Marginal Cost – 2 countries TC 1 = 10 + Q 1 + (1/2)Q 1 2 TC 2 = 20 + 2Q 2 + Q 2 2 MC 1 =  TC 1 /  Q 1 = 1 + Q 1 MC 2 =  TC 2 /  Q 2 = 2 + 2Q 2

225 224 MC for Monopolist- Horizontal Sum Firm 1 MC 1 = 1 + 1Q 1 MC Q1Q1 1 Q2Q2 2 Firm 2 MC 2 = 2 + 2Q 2 MC 1 MC 2 Given MC sum the Q MC 1+2 MC= 1 + 1Q 0 < Q < 1 Q = Q 1 +Q 2 Q >1

226 225 MC Above Kink Firm 1 MC 1 = 1 + 1Q 1 MC Q1Q1 1 Firm 2 MC 2 = 2 + 2Q 2 MC 1 Given MC sum the Q Q 1 = -1 + MC 1 Q 2 = -1 + (1/2)MC 2 Q 1 + Q 2 = -2 + (3/2)MC Q = -2 + (3/2)MC MC = 4/3 + 2/3Q MC 1+2 Q = Q 1 +Q 2, Q > 1 2

227 226 Now Add Demand What Should Monopolist Do? P Q 1 2 MC 1 MC 1+2 1 P=75-0.5Q MR = MC MR= 75-Q=4/3+2/3*Q Q=44.2 P=75-0.5Q = 75-0.5*44.2 = 52.9 MC=MR = 75-44.2=30.8 Q 1 = -1 + MC = -1 + 30.8 = 29.8 Q 2 = -1 + (1/2)MC =-1 + (1/5)30.8 = 14.4 D MR QmQm =44.2 PmPm 52.9= MC m

228 227 Sum Up Competitive Market Short Run Supply Competitive Market P = MC above AVC P Q1Q1 P Q2Q2 P Q3Q3 MC 1 MC 2 MC 3 ΣMC i MC i =f i (Q i ) Invert Q i = f i -1 (MC i ) Horizontal Sum Q 1 +Q 2 +Q 3 = f 1 -1 (MC 1 )+ f 2 -1 (MC 2 ) + f 3 -1 (MC 3 ) Set Q = Q 1 +Q 2 +Q 3 and MC i =MC j Q

229 228 Competitive Market Long Run Supply With Entry and Exit Increasing, Constant, Decreasing Cost Industry P Q S lri S lrc S lrd

230 229 2 Order Conditions P Q 1 MC 1 MC 1+2 MR – MC = 0  MR –  MC <0  Q  Q MR = 75-Q MC = 4/3+2/3*Q  MR = -1  Q  MC = 2/3 Q Q  MR –  MC = -1 – (2/3) < 0  Q  Q D MR QmQm PmPm MC m

231 230 Individual Producer's Profits Profits  1 = P*Q 1 - TC 1 = P*Q 1 - 10 - Q 1 - (1/2)Q 1 = 52.9*29.8 - 10 – 29.8 - (1/2)29.8 2 =1751  2 = P*Q 2 – TC 2 = P*Q 2 – 20 - 2Q 2 – Q 2 2 = 52.9*14.4 - 20 - 2*14.4 – (14.4) 2 =824

232 231 Competitive Model (P=MC) P Q MC 1 MC 1+2 D Q pc P pc Q2Q2 Q1Q1 Be able to solve for P pc, Q pc, Q 1, Q 2,  1,  1

233 232 Market Failure from Monopoly P Q D MR QmQm PmPm Q pc P pc MC Social Losses? redistributed from consumers to monopolist Efficiency Distribution

234 233 Sources of Cost - economic model if competitive market supply = marginal cost fit function to data P Q

235 234 Where to Get Demand Q d = f(P d, Y, P s, P c,..., etc. ) Collect data on Q d, P d - all variables that change Fit a function using statistical techniques Simplified Two Variable Illustration Q t =  1 +  2 P t + e t (truth) R.V. = e t ~ 0,  2 êtêt etet P Q

236 235 MRP = Factor Demand P E E o = P o MRP = P o marginal revenue product must slope down PEEoPEEo PoPo P o2 P o1 O2O2 O1O1 Can compute demand if know E o Q

237 236 Abdel Reviewed Competitive Short Run Supply P Q AVC 1 MC 1 P Q AVC 2 MC 2 Market is horizontal sum S i = MC i above AVC Q S D P sr P Q1Q1 Q2Q2 Q 1 +Q 2

238 237 Competitive Long Run Supply With Entry and Exit D  sr MC i = S sr P Q S lr D'

239 238 Market Failure from Monopoly P Q D MR QmQm PmPm Q pc P pc MC Social Losses? redistributed from consumers to monopolist Efficiency Distribution

240 239 Supplier Oil Price and Transport Cost Demand and Supplier Separated by Transport Cost tr P s1 Q S D S+ tr 1 S+ tr 2 Q2Q2 Q1Q1 P P s2 Price lower the farther from the market

241 240 Location - Supplier Price and Arbitrage $70 $1 >$69? $69 <$69? D S2S2 S1S1

242 241 Supplier Price and Arbitrage $70 $2 $1 >$68? $69 <$68? D S2S2 S1S1 Prices can only differ by transport and transaction cost

243 242 Income Redistribution If Producer Exports all of Product Income distribution before tax MR MC Consumer Surplus Producer Surplus P Q D PmPm QmQm

244 243 Sum Up: Horizontal Sum MC Competitive Supply, MC for multi-plant Monopoly P MC

245 244 Sum Up: Factor Demand = MRP = P Q MP E Horizontal sum from individual to Market P Q D

246 245 Sum Up: Add Demand to MC What Should Monopolist Do? P Q 1 2 MC 1 MC 1+2 1 D MR QmQm PmPm MC m Put together Demand and MC MR = MC

247 246 World Demand Supply of fringe QsQs P Q QwQw Q o = Q w – Q s call on OPEC horizontal difference MR L MR U Next Horizontal Difference: Dominant Firm's Demand

248 247 What should OPEC Do? Add MC – Case 1 QsQs P Q QwQw MR MC o MC f  QoQo PoPo PfPf Fringe? 2 places

249 248 What should OPEC Do? Add MC – Case 2 P Q QwQw MR MC o MC f QoQo PoPo

250 249 What should OPEC Do? Add MC – Case 3 P Q QwQw MR MC o MC f QoQo PoPo

251 250 More on Price and Elasticity P = MC (1-1/|  p |) One other implication What if  p inelastic = -1/2 Then |  p | = 1/2 Formula say P = MC = MC = -MC (1-1/(1/2)) (1 - 2) Whoops - negative price? conclusion monopolist not in inelastic range of demand

252 251 Numerical Example - 2 Country OPEC Costs OPEC MC 1 = 2 + Q 1 MC 2 = 2 + 2Q 2 World Demand Q w = 30 - 0.5P Supply fringe Q f = -10 + P QsQs P Q QwQw MR L MR U MC

253 252 Numerical Example - 2 Country OPEC OPEC MC Marginal costs MC 1 = 2 + Q 1 MC 2 = 2 + 2Q 2 Horizontal sum Q Invert (let MC 1 = MC 2 = MC) Q 1 = -2 + MC Q 2 = -1 +(1/2)MC add Q's, Q 1 +Q 2 = Q = -3 + (3/2)MC invert back => MC = 2 + (2/3)Q P Q MC 2

254 253 OPEC Demand - Find Kink MC fringe or supply fringe is MC f = P = 10 + Q => Q f = -10 + P Inverse Demand World P = 60 - 2Q w => Q w = 30 - 0.5P Kink P Q f = 0 = - 10 + P => P = 10 Kink Q World demand = Q = 30 - 0.5(10) = 25 QsQs P Q QwQw 25 10

255 254 OPEC Demand Above kink P > 10 and Q < 25 Q w - Q f Q o = 30 - 0.5P - (-10 + P) = 40 -1.5P Below Kink P 25 Q o = Q w = 30 - 0.5P QsQs P Q QwQw 25 10

256 255 Marginal Revenue Above kink P > 10 or Q < 25: Q w - Q f Q o = 40 -1.5P Invert P = 40/1.5 -2/3Q MR = 40/1.5 - 4/3Q Below Kink Q o = 30 - 0.5P Invert P = 60 -2Q MR = 60 - 4Q QsQs P Q QwQw

257 256 Solution - 3 choices Try above the kink MR = MC MR = 40/1.5 - 4/3Q MC = 2 + (2/3)Q 40/1.5 - 4/3Q = 2 + (2/3)Q 74/3 = (6/3)Q 6Q = 74 Q = 12.333 less than 25 P = 40/1.5-2/3(12.333) = 18.444 QsQs P Q QwQw MR L MR U MC 25 10

258 257 Income Distribution Affect in Monopoly Market MR MC Consumer Surplus Producer Surplus P Q D PmPm QmQm

259 258 Tax in Monopoly Market: Global Changes New P m ' and Q m ' New CS, Producer G tax revenue, New PS MR MC P Q Qm'Qm' P m' MC+t QmQm PmPm Same Effect - unit tax Tax Consumer MR-t = MC Tax Producer MR = MC + t

260 259 Income Distribution in Monopoly Market Assume Producer Exports all of Product Income distribution before tax MR MC Consumer Surplus Producer Surplus P Q D PmPm QmQm

261 260 Tax in Monopoly Market: Global Changes New P m ' and Q m ' New CS, Producer G tax revenue, New PS MR MC P Q Qm'Qm' P m' MC+t QmQm PmPm Same Effect - unit tax Tax Consumer MR-t = MC Tax Producer MR = MC + t

262 261 Tax in Monopoly Market: Global Changes New P m ' and Q m ' New CS, Producer G tax revenue, New PS 261 MR MC P Q Qm'Qm' P m' MC+t QmQm PmPm

263 262 Tax in Monopoly Market Tax Producer Government Consumer Country Loss MR MC P Q Qm'Qm' P m' MC+t QmQm PmPm transfer to producer government loss tax revenues to producer government

264 263 Tax in Monopoly Market Effect on Producers MR MC P Q Qm'Qm' P m' MC+t QmQm PmPm Producer Losses Transfer to Producer Government

265 264 Tax in Monopoly Market Net Effect on Producer Country MR MC P Q Qm'Qm' P m' MC+t QmQm PmPm 1. Producer DW Losses 2. Tax revenues from consumer country Change in Producer Country Welfare = 2-1

266 265 Numerical Example- P&Q Before Tax P = 50 -2Q MC = 1 + 3Q MR = 50-4Q MR = MC 50 – 4Q = 1 + 3Q 7Q = 49 Q = 7 P = 50 – 2*7 = 36 MC = 1 + 3*7=22 MR MC P Q D PmPm QmQm 50 36 = 1 7= 22

267 266 Numerical Example- CS & PS Before Tax Consumer Surplus =0.5(50-36)7=49 Producer Surplus =(36-22)*7+0.5*(22-1)*7 =171.5 MR MC P Q D PmPm QmQm 50 36 = 1 7= 22

268 267 Tax in Monopoly Market Producer Tax of 7 MR MC P Q 6 36 MC+t P = 50 -2Q MC = 1 + 3Q MR = 50-4Q MR = MC+7 50 – 4Q = 1 + 3Q+7 7Q = 42 Q = 6 P = 50 – 2*6 = 38 MC = 1 + 3*6=19 50 1 19  38

269 268 Welfare Effects Producer Tax of 7 Before CS = 49 Now consumer surplus CS = 0.5(50-38)*6= 36 Change in consumer surplus 49 - 36 = 13 Tax from Consumer transfer to producer Gov (38-36)*6=12 Consumer DWL = 13-12 = 1 MR MC P Q 6 38 MC+t 50 1 19  36

270 269 Welfare Effects Producer Tax of 7 Before PS = 171.5 Now Producer Surplus PS = (31-19)*6 + 0.5*(19-1)*6 = 126 Change in PS 171.5-126 = 45.5 Tax from Producer (7-2)*6 = 30 DWL Producer 45.5-30 = 15.5 MR MC P Q 6 38 MC+t 50 1 19  36

271 270 Welfare Effects Producer Tax of 7 Net Effect for Producing Country DWL from producers = 15.5 Tax revenue gain from consumer country = 12 Net effect = Loss of 15.5 - 12

272 271 Total Global Welfare Effects Before CS = 49 Now CS = 36 PS = 171.5 PS + TR = 126 + 42 Total= 204 = 168 MR MC P Q D PmPm QmQm Total Losses 220.5-204 = 16.5 QtQt

273 272 Do for Tax by Consumer Government Demand P= 50 - 2Q MC = 1 + 3Q Subtract tax from demand P - t = 50 -2Q - 7 = 34 - 2Q Create MR from new demand MR = 43 - 4Q Set MR = MC and solve See file ch06-Monopoly Tax to check problems for producer tax consumer tax both taxes

274 273 Consumer Tax in Monopoly Market Consumer Government Adds a Tax MR t MC P Q QtQt PtPt P-T P MR

275 274 What should OPEC Do? Add MC – Case 1 QfQf P Q QwQw MR MC o MC f  QoQo PoPo PfPf Fringe? 2 places QfQf Q d -Q f QfQf

276 275 What should OPEC Do? Add MC – Case 2 P Q QwQw MR MC o MC f QoQo PoPo

277 276 What should OPEC Do? Add MC – Case 3 P Q QwQw MR MC o MC f QoQo PoPo

278 277 Quiz - left of Kind QfQf P Q QwQw MR MC o QoQo PoPo QfQf N.B. 1. read price off of OPEC demand not world demand 2. Price is the same for OPEC and the fringe

279 278 Quiz OPEC at Kink Q f = 0 P Q QwQw MR MC o MC f QoQo PoPo

280 279 Quiz - OPEC to right of kink Q f = 0 P Q QwQw MR MC o MC f QoQo PoPo

281 280 Quiz Key -Graphically there are two ways to show economic profits Profits = producer surplus = area below price and above marginal cost Profits = (P-ATC)*Q

282 281 Numerical Example - OPEC Optimum Pick to right or left Pick Left MR = MC Left of kink Q< 123 P = 92 - (2/3)Q MR = 92 - (4/3)Q OPEC Marginal Cost MC = 2 + (2/3)Q o MR = MC 92 - (4/3)Q o = 2 + (2/3)Q o 90 = (6/3)Q o Q o = 45 QsQs P Q QwQw MR MC 10 123 45

283 282 What Else do We Know About the Market Q o = 45 P = 92 - (2/3)Q = 92 - (2/3)45 = 62 Q f = -10 + P = -10 + 62 = 52 QsQs P Q QwQw MR MC 10 12345 62 52

284 283 OPEC Quotas MC = 2 + (2/3)Q = 2 + (2/3)45 = 32 OPEC Quotas MC 1 = 2 + Q 1 32 = 2 + Q 1 Q 1 = 30 MC 2 = 2 + 2Q 2 32 = 2 + 2Q 2 Q 2 = 15 P Q QwQw MR MC 62 MC 1 MC 2 32 15 30 45

285 284 Industry Profits OPEC π= P*Q - TC = P*Q - ATC*Q = P*Q -  0 Q MC(Q)dQ Solve = 62*45 -  0 45 (2 + (2/3)Q)dQ = 2790 - (2Q + (1/3)Q 2 )| 0 45 = 2790 -[2*45 + (1/3)45 2 - 2*0 + (1/3)0 2 ] = 2025 Individual OPEC countries or the fringe = P*Q i -  0 Qi MC i (Q i )dQ

286 285 Adjust for Technical Change and Depletion AC Cumulative Q learning curve depletion curve

287 Chapter 7

288 287 Price control versus Quantity control Q contract Q P Q P Q capacity PcPc D2D2 D3D3 D1D1 D2D2 D3D3 D1D1 shortage

289 288 Opportunism - quasi rent P Q MC ATC AVC rent quasi-rent P1P1 P2P2

290 Chapter 8

291 290 Policy - Negative Externalities on Supply? P Q D pv S pv S soc Q pv P pv P soc Q soc

292 291 Numerical example Q d = 90 - P d Q s = 2P s Externality X = 9 Solve for equilibrium Q d = 90 - P d = Q s = 2P s Drop subscripts and solve P = 30 Q = 90-30 = 2*30 = 60

293 292 Numerical example Externality = 9 Invert Q d and Q s P d = 90 - Q d P s = Q s /2 Add negative externality to P s P d = P s + X drop subscripts 90 - Q = Q/2 + 9 Q = 54, P = 90 - 54 = 36 30 60 P d = 90 - Q d P s = Q s/ 2

294 293 Numerical example- Social Costs 30 60 Pd = 90 - Qd Ps = Qs/2 54 9 P d = P s + X = 30 + 9 = 39 Welfare loss 0.5(39-30)(54-60)=27 units? = units of PQ price in $/ton quantities - millions of tons P*Q = $ * millions tons ton = millions $ 39

295 294 Social Loss - positive externality on supply P Q D pv S soc S pv

296 295 MB = MC X MB,MCMCMC MB Xo

297 296 Model Two Pollution - Optimal Level MB of Pollution MC of Pollution ABCD E $ Benefits Costs X Pollution G

298 297 Polluter has Property Rights? What are Social Losses? MB of Pollution MC of Pollution ABC F D E X Pollution $ MB,MC G

299 298 One Who Suffers has Property Rights? You Show Social Losses? MB of Pollution MC of Pollution ABC F D E X Pollution $ MB,MC G

300 299 Coase’s Law No Transaction Costs Suppose Dow has property rights MB of Pollution MC of Pollution ABC F D E X Pollution $ MB,MC G Dow Exxon Most benefit

301 300 Distribution Affects Polluter Has Property Rights MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G social loss polluter benefits from pollution

302 301 Command and Control You can only emit C MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G at C no social loss Pollution at CPolluter clean up cost

303 302 What Happens if Pollution Tax = T 1 MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G T1 pollution

304 303 What Happens if Pollution Tax = T 2 MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G T2

305 304 Optimal Pollution Tax MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G T3 pollution taxes

306 305 Polluter Had Property Rights Redistribution Affects With Tax MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G society gains back polluter losses from tax tax abatement

307 306 Distribution Affects Sufferer Had Property Rights MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G social loss polluter benefits H I

308 307 Distribution Affects With Tax MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G polluter gains JIG-AEJB H I fix tax J

309 308 Issue Marketable Permit of AC MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G H I P1 J polluters will want to buy AJ PP Price will go to AE

310 309 Polluter Had Rights Subsidize Clean Up MB of Pollution MC of Pollution A C F D E $ Benefits Costs Q Pollution G H Sb Total Subsidy

311 310 Distribution Affects from Subsidy Polluter Had Property Rights MB of Pollution MC of Pollution AB C F D E $ Benefits Costs Q Pollution G social loss K Total Subsidy Total Polluter Benefits AHGC+GKD H

312 311 Which Policy Does Polluter Prefer

313 312 Model 3 Optimal level of Abatement CD Optimal Level – Pollution AC MB of Pollution MC of Pollution ABC F D E $ Benefits Costs Q Pollution G

314 313 Model 3 Abatement over to firms of CD MC2 MC1 A1 A2 MC needed abatement CD Price of permits P2 P3 What happens at P2? P3? A1 ’ A2 ’

315 Chapter 9

316 315 Public Good Quantitative Separate Players MC = 6 MB 1 = 30 - 3A 1 MB 2 = 20 - 2A 2 MC= MB 1 6 = 30 - 3A 1 3A 1 = 30 - 6 = 24 A 1 = 24/3 = 8 MC = MB 2 6 = 20 - 2A 2 2A 2 = 14 A 2 = 7 MC MB MB 1 =30-3A 1 MC=6 MC MB MB 2 = 20 - 2A 2 MC=6 A1A1 A2A2 A 1o A 2o =8 =7

317 316 Public Good Quantitative Gaming the System Non-excludeable A 1 wants A 2 to produce? 7 A 1 will produce 1 A 2 wants A 1 to produce? 8 A 2 will produce 0 Each will want to free ride MC MB MB 1 =30-3A 1 MC=6 MC MB MB 2 MC A1A1 A2A2 A 1o A 2o =7 MB 1 +MB 2 A so =8

318 317 Public Good Quantitative Social Optimum Since non-rivalrous benefits MB 1 + MB 2 MC = 6 MB 1 = 30 - 3A 1 MB 2 = 20 - 2A 2 MB = 50 - 5A MB = MC 50-5A = 6 5A = 44 A = 44/5 = 8.8 MC MB MB 1 =30-3A 1 MC=6 MC MB MB 2 MC A1A1 A2A2 A 1o A 2o =7 MB 1 +MB 2 A so =8 =8.8

319 318 Value of life Occupation increases the probability of dying by 1/1000 = 0.001 Salaries are 5,000 higher in this occupation How are they valuing their lives Die lose = V Don't die from work accident loss = 0 0.001V + 0.999*0 = 5000 V = 5,000,000

320 319 Conservation – levelized costs 75-watt incandescent bulb (75/1000 = 0.075 kilowatts) lasts 600 hours buy packs of two $1.40 more than 90% of energy lost to heat 20-watt (20/1000 = 0.020 kilowatts) compact fluorescent bulb same amount of light lasts around 8,400 hours costs around $14.50

321 320 Conservation – levelized costs Suppose lights will run 1200 hours per year electricity costs $0.10 per kilowatt-hour interest rate is 12% compounded once a month operating costs/hour for incandescent bulb (o i ) = kilowatts per bulb X costs per kilowatt hour = (0.075)*0.10 = $0.0075 per hour operating costs o f /hour for fluorescent = (0.020)*0.10 = $0.0020/hr

322 321 Levelized Capital Costs for each Bulb a bit harder to compute. X monthly output of light (1200/12) lasts for n years K is initial capital costs, let $ equal levelized cost K = $X + $X +.... $X (1+r/12) (1+r/12) 2 (1+r/12) n*12 Then K = $X  i=1 n*12 (1/(1+r/12) i ) Solving for $ = (K/X)/  i=1 n*12 (1/(1+r/12) i )

323 322 Levelized Cost Fluorescent & Incandescent Package of incandescents costs K = $1.40 n=1 year, X = 100 hours per month $ i =  i=1 n*12 (1/(1+r/12) i ) = (1.40/100)/11.255 = $0.0012 capital costs per unit of light lower than operating for incandescent compact fluorescent cost K =$12.00 n=7 years, X = 100 hours per month $ f = (12/100)/Σ i=1 12*7 (1/(1+0.1/12) i ) = (12/100)/56.648 = $0.0021 compact fluorescent operating costs lower than capital costs

324 323 Total unit Cost Fluorescent & Incandescent Adding capital and operating costs total incandescent costs $ i + o i = $0.0012 + $0.0075 = $0.0087 total compact fluorescent costs $ f + o f = $0.0021 + $0.0020 = $0.0041 Total Formula unit cost = kilowatts*P e + $ i = (K/X)/  i=1 n*12 (1/(1+r/12) i )

325 324 Market Power P DD Q MC P Q Competition Monoply MR = MC Competition Buyer Power Seller power

326 325 Graph the Decision Process MRP = P P L & MRP L LL1L1 P L-1 D = MRP L P L-2 L2L2

327 326 Marginal Factor Cost from Supply Market Power of Buyer  = P E E(L) – P L (L)*L  L = P E E L – (P L + dP L L)= 0 dL MRP - MFC =0 Example: L = -10 + 2P L supply P = 5 + 0.5L

328 327 Numerical Example of Marginal Factor Cost to Monopsonist TC = P L L P L = 5 + 0.5L TC = P L L = (5 + 0.5L)L = 5L + 0.5L 2 MFC= TC L = 5 + 2*0.5L = 5 + L

329 Chapter 10

330 329 Sum Up Factor Demand

331 330 Monopsony Outcome

332 331 Bilateral Monopoly Assume Both Want the Same Quantity

333 332 Bilateral Monopoly Assume Both Want the Same Quantity

334 333 Reservation Prices

335 334 Graph the Decision Process MRP = P P L & MRP L LL1L1 P L-1 D = MRP L P L-2 L2L2

336 335 MRP = D (Marginal Benefit) Need Buyer Marginal Cost P L & MRP L L =Seller Marginal production cost SLSL MFC L L ms P L-ms D = MRP L

337 336 Numerical Example – Monopsony Market Sell Electricity P E = $10 per megawatt Produce electricity from LNG (let Lng = L) E = 8L – 2L 2 Buy LNG supply L = -10 + 0.5P L  P L = 20 + 2L Maximize profits P E *E - P L L  = 10(8L – 2L 2 ) – (20+2L)L  L = 80 – 10*4L - 20 – 4L = 0 80 – 10*4L = 20 + 4L MRP = MFC P L & MRP L LNG MFC L L ms P L-ms D = MRP L =80 – 10*4L P L = 20+2L = 20+4L L

338 337 Numerical Example – Monopsony Market P L & MRP L LNG MFC L P L-ms =22.72 D = MRP L =80 – 10*4L P L = 20+2L = 20+4L L MRP = MFC 80 – 10*4L = 20 + 4L 44L = 60 L = 60/44 = 1.36 P L = 20 + 2L = 20 + 2*1.36 = 22.72 E = 8L – 2L 2 = 8*1.36 – 2*1.36 2 = 8.156 L ms=

339 Chapter 11

340 339 Duopoly theory – Cournot model Two Players Choose quantity to maximize profits given the other firms output Inverse demand function demand P = 100 - 0.5(q 1 + q 2 ) C 1 = 5q 1, C 2 = 0.5q 2 2 Profit functions  1 = (100 - 0.5(q 1 + q 2 ))q 1 - 5q 1  2 = (100 - 0.5(q 1 + q 2 ))q 2 - 0.5q 2 2

341 340 Firm 1  1 /  q 1 =(100 - 0.5(q 1 + q 2 )) - 0.5q 1 - 5 = 0 rearranged to = 95 - q 1 - 0.5q 2 = 0 reaction function q 1 = 95 - 0.5q 2 Firm 2  1 /  q 2 = (100 - 0.5(q 1 + q 2 )) - 0.5q 2 - q 2 rearranged to = 100- q 2 - 0.5q 1 - 2q 2 reaction function q 2 = 100/2 - (0.5/2)q 1 = 50 - 0.25q 1 Duopoly theory – Cournot model First order conditions

342 341

343 342 second equation into the first. q 1 = 95 - 0.5(50 - 0.25q 1 ) = 95 – 25 + 0.125q 1 = 70 + 0.125q 1 q 1 - 0.125q 1 = 70 q 1 (1-0.125) = q 1 *0.875 = 70 q 1 = 70/(0.875) = 80 Then q 2 = 50 - 0.25*80 = 30 Price from P= 100 - 0.5(80 + 30) = 45 Equilibrium Solve Where Reaction Functions Cross

344 343 What If Out of Equilibrium q 1 = 95 - 0.5q 2 q 2 = 50 - 0.25q 1

345 344  1 = Pq 1 - C 1 = 45*(80) - 5*80 = 3200  2 = Pq 2 - C 2 = 45*(30) - 0.5*30 2 = 900 Profits

346 345 P = 100 - 0.5(q 1 + q 2 )C 1 = 5q 1, C 2 = 0.5q 2 2 MC 1 = 5 MC 2 = q 2 Competitive Model P = MC

347 346 P = 5 = 100 - 0.5(q 1 +q 2 ) q 1 + q 2 = 190 MC 2 = P = 5 = q 2 q 2 = 5 q 1 = 190- 5 = 185  1 = 185*5 - 5(185) = 0 no Ricardian rents normal rate of return  2 = 5*5 - 0.5*(52)= 12.5 Ricardian rents Competitive Model P = MC

348 347 What if equal n-opolist P = a –bnq i TC = c + dq i If act as competitors P = MC a –bnq i = d => q i = (a-d) P = a –bn(a-d) = d bn bn If act as duopolist  i = (a-b[(n-1)q j +q i ])q i – c – dq i = 0  i = aq i -b(n-1)q j q i +q i 2 – c – dq i = 0

349 348 What if equal n-opolist  i /  q i = a – b(n-1)q j +2bq i – d = 0 a – b(n+1)q i – d = 0 q i = (a-d) 2b(n+1) P= (a-b[a-d/2b(N+1)_= 0

350 349 What if sold gas on a monopoly market? MR= MC 100 - (q 1 +q 2 ) = 5 q 1 + q 2 = 95 P = 100 -0.5(95) = 52.5

351 350 MC 2 = 5 = q 2 q 2 = 5 q 1 = 95-q 2 = 95-5 = 90  1 = 90*52.5- 5(52.5) = 4462.5  2 = 5*52.5 - 0.5*(5 2 )= 250 Monopoly rents How much does each player produce?

352 351 Perfectly Price Discriminating Monopolist

353 352 one firm more information or more dominant optimizes given the other firm’s reaction function In the above, suppose 1 is the dominant firm  1 = (100 - 0.5(q 1 + q 2 ))q 1 -5q 1 but knows that firm 2’s reaction function is q 2 = 50 - 0.25q 1  1 = (100 - 0.5(q 1 + (50 - 0.25q 1 ))q 1 - 5q 1  1 = 100q 1 - 0.5q 1 2 - 25q 1 + 0.125q 1 2 - 5q 1  1 /  q 1 = 100 – q 1 - 25 + 0.25q 1 – 5 = 0 0.75q 1 = 70 => q 1 = 70/0.75 = 93 1/3 Stackleberg solution q 1

354 353 q 2 ’s reaction function is the same as before q 2 = 50 - 0.25q 1 = 50 - 0.25*93.33 = 26.67 Stackleberg Cournot PC Monopoly q1 93.33 80 185 90 q2 26.67 30 5 5 P = 40.00 45 5 52.5 profit 1= 3266.67 3200 0 4275 profit 2=711.11 900 12.5 250 Stackleberg solution q 2

355 354 Firm 1 produces q 1 = 93 1/3 expecting q 2 = 26.67 Firm 2 maximizes  2 = (100 - 0.5((95 - 0.5q 2 ) + q 2 ))q 2 - 0.5q 2 2 = 52.5q 2 - 0.75q 2 2  2 /  q 2 = 52.5 - 1.50q 2 = 0 q 2 = 35 expecting q 1 = 95-0.5(35) = 77.5 P = 100 – 0.5*(93.33+77.5) = 15  1 = 933  2 =-88 Not a stable equilibrium What If Both Try to be Leader

356 355 Bilateral Monopoly Model Quantity agreed upon – X c = 1 c 1 reservation price of seller b reservation price of buyer Price between c 1 and b

357 356 Add a Second Supplier with a Reservation (c 2 ) c 1 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3439909/slides/slide_357.jpg", "name": "356 Add a Second Supplier with a Reservation (c 2 ) c 1

358 357 Possible Rents at P If 1 sells  1 = p-c 1 = rent supplier 1 If 2 sells  2 = p-c 2 = rent supplier 2  3 = b-p = rent buyer Find core no coalition can block

359 358 Core no coalition can block 1.  1 +  2 +  3 = b - c1 Core = 5.  i > 0 6.  1 +  2 > 0 7.  1 +  3 > b - c 1 8.  2 +  3 > b - c 2

360 359 1.  1 +  2 +  3 = b - c 1 Core = 2.  i > 0 3.  1 +  2 > 0 4.  1 +  3 > b - c 1 5.  2 +  3 > b - c 2 1 & 3 => 6.  2 = 0 (insight #1) 5 & 6 => 7.  3 > b - c 2 Substituting 6 into 1 8.  1 + 0 +  3 = b - c 1 Rearranging 8 9.  1 = b - c 1 -  3 Using 2 and 7 10. 0 <  1 < b - c 1 - (b - c 2 ) 11. 0 <  1 < c 2 - c 1 (insight #2) Core no coalition can block

361 360 Case 2: c 1 < c 2 best that Firm 1 can do is difference between its costs and rival 2 if Firm 1 charges slightly lower price will get all sales Redo for one seller and two buyers

362 361 Limit Pricing Model

363 Chapter 12

364 363 first order conditions (foc) International Energy Workshop collected forecasts

365 364 Graph - Two Period Model 2 periods – now and next year Q = 10 – 2.5P + 0.1Y Res = 50 no income growth r = 0.2 no costs Y = 500 Q = 10 -0.5P + 0.1(500) = 60 - 0.5 Q Inverted Demand P = 200 – 2Q

366 365 Demand Now

367 366 Demand Now and Next Year

368 367 Discount Next Year

369 368 Mathematical Solution Basic Model Model Po = P1/(1+r) R = Qo + Q1 r = 0.2 Res = 50 Solution 120 – 2Qo = (120 – 2Q1)/(1+r) 120 – 2Qo = (120 – 2(50-Q1)/(1+0.2) Solve for Qo = 28.18 Q1 = 50 – 28.18 = 21.82 Po = 120-2*28.18 = 63.636 P1 = 120-2*21.82 = 76.364

370 369 Two Period Model with Income Growth 2 periods – now and next year Q = 10 – 2.5P + 0.1Y Res = 50 income growth 25% r = 0.2 no costs Y = (1+0.25)600 = 625 Q = 10 -0.5P + 0.1(625) = 72.5 - 0.5Q Inverted Demand P = 145 – 2Q

371 370 Basic Model – Increase Income Green for More Money

372 371 Increasing Income Period

373 372 Discount P1

374 373 Two Period Model with Higher Interest Rate 2 periods – now and next year Q = 10 – 2.5P + 0.1Y Res = 50 income growth 25% r = 0.4 no costs Y = (1+0.25)500 = 625 Q = 10 -0.5P + 0.1(625) = 72.5 - 0.5Q Inverted Demand P = 145 – 2Q

375 374 Raise Interest Rate Green for More Interest

376 375 Discount Future More

377 376 Mathematical Solution Raise Interest Model Po = P1/(1+r) R = Qo + Q1 r = 0.4 Res = 50 Solution 120 – 2Qo = (145 – 2Q1)/(1+r) 120 – 2Qo = (145 – 2(50-Q1))/(1+0.2) Solve for Qo = 26.67 Q1 = 50 – 26.67 = 23.33 Po = 120-2*26.67 = 66.67 P1 = 1450-2*23.33 = 73.33

378 377 Model 4: Raise Reserves Green for More Reserves

379 378

380 379 Add Demand Next Period

381 380 Discount Next Period

382 381 Mathematical Solution Raise Interest Model Po = P1/(1+r) R = Qo + Q1 r = 0.4 Res = 75 Solution 120 – 2Qo = (120– 2Q1)/(1+r) 120 – 2Qo = (120 – 2(75-Q1))/(1+0.2) Solve for Qo = 39.55 Q1 = 50 – 39.55 = 35.45 Po = 120-2*39.550 = 40.91 P1 = 120-2*35.45 = 49.09

383 382 Add Constant Costs to the Model = 20

384 383 Red = Marginal Cost 20

385 384 Add Demand Next Period and Discount for Basic Model

386 385 P1 – MC!

387 386 Discount P1 - MC

388 387 Mathematical Solution Marginal Cost = 20 Model Po - MCo= (P1-MC1)/(1+r) R = Qo + Q1 r = 0.2 Res = 50 Solution 120 – 2Qo – 20 = (120 – 2Q1 – 20)/(1+r) 120 – 2Qo – 20 = (120 – 2(50-Q1) – 20 /(1+0.2) Solve for Qo = 27.27 Q1 = 50 – 27.27 = 22.73 Po = 120-2*27.27 = 65.45 P1 = 120-2*22.73 = 74.55

389 388 Costs a Function of Current Production MC i = a + bQ i b > 0 = increasing cost industry b = 0 constant cost industry b< = decreasing cost industry MC o = a + bQ o MC 1 = a + bQ 1 Purple = Cost

390 389 Costs Increase with Production MCo

391 390 Add Po

392 391 Po - MCo

393 392 MC1

394 393 PQ and MC1

395 394 P1 - MC1

396 395 Put Two Sides Together

397 396 Find Po-MCo=(P1-MC1)/(1+r)

398 397 Quantitative NPV Consumer Surplus =  P o dQ o +  P 1 dQ 1 - NPV  = 0.5*(200-142.86)*28.57 + 0.5*(200-157.14)*21.43/(1.1) -7142.88 = 1233.74

399 398 Mathematical Solution MC = 10+1.5Qi Model:Po – MCo = (P1-MC1) / (1+r) R = Qo +Q1r = 0.2 MCo = 10 + 1.5 QoMC1 = 10 + 1.5Q1 120-2Qo –(10 +1.5Qo) = 120-2Q1 – ( 10- 1.5Q1)/(1+0.2) 120-2Qo –(10 +1.5Qo) = (120-2Q1 – ( 10- 1.5 (50-Qo))) (1+0.2) Solution: Qo = 25.58 Q1 = 50 - 25.58 = 24.42 Po = 120 - 2*25.58 = 68.83 P1 = 120 - 2*24.42 = 71.17

400 399

401 400 MCo = 20 + Qo

402 401 Po-MCo 120- 2Qo – ( 20 + Qo)

403 402 Next Period MC1 = 20 + Qo

404 403

405 404

406 405 Income increases

407 406 Change Interest Rate

408 407 Non-Scarce resources

409 408 What is socially optimal use of resources? instead of maximizing NPV profits maximize NPV of social welfare consumer + producer surplus s.t. resource constraint

410 409 Social Welfare

411 410 5. Model 3 + three cases for MC a. MC = constant = 20 b. MC = function of current production MC o = 2 + 0.2Q o MC 1 = 1 + 0.2Q 1 technical progress in 2 that lowers costs c. MC = function of cumulative production MC o = 2 + 0.2Q o MC 1 = 2 + 0.2Q o 5a. P 0 -MC o = P 1 - MC 1 (1+r)

412 411 Model with Costs

413 412 P o -MC o = P 1 - MC 1 (1+r) 200-2Q o -20= 206-2Q 1 - 20 (1+0.05) substitute in the constraint 200-2Q o -20= 206-2(50-Q o ) - 20 (1+0.05) Q o /Q 1 = 25.12/24.88 P o /P 1 = 149.76/156.24 NPV Net Rev =6487.80 NPV Cons Surp =2196.864

414 413 Compare to case 3 Q o /Q 1 = 25.37/24.63 P o /P 1 = 149.26 /156.74 Reduce current consumption higher costs delays consumption 5b. MC = function of current production MC o = 20 + 0.2Q o MC 1 = 10 + 0.2Q 1 technical progress in 2 that lowers costs

415 414 Model 6. Back Stop Fuel - Sweeney 1989 (LA) @10% Example (update) gasoline => $31.50 NG => methanol $45 per barrel coal => methanol $52 wood => methanol $73 compressed NG => $33 corn => ethanol $65 oil shale => oil $42 tar sands => oil $41

416 415 Back Stop Case 1: Po = 200 - 2Qo no Y grow r = 10% R = 50 MC = 0

417 416 Backstop @ 125 P 1 = 125 P o = 125/(1.1)= 113.64 Q0 = 43.18 Q0 resource43.18 Q0 bkstop0.00 Q137.50 Q1 resource6.82 Q1 bkstop30.68 Resource price will gradually approach backstop price.

418 417 Backstop Analysis

419 418 Shortage Case P = 200 - 2Q MR = 200 - 4Q 200-4Q o = 100 - 4Q 1 (1+.1) 200-4Q o = 100 - 4(50-Q o ) (1+.1) Qo26.19 Q123.81 P0147.62 P1152.38 Compared to PC Qo28.57 Q121.43 P0142.86 P1157.14 Market imperfections 1. r social < r private 2. externalities 3. taxes 4. monopoly Static PC P = MC Dynamic PC P o -MC o = P 1 -MC 1 (1+r) Static Monopoly MR = MC Dynamic Monopoly MR o -MC o = MR 1 -MC 1 (1+r) Two views of resources Adelman never use them all race between depletion and technology so far technology has been winning Hotelling exhaustible Application ad valorem vs unit tax bonus bidding versus royalty disruption 1 period, normal the next Pindyck Gams TD = 1 - 0.13 P t + 0.87td t-1 + 2.3 (1.015) t S t = 1.1 + 0.10P t (1.02) CS/7 + 0.75S t-1 S t = supply of the competitive fringe CS = cumulative production competitive fringe CS t = CS t-1 + S t D t = Td t - S t Opec Demand R t = R t-1 - D t N max W =  ((1+  ) t ) -1 (P t - 250/R) * D t=1 Gams indices given data decision variables constraints objective function sets parameters, tables, scalars variables equations model Solve (LP, NLP, MIP)

420 419 Shortage Other cases - n periods P o = P 1 + P 2 + P 3 = P n (1+r) (1+r) 2 (1+r) 3 (1+r) s.t. Q 0 + Q 1 + Q 2 + + Q n = R Example: No income growth Case 1: P = 200 - 2Q Q = 100 - 0.5P Maximum price = 200 QReserves left Q = 100 - 0.5P P n 200.000.001000 P n-1 = P n /(1+r)181.829.09990.91 P n-2 =P n-1 /(1+r)165.2917.36973.55 P n-3 150.2624.87948.69 P n-4 136.6031.70916.99 P n-5 124.1837.91879.08 P n-6 112.8943.55835.53 P n-7 102.6348.68786.84 P n-8 93.3053.35733.49 P n-9 84.8257.59675.90 P n-10 77.1161.45614.46 P n-11 70.1064.95549.51 P n-12 63.7368.14481.37 P n-13 57.9371.03410.34 P n-14 52.6773.67336.67 P n-15 47.8876.06260.61 P n-16 43.5378.24182.37 P n-17 39.5780.22102.16 P n-18 35.9782.0120.14 Keep going until R is gone What if Y grows don’t know highest price need NL programming model max max PS + CS or max NPV subject to reserve constraints 9. Monopoly Model Maximize NPV  = P(Q o )Q o + P(Q 1 )Q 1 (1+r) s.t. R = Q o + Q 1  = P(Q o )Q o + P(Q 1 )Q 1 + (R - Q o - Q 1 ) (1+r)  Qo = P(Q o )+ dPo Qo - = MR o - = 0 dQ o  Q1 = P(Q 1 ) + dP 1 Q 1 - = MR 1 - = 0. (1+r) dQ 1 (1+r) (1+r) MR o = MR 1 (1+r)

421 420 Monopoly

422 421 Price Control P = Pmax = P 1 = $112.00 P o = $101.82 Q 0 = 49.09 Q 0 resource = 49.09 Q 0 backstop = 0.00 Q 1 = 44.00 at $101.82 but Q 1 resource = 0.91 Then jumps to the backstop price P1 = $150

423 422 Compare to competitive case

424 423 Calculus of Variation Chiang pick a time path that optimizes a function  0 T F(t, y(t),y’(t))dt y(0) = A Y(T) = Z know Z and T y might be oil production F could be discounted profits from the mine y’(t) is how production is changing

425 424 CV – Objective Function  =  P o dQ o -  MC o dQ o +  P 1 dQ 1 -  MC 1 dQ 1 + (R-Q o -Q 1 ) (1+r) (1+r)  = R-Q o -Q 1 =0  Qo = P 0 - MC o - = 0  Q1 = P 1 - MC 1 - = 0 (1+r) P 0 -MC o = P 1 - MC 1 = (1+r)

426 425 Backstop

427 426 Backstop Quantitative Q = 60 – 0.5P Res = 60 P = 120 – 2Q r = 0.2 Backstop = $42 P 1 = 42 P o = 42/1.2 = 35 Q o = 60 – 0.5*35 = 42.50 Q 1 = 60 – 0.5*42 = 39.00 Q 1 + Q o = 81.50 Q o + Q 1 – Res = backstop consumption = 81.50 - 60 = 21.50

428 Chapter 13

429 428 Above Ground Costs - continuous Suppose R o = 100 and  = 0.10 (decline rate of  ): Q o = 0.10*100 = 10 =  R o Q 1 = e.10*t  R o = e -0.10*1   Q 2 = e.10*t  R o = e -0.10*2 ... Q 20 = e.10*t  R o = e -0.10*2    Q 100 = e.10*t  R o = e -.10*2 

430 429 Above Ground Costs - continuous (decline rate of  ): Q o =  R o K =  $Q t e -rt dt =  $  R o e -  t e -rt dt $ = (K/(R o  /(  o  e (-  -r)t dt) = denominator = [(e (-  -r)t /(-  -r)]| o  = [(e (-  -r)  /(-  -r) - (e (-  -r)0 /(-  -r)] = [(0)(-1)/(-  -r) = 1/(  +r) Solving $ = (K/R o )(  +r)/  (K/Q o )(  +r) K/Q o is referred to as capacity cost

431 430 Oil Costs Example:Decline rate 0.13, r = 0.10 $1 billion, R = 200 million $ = (1000/200)(0.13+0.10)/0.10 = $11.5

432 Nuclear Policy 431

433 Hubbert: 1962 used logistics curves on US reserves Q t = Q  (1+  e (-  (t-to) ) Q t = cumulative production Q  = total reserves that will ever be produced

434 Chapter 14

435 434 Whole Blending Problem max  = $0.08*X 1 + $0.09*X 2 s.t. 0.4X 1 + 0.57143X 2 < 100,000 straight run 0.8X 1 + 0.57143X 2 < 140,000 cracked Graph in X 1 X 2 space constraint 1 constraint 2 X 1 = 0 => X 2 = 175,000 X 1 = 0, X 2 = 245,000 X 2 = 0 => X 1 = 250,000 X 2 = 0, X 1 = 175,000

436 435 Graph Constraints constraint 1 constraint 2 X 1 = 0 => X 2 = 175,000 X 1 = 0, X 2 = 245,000 X 2 = 0 => X 1 = 250,000 X 2 = 0, X 1 = 175,000

437 436 Objective Function  = 0.08X 1 + 0.09X 2 X 2 =  /0.09 - (0.08/0.09)X 1 Find highest line on constraint -slope dX 2 /dX 1 = -0.8888

438 437 For this Shaped Constraint Set Always on Corners Check profits A, B, C  (A) = 0.08X 1 +.09X 2 = 0.08*(0) + 0.09(175,000) = 15750  (C) = 0.08X 1 +.09X 2 = 0.08*(175,000) + 0.09(0) = 14000  (B) need to find what X 1 and X 2 are.

439 438 Solve simultaneously (1)0.4X 1 + 0.57143X 2 = 100,000 (2)0.8X 1 + 0.57143X 2 = 140,000 Solve 1 for X 1 (3)X 1 = (100,000 - 0.57143X 2 )/0.4 Plug (3) into (2) (4) 0.8(100,000 - 0.57143X 2 )/0.4 + 0.57143X 2 = 140,000 (5) X 2 =105,000 (6) X 1 = (100,000 - 0.57143*105000)/0.4 = 100,000

440 439 Solve simultaneously X 2 = 105,000 X 1 = 100,000  = 0.08X 1 + 0.09X 2  (B) = 0.08(100000) +0.09(105000) = 17,450  (A) = 0.08*(0) + 0.09(175,000) = 15750  (C) = 0.08*(175,000) + 0.09(0) = 14000

441 440 How to Blend X 2 = 105,000 X 1 = 100,000 straight run 0.40(100,000) + 0.57143(105,000) = 100,000 cracked 0.70(100,000) + 0.57143(105,000) = 140,000 u 1 = 40,000 to grade 1 = 60,000 to grade 2 u 2 = 80,000 to grade 1 = 60,000 to grade 2

442 441 Transport Problem five supply points for crude oil A, B, C, D, E available are 10, 20, 30, 80, 100 three refineries X, Y, Z crude oil requirements of 40, 80, 120

443 442 Transport Costs

444 443 Math Formulation of Problem Objective: Minimize TTC =  *  = C ij *A ij. supply shipments  j A ij < Y i for all i refinery satisfy crude oil needs  i A ij = X j for all j Set up in Excel Solver

445 444 Simple Example max  = $0.08*X 1 + $0.09*X 2 s.t. 0.4X 1 + 0.57143X 2 < 100,000 straight run 0.8X 1 + 0.57143X 2 < 140,000 cracked

446 Blending Model Profit Function two processes grade 1 X 1 = 2.5 min (u 1, u 2 /2) grade 2 X 2 = 1.75 (u 1,u 2 ) u 1 = straight run (100,000) u 2 = cracked gasoline (140,000)  1 = $0.08/gal X 1  2 = $0.09 / gal X 2  = $0.08*X 1 + $0.09*X 2

447 What are technical constraints – u 1 2.5 gallon of grade X 1 requires 1 gallon of u 1 gallon of u 1 per gallon of X 1 u 1 /X 1 = 1/2.5 = 0.4 1.75 gallon of grade X 2 requires 1 gallon of u 1 gallon of u 1 per gallon of X 2 u 1 /X 2 = 1/1.75 = 0.57 0.4X 1 + 0.57X 2 < 100,000 u 1 constraint

448 Total requirements of u 2 for X 1 and X 2 2.5 gallon of grade X 1 requires 2 gallon of u 2 gallon of u 2 per gallon of X 1 u 2 /X 1 = 2/2.5 = 0.8 1.75 gal of grade 2 requires 1 gallon of u 2 gallon of u 2 per gallon of X 2 u 2 /X 2 = 1/1.75 = 0.57 0.8X 1 + 0.57X 2 < 140,000 u 2 constraint

449 Whole problem max  = $0.08*X 1 + $0.09*X 2 s.t. 0.4X 1 + 0.57143X 2 < 100,000 straight run 0.8X 1 + 0.57143X 2 < 140,000 cracked Graph in X 1 X 2 space Constraint 1 Constraint 2 X 1 = 0 => X 2 = 175,000X 1 = 0, X 2 = 245,000 X 2 = 0 => X 1 = 250,000 X 2 = 0, X 1 = 175,000

450 Constraint Set  = 0.08X 1 + 0.09X 2 X 2 =  /0.09 - (0.08/0.09)X 1

451 Finding highest line that touches constraint set with slope dX 2 /dX 1 = -0.8888 Will be one of points A, B, C Check the profit at each point  (A) = 0.08X 1 +.09X 2 = 0.08*(0) + 0.09(175,000) = 15,750  (C) = 0.08X 1 +.09X 2 = 0.08*(175,000) + 0.09(0) = 14,000  (B) need to find what X 1 and X 2 are

452 Solve simultaneously 0.4X 1 + 0.57143X 2 = 100,000 0.8X 1 + 0.57143X 2 = 140,000 If know matrix algebra 0.4 0.57143 X 1 = 100,000 0.8 0.57143 X 2 140,000 Invert and multiply -2.5 2.5 100,000 = 100,000 3.49999 -1.75 140,000 105,000  (C) = 0.08(100,000) +0.09(105,000) = 17,450

453 How much u 1 and u 2 to blend to get X 1 = 100,000 and X 2 = 105,000 straight run 0.40(100,000) + 0.57143(105,000) = 100,000 u 1 cracked 0.80(100,000) + 0.57143(105,000) = 140,000 u 2 u 1 = 40,000 to grade 1 60,000 to grade 2 u 2 = 80,000 to grade 1 60,000 to grade 2

454 Transport Problem five supply points for crude oil A, B, C, D, E available are 10, 20, 30, 80, 100 three refineries X, Y, Z crude oil requirements of 40, 80, 120

455 Transport Costs

456 Math Formulation of Problem Objective: Minimize TTC =  *  = C ij *A ij. supply shipments  j A ij < Y i for all i refinery satisfy crude oil needs  i A ij = X j for all j Set up in Excel Solver

457 Chapter 15

458 457 Cash Market Ignore transaction and storage costs Trader has a barrel of crude in transit Current spot price is S t = $18 Trader is paid the spot price upon delivery at S T S T Gain or Loss Value $18 17 -1 18 0 19 1

459 458 Trader Wants to Hedge Suppose F T = $18 to deliver oil at time T Sells one futures At time T the contract is worth F T – S T Futures Cash 18 - S T T S T Contract Sold $18 17 1 -1 18 0 0 19 -1 1

460 459 Hedging When Futures Price Different than Current Spot (Mia and Edwards 111-117) Heating Oil Distributor 1-Oct Cost of Crude0.51 Spot price0.54 Cost of carry/gal/month0.008 Contract for Delivery Dec 15420,000 at market price Delivery in 2.5months If sell at current spot on Dec 15, profits would be (0.54 - 0.51 - 2.5*0.008)*420,000 = 4,200 Expecting price to rise but not certain If price 0.54 more profits

461 460 Short with Zero Basis Risk Suppose Futures price is for Dec. 15 = $0.56 Basis = Cash Price – Minus futures Price = 0.54 – 0.56 = -$0.02 If products same –basis should go to zero at delivery But if using another product to hedge basis may not go to zero Suppose the basis stays constant You want to hedge to lock in profits

462 461 Distributor Short Crude1/12/15 Agreed to Deliver Crude Price0.55 Spot price 0.535 Cost of carry/gal/month 0.008 Deliver Crude March 15 420,000 Delivery in 1.5 month If buy at current spot, hold and sell at contract rate profits: (0.55 - 0.535 - 1.5*0.008)*420,000 =$1,260 Could wait to buy in March for delivery Suspects price will be lower, more profit But price may be higher – doesn't want to take risk Long Hedge

463 462 Convenience Yield < Storage plus Interest Rate example r = 1%,  = 1%,  = 1%, S t = 20. F T = S t e (r+  -  )T = S t e (0.01 + 0.01 - 0.01)T T F T = $21.025 5 F T = $22.103 10 Further out is T the higher is F T Normal market - contango

464 463 Convenience Yield > Storage Plus Interest Rate (r+  -  ) r+  <  Example: r = 1%,  = 1%,  = 3%, S t = 20. T F T = $19.025 5 F T = $18.097 10 Further out is T the lower is F T Backwardation or inverted market

465 464 Futures Markets in Contango (normal) and Backwardation (inverted)

466 What Determines Energy Future Prices Q2Q2 P1P1 Q1Q1 QeQe QdQd QcQc QbQb PTPT PTPT P2P2 S1S1 S2S2 D1D1 D2D2 a b c de f importsexports 2 Market 1Market 2

467 Optimal Hedge Ratio United Airlines will buy 500,000 gallons of jet fuel There is no futures market for jet fuel σ s,jet = 0.028 σ f,heating = 0.05 ρ = 0.9 United Airlines should buy 250,000 gallons of heating oil at the futures market to hedge their risk 3

468 One small wrinkle to the spark spread An electricity contract is 736 mWh Gas contracts are in 10,000 MMBtu h = 0.59 ≒ 0.6 lowest common denominator 3 gas contracts for every 5 electricity contracts 5

469 468 Hedging When Futures Price Different than Current Spot (Mia and Edwards 111-117) Heating Oil Distributor 1-Oct Cost of Crude0.51 Spot price0.54 Cost of carry/gal/month0.008 Contract for Delivery Dec 15420,000 at market price Delivery in 2.5months If sell at current spot on Dec 15, profits would be (0.54 - 0.51 - 2.5*0.008)*420,000 = 4,200 Expecting price to rise but not certain If price 0.54 more profits

470 Chapter 16

471 470 Value of Call at Expiration

472 471 Value of Put at expiration

473 472 Value before expiration depends on following variables Increase Call Put 1. Underlying Asset Price K K Value STST STST

474 473 Value before expiration depends on following variables Increase Call Put 2. Exercise Price K K Value STST STST

475 474 Value before expiration depends on following variables Increase Call Put 5. Stock Risk K K Value STST STST

476 475 Single Period Binomial Pricing Model European Call know percentage rise or fall

477 476 Buy a stock and bond portfolio equivalent to C Let risk free rate = 6% Bond matures in one period Sell a bond

478 477 Buy a stock and bond portfolio equivalent to C Buy a Stock

479 478 After a Year If the stock price goes up you have 55-45 = $10 If the stock price goes down you have 45 – 45 =0 same portfolio as buying a call must be worth the same otherwise arbitrage Value of portfolio now $50 - 42.45 = $7.55

480 479 Solve for N anf B t Port uT = N*U*S t + R*B t = c u = S T - K = 10 Port dT = N*D*S t + R*B t = c d.= 0 N = (c u - c d )/[(U - D)*S t ], B t = [c u *D - c d *U]/[(U - D)*(-R)] N = (10 - 0)/[(1.1 - 0.9)*100] = 0.5, B t = [(10*0.9) - (0*1.1)]/[(1.1 - 0.90)*(-1.06)] = -42.45. buy (+) half a stock sell (-) $42.45 worth of bonds Value of the portfolio is, as before, N*S t + B t = 0.5S t - 42.45 = $50 - $42.45 = $7.55.

481 480 What is Value of Your Portfolio? If risk neutral in the above example then 1.1 (p) + 0.9(1-p) = 1.06 1.1p - 0.9p = 1.06-0.9 p =0.16/0.20 = 0.8 value of call 0.8*(10) + 0.2(0) = $7.55 (1.06) Same value so can act as if risk neutral

482 481 P for general case 1.1 (p) + 0.9(1-p) = 1.06 (p)*US t + (1 - p)*DS t = (1 + r)*S t = R*S t Solving we get p = (R - D)/(U - D)

483 482 What is Value of Your Portfolio? What if two periods to maturity 0.8 (1.1) 2 100 S = 121, C = 21 0.8 (1.1)*100 0.2 100 (2)0.2*0.8 (1-.1)(1+.1)100 S = 99, C=0 0.2 (1-.1)*100 0.2 2 (1-.1)(1-.1)100 S = 81, C= 0 Value today 0.8*0.8*21 + 2*(0.8*0.2)0 + 0.2*0.2(0) = $11.962 1.06 2

484 483 Finish - Value of 2 period American Call What if two periods to maturity 0.8 (1.1) 2 100 S = 121, C = 21 0.8(1.1)*100 0.2 100 (2)0.2*0.8(1-.1)(1+.1)100 S = 100, C=0 0.2(1-.1)*100 0.2 2 (1-.1)(1-.1)100 S = 81, C= 0 Value today 0.8*0.8*21 + 2*(0.8*0.2)0 + 0.2*0.2(0) = $11.962 1.06 2

485 Chapter 17

486 Chapter 18

487 Chapter 19

488 487 Input Output Model - Leontief 0.05 basic materials (B) per B (tons) (B/B) 0.01 B per manufacturing (M) (tons) (B/M) 0.09 B into E (10 6 BTU) (E) (B/E) 0.00 M per B (M/B) 0.50 M per M (M/M) 0.01 M per E (M/E) 0.20 E per B (E/B) 0.07 E per M (E/M) 0.15 E per E (E/E)

489 488 Write in Equations: B = 0.05 B + 0.01M + 0.09E + 1 M = + 0.5M + 0.01E + 2 E = 0.20 B + 0.07M + 0.15E + 10 M all constants to the right and convert to matrices 1 0 0 0.05 0.01 0.09 B 0 1 0 - 0.00 0.50 0.01 M = d 0 0 1 0.20 0.07 0.15 E I A x d (I-A)x = d

490 489 Last Time I-O Regionalize – Consumption, Investment, Government, Net Exports a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24... a 91 a 92 a 93 a 94 Region X Sector Sector X Sector a 11 C a 12 I a 13 G a 14 (X-M) a 21 C a 22 I a 23 G a 24 (X-M) AU=... a 91 C a 92 I a 93 G a 94 (X-M) Region X Sector C000C000 0I000I00 00G000G0 00 0(X-M)

491 490 Last Time Keeping Track of Pollution (1) f ij is the amount of pollutant i per unit of good j Total amount of pollution i is P i =  j (f ij X j ) Example 2 pollutants (P 1,P 2 ), 3 goods (X 1, X 2, X 3 ) pollutant 1 pollutant 2 f 1j f 2j X j X 1 = coal25250 X 2 = gas14340 X 3 = oil19430

492 491 Last Time Keeping Track of Pollution (2) pollutant 1 pollutant 2 f 1j f 2j X j X 1 = coal25250 X 2 = gas14340 X 3 = oil19430 P = P 1 P 2 F X P = F'X

493 492 Last Time Keeping Track of Pollution (2) f 1j f 2j X j 25250 14340 19430 P = F'X P = 25 14 19 50 2 3 4 40 30 = 25*50 + 14*40 + 19*30 = 2380 2*50 + 3*40 + 4*30 = 340

494 493 Last Time Units Sector output input energy (a) other (b) energy (a)0.200.20 other (b) 0.300.10 a/a a/b b/a b/b a BTU b tons outputs = BTU and tons

495 494 If a and b both $ output input energy other energy 0.200.20 other 0.400.10 Sum0.600.30 1- sum = value added 1-0.6=0.4 1-0.3=0.7

496 495 Three Sectors - $ E MOd E 0.200.300.25 10 M 0.350.100.1520 O 0.450.150.4030 x = (I-A) -1 d x = 81.45 75.56 129.98

497 496 Add Regulation - Change Coefficient E MOd E 0.300.300.25 10 M 0.350.100.20 20 O 0.500.250.4030 x = (I-A) -1 d x r = 158.27 136.86 238.92 x r -x = 158.27 - 81.45 = 76.82 136.86 - 75.56 = 61.31 238.92 - 129.98 = 108.94 sum = 247.06

498 497 How measure Above in dollars - $247.06 as % of GDP 247.06/10000*100 ~ 2.5% If E, M, O measured in tons E = 76.82 M = 61.31 O = 108.94 Value = Price'X = [1, 4, 3] 76.82 61.31 108.94 = 705.88

499 498 How Can the Rules be Written How much to clean up quantity Z = 7 % of Pollution Z = αP How much you can pollute quantity P a = 2 = P - Z %of pollution P a = αP

500 499 More complicated - model control industry new control industry Z produce 0.05 lbs of pollutant/$ of energy current pollution = 0.05*X1 = 0.05* 197.146 =9.857 regulation remove 90% Z = 0.9*0.05*X1 But it takes resources to remove pollution $0.02 energy / lb removed $0.05 mineral / lb removed $0.12 mfg / lb removed

501 500 More complicated - model control industry New industry Z X 1 = 0.2X 1 +0.2X 2 +0.25X 3 + 0.15X 4 + 0.02Z +10 X 2 = 0.0X 1 +0.1X 2 +0.15X 3 + 0.17X 4 + 0.05Z+20 Fill in for X 3 to X 4 X 3 = X 4 = Add regulation Z = 0.9*0.05*X 1 Notice: Old A plus extra row and extra column x includes Z x = A g x + d 0.3X 1 +0.3X 2 +0.30X 3 + 0.10X 4 + 0.12Z+100 0.3X 1 +0.1X 2 +0.20X 3 +0.40X 4 + 30

502 501 Variables left, constants right X 1 - 0.2X 1 -0.2X 2 -0.25X 3 - 0.15X 4 - 0.02Z =10 X 2 - 0.0X 1 -0.1X 2 -0.15X 3 - 0.17X 4 - 0.05Z =20 X 3 - 0.3X 1 -0.3X 2 -0.30X 3 - 0.10X 4 - 0.12Z=100 Compute for X 4 X4X4 Z - 0.9*0.05*X 1 =0 Let's write as [I-A g ]x = d - 0.3X 1 -0.1X 2 -0.20X 3 - 0.40X 4 = 30

503 502 Write as [I-A g ]x = d 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0.200.200.25 0.15 0.02 0.000.100.150.17 0.05 0.300.300.300.10 0.12 0.300.100.200.40 0.00 X1X2X3X4ZX1X2X3X4Z 0.0450.0 = 10 20 100 30 0 - Solve:(I-A g )x = d  x = (I-A g ) -1 d

504 503 Opportunity Cost of Pollution Regulation Before After Opportunity Cost X1197.146199.3232.177 X2128.245129.8011.556 X3321.918325.4143.496 X4277.253279.7662.513 Z0.000 8.9708.970 pollution before = 0.05*X 1 =0.05*197.146=9.8573 pollution after = 0.05*199.323 - 8.97= 1.00 cost as percent of GDP =

505 Cradle to Grave x = Ax + d x 1 = a 11 x 1 + a 12 x 2 + d 1 x 2 = a 21 x 1 + a 22 x 2 + d 2 x = (I-A) -1 d x 1 = τ 11 d 1 + τ 12 d 2 x 2 = τ 21 d 1 + τ 22 d 2 cradle to grave use of x 1 to get 1 more d 1 dx 1 /dd 1 = τ 11 cradle to grave use of x i to get 1 more d k = τ ik 504

506 505 Set up the Problem B = 0.05B/B*B + 0.01B/M*M + 0.09B/E*E + 1

507 506 Solve B = 0.05 B + 0.01M + 0.09E + 1 (1) M = + 0.50M + 0.01E + 2 (2) E = 0.20B + 0.07M + 0.15E + 10 (3) Solve (1) - (3) simultaneously From equation 2, solve for M M - 0.50M = 0.01E + 2 M = (0.01E + 2)/(1-0.5) = 0.02E+4 Substitute M into equations 1 and 3

508 507 Solve Substitute for M in 1, 3 B = 0.05 B + 0.01(0.02E+4) + 0.09E + 1 (4) E = 0.20B + 0.07(0.02E+4) + 0.15E + 10 (5) Rearrange some terms and simplify B - 0.05 B - 0.0002E - 0.04 - 0.09E = 1 E - 0.20B - 0.0014E - 0.28 - 0.15E = 10 Further combine terms 0.95 B - 0.0902E = 1.04 - 0.20B + 0.8486E = 10.28

509 508 Solve Solve 2 equations with 2 unknowns 0.95 B - 0.0902E = 1.04(1) -0.20B + 0.8486E = 10.28(2) From eq. (1) B = (0.0902E +1.04)/0.95 = 0.0940E + 1.095 Substitute B into eq. 0.968 (2) -0.20(0.0940E + 1.095) + 0.8486E = 10.28 Solve for E -0.0181E - 0.219+ 0.8436E = 10.28 0.8255E = 10.499 →E = 12.718

510 Solution Total E, M, B to support end-use demands of 10, 2, 1 E = M = 0.02E+4 = 0.02*12.718 + 4 = B = = 0.0940*E + 1.095 = 0.0940*12.718 + 1.095 = 509 12.718 4.254 2.244

511 Let's Rewrite Technical Coefficients as Input per unit Output Matrix From Slide 33 0.05 basic materials (B) per B (tons) (B/B) 0.01 B per manufacturing (M) (tons) (B/M) 0.09 B into E (10 6 BTU) (E) (B/E) Inputs down/outputs across B M E B M E 510 0.050.010.09 0.0 0.50 0.01 0.20 0.07 0.15

512 511 Cradle to Grave, Wells to Wheels A= 0.050.010.09 B = 2.30 tons 0.000.500.01 M = 4.25 tons 0.20 0.070.15 E = 12.66 X 10 6 BTU Why can't diagonal elements > 1? Cradle to grave use of energy to produce mfg Direct E*M = M = 0.07*4.25 = 0.30

513 512 Continue Solution A= 0.050.010.09 B = 2.30 tons 0.000.500.01 M = 4.25 tons 0.20 0.070.15 E = 12.66 X 10 6 BTU First Order Indirect (B) E*B M B M = 0.2*0.01*4.25 = 0.01 First Order Indirect (E) E*E*M = 0.15*0.07*4.25 = 0.04 E M Total = 0.30 + 0.01 + 0.04 = 0.350

514 513 Cradle to Grave, Wells to Wheels A= 0.050.010.09 B = 2.30 tons 0.000.500.01 M = 4.25 tons 0.20 0.070.15 E = 12.66 X 10 6 BTU Why can't diagonal elements > 1? Cradle to grave use of energy to produce mfg Direct: 0.07 per unit of M Total first order direct E*M = 0.07*4.25 = 0.30 M But first order indirect: need B to produce M which needs E need E to produce E which needs E

515 Take a Look Back Last Time Input Output A inputs from one industry to another k industries how much x to produce to get d solution for general case x = (I-A) -1 d (kX1) (kXk)(kX1) 530-11f-19m.xlsx CA2 - hand in sheet with names and answers 514 x = (I-A) -1 d

516 Take a Look Back (b) When does solution exist When does solution make economic sense Disaggregate models (i regions, j products) a ij = region i's share of total product j (A)(27X3) European Union (27) Fossil fuels (3) (Coal,Oil,Ngas) FF (3X3) i = 27, j = 3, FF_Region (27X3) a 1,1 a 1,2 a 1,3 Coal 0 0... 0 Oil 0 a 27,1 a 27,2 a 27,3 0 0 Ngas 515

517 Take a Look Back (c) A times FF = FF_Reg a 1,1 a 1,2 a 1,3 Coal 0 0... 0 Oil 0 a 27 a 27,3 a 27,3 0 0 Ngas FF_Reg = consumption of each fuel by region = a 1,1 Coal a 1,2 Oil a 1,3 Ngas a 2,1 Coal a 2,2 Oil a 2,3 Ngas … a 27,1 Coal a 27,2 Oil a 27,3 Ngas 516

518 Take a Look Back (d) f ij is the amount of pollutant i per unit of good j i = 5 pollutants - O 3, PM, CO, Nox, Sox j = 3 products electricity (E), metals (M), Pulp&Paper (PP) 517

519 Take a Look Back (f) 518 x = total output of the three products What do you want to know? I = total of each pollutant I = Fx

520 Take a Look Back (g) Pollution by industry p ij = total pollution i from good j P = F*X 519

521 520 Direct Inputs from One Industry to Another A= 0.050.010.09 0.000.500.01 0.20 0.070.15 x = (I-A) -1 d = 2.30 tons (B) 4.25 tons (M) 12.66 X 10 6 (E) Direct E into each industry 530-11f-19m.xlsx, IO!A20:A22 E*B = a EB *B E*M = E*E B M E = 0.20*2.3 = 0.46 = 0.07*4.25 = 0.30 = ?*?

522 Go to Excel - 530-11f-19m.xlsx You can change yellow, solution is in red 521

523 Solve In Excel 522

524 Multiply (I-A) -1 d Highlight c5:c6 Type in =MMULT(A5:B6,C2:C3) Ctrl Shift Enter Should show { } if it’s a matrix Excel will not allow you to change single elements inside matrix 523

525 Input Output in $ a ij = $ of input i for 1 dollar of output j A = input output matrix of technology matrix B M E B M E When B sells $1 of output B buys 0.10 +0.30 + 0.05 = 0.45 from other industries/$ The remainder 1-0.45 = 0.55 is called value added Value added for M = 1-0.40-0.20-0.25 = 0.15 Value added for E = ? 524

526 525 More complicated - model control industry new control industry Z produce 0.05 lbs of pollutant/$ of energy current pollution = 0.05*X1 = 0.05* 197.146 =9.857 regulation remove 90% Z = 0.9*0.05*X1=0.045X1 But it takes resources to remove pollution $0.02 energy / lb removed $0.05 mineral / lb removed $0.12 mfg / lb removed

527 526 More complicated - model control industry New industry Z X 1 = 0.2X 1 +0.2X 2 +0.25X 3 + 0.15X 4 + 0.02Z +10 X 2 = 0.0X 1 +0.1X 2 +0.15X 3 + 0.17X 4 + 0.05Z+20 Fill in for X 3 to X 4 X 3 = X 4 = Add regulation Z = 0.45*X 1 Notice: Old A plus extra row and extra column x includes Z x = A g x + d 0.3X 1 +0.3X 2 +0.30X 3 + 0.10X 4 + 0.12Z+100 0.3X 1 +0.1X 2 +0.20X 3 +0.40X 4 + 30

528 527 Variables left, constants right X 1 - 0.2X 1 -0.2X 2 -0.25X 3 - 0.15X 4 - 0.02Z =10 X 2 - 0.0X 1 -0.1X 2 -0.15X 3 - 0.17X 4 - 0.05Z =20 X 3 - 0.3X 1 -0.3X 2 -0.30X 3 - 0.10X 4 - 0.12Z=100 Compute for X 4 X4X4 Z - 0.9*0.05*X 1 =0 Let's write as [I-A g ]x = d - 0.3X 1 -0.1X 2 -0.20X 3 - 0.40X 4 = 30

529 528 Write as [I-A g ]x = d 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0.200.200.25 0.15 0.02 0.000.100.150.17 0.05 0.300.300.300.10 0.12 0.300.100.200.40 0.00 X1X2X3X4ZX1X2X3X4Z ??? 0.045 ?? 0.0 = 10 20 100 30 ? 0.0 - Old A A g Solve:(I-A g )x = d  (I-A g ) -1 (I-A g )x = (I-A g ) -1 d x = (I-A g ) -1 d Z - 0.045*X 1 =0

530 529 Clean Up More Compliucated Model with Control Industry (I-A g )x = d 0.8 -0.2 -0.25 -0.15 -0.02 X 1 = 10 -0.0 0.9 -0.15 -0.17 -0.05 X 2 20 -0.3 -0.3 0.70 -0.10 -0.12 X 3 100 -0.3 -0.1 -0.20 0.60 -0 X 4 30 -0.045 0 0 0 1 Z 0 (I-A g ) with augmented row and column

531 530 X with pollution control industry 2.126 1.010 1.270 1.029 -0.245 10 = 199.323 0.543 1.551 0.725 0.696 -0.175 20 129.801 1.391 1.275 2.521 1.129 -0.394 100 325.414 1.617 1.188 1.596 2.674 -0.283 30 279.766 0.096 0.045 0.057 0.046 -1.011 0 8.970 pollution after = 0.05*199.323 - 8.970= 0.997 0.05*199.323 -.97 = 0.997 x = (I-A g ) -1 d

532 531 Opportunity Cost of Pollution Regulation Before After Opportunity Cost X1197.146199.3232.177 X2128.245129.8011.556 X3321.918325.4143.496 X4277.253279.7662.513 Z0.000 8.9708.970 pollution before =.05*X 1 =0.05*197.146=9.025 pollution after = 0.05*199.323 - 8.97 = 0.996 cost in extra output industry 2.177+1.556+3.496+2.513 = 9.742 sometimes as a percent of GDP = 9.742/GDP

533 Chapter 20


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