# Chapter 2.

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Chapter 2

Shifts in Supply and Demand Influence Price

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

World Coal Use by Sector

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

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

Sum Up Where are coal reserves Conversions
E1 in unit 1, (u1) or E2 in unit 2 (u2) conversion is units of 1 per unit of (u1/u2) Energy Content

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

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/Q2 = - TC2/Q2 = - MC(Q)/ Q<0 MC(Q)/ Q>0 operate where price equals marginal variable cost short run supply equals marginal cost curve

Typical Competitive Firm Cost Short Run Supply
Psr AVC1 AVC2 D Q Q1 Q2 Q Q Q1 +Q2 Si = MCi above AVC Market is horizontal sum

Where They Cross Determines P & Q Supply = Demand
Pe monopsony, monopoly, both Model Building Blocks D Qe Q 11 11

Out of Equilibrium P S Price too high PH PL Price too low D Q Qs Qd
monopsony, monopoly, both Price too low D Q Qs Qd 12

Shift in D Change in Qs – movement along S
P S Pe' Pe monopsony, monopoly, both ^ Pe" D"(decrease)← D D' (increase)→ Q Qe" Qe Qe' 13

Shift in S Movements along the D curve
S"(decrease)← P S S' (increase)→ Pe" Pe monopsony, monopoly, both ^ Pe' D Qe' Qe" Qe Q 14

More than one Change Coal Mine Productivity Per Miner Increases
Q PQ Qe Qe' 15 15

1.Chinese Coal Mine Productivity  2. Plus Cheaper Sequestration
P ↓ Q ↑ P ↑ Q↑ S S P S' Pe' Pe monopsony, monopoly, both Pe Pe' D D →D' Q Q Qe Qe' Qe Qe' 16 16

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

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

Supply and Demand Building Blocks Dynamic -Two time periods
n time periods

Trade Models- Two Areas in World
P S1 S1+S2 D1+D2 D1 D D2 Qw Q1 Q2

Market Power Seller P S monopsony, monopoly, both D Q

Market Power Buyer P S monopsony, monopoly, both D Q

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 23

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 = Pl = price of labor = 40 Pnr = price of other natural resources used in production of coal = 10

Qd = Pcd + Psb - 2Pcm + 0.1Y Qs =30 + 1Pcs – 1Pk - 0.2Pl - 0.4Pnr Qd = Pcd *10+0.1*200 = Pcd Qs = 30 + Pcs – 1* * *10 = Pcs

Model02.xls: Worksheet S&D

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

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 = = 37 P 60 S 40 P = 37 20 D 50 100 Q = 26 Q

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

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

General Equilibrium Model (1) Think about but not to be tested
Markets for all products all factors or production consumers buy m final goods: a,b,c,…. at prices pb, pc, pd,…. their demand for final goods: db, dc, dd,…. consumers own and sell n factors of production: qt, qp, qk, …. at prices pt, pp, pk, … their supply/of n factors: st, sp, sk,… m + n unknown prices

General Equilibrium Model (2) Think about but not to be tested
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 stpt + sppp+ skpk + …. = da + dbpb+ dcpc + …. income = expenditure If holds for each household, holds for market

General Equilibrium Model (3) Think about but not to be tested
producers buy n factors of production demand: dt, dp, dk, producers produce m end use goods s demand: st, sp, sk, m commodities and n factors there are m+n unknowns quantities m + n - 1 unknown prices total: 2m + 2n - 1 unknowns

General Equilibrium Model (4) Think about but not to be tested
Consumer Demand for goods (m-1 independent) da = da(pt, pp, pk, …, pb, pc, pd,…) Supply of factors (n) st = st(pt, pp, pk, …, pb, pc, pd,…) Producers Demand for factors (n) dt = dt(pt, pp, pk, …, pb, pc, pd,…) Supply of goods m sa = sa(pt, pp, pk, …, pb, pc, pd,…)

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

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 P 60 S Ps = 51 P = 30 D Qs = 19 Qd = 40 100 Q

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

Demand Price Elasticity
Q responsiveness to price P P2 P1 Dlr D1 Qlr Q Q2 Q1 may change over time

Back to 1973 Oil Market S79 OPEC Supply Shocks 73&79 P S73 P79 Dlr D1
Q82 Q Q79

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%

Let’s Develop Formal Definition
d = % change quantity % change in price Qd *100 d = Qd  Pd *100 Pd Q2-Q1 = Q1 P2 - P1 P1

Suppose We Have Price Increase
P Q \$2.00/g  106 g/d \$3.00/g  106 g/d Qd d = Qd Pd Pd (400  106 g/d – 500  106 g/d) g/d (\$3.00 g – \$2.00 g) \$2.00 /g = -0.20/0.5 = (no units)

Lets Go Back to Lower Price
P Q \$2.00/g  106 g/d \$3.00/g  106 g/d Q2 – Q1 d = Q1 P2-P1 P1 (500 – 400) = (1/4) = (2– 3) (1/3) 3 = -(1/4)(3/1) = - 3/4 =

Sum Up Computing Arc Elasticities
d = % change quantity % change in price Qd d = Qd  Pd Pd Q2-Q1 = (Q1+ Q2)/2 P2 - P1 (P1+ P2)/2

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

More Convenient for Elasticity
Qs and Qd 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

Where do they come from? Estimate whole function market data
Qd = f(Pd, Y, Ps, Pc, . . ., etc. ) εp = Q P P Q Function Forms linear: Q = a – bP εp= -b(P/Q)

Linear Function P |Elastic| > 1 a/b D |Unit Elastic| = 1 =- =-1
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 a/2b |Inelastic| (1,0) =0 a Q a/2

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+εp) TR/P < 0 when (1+εp)<0 subtract -1 from both sides εp<-1 = Q(1+εp) (elastic) Raising price lowers revenue Lowering price raises revenue

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

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

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

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 inelastic -1 2

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 inelastic -1 2

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

Be Able to Compute for Other Functional Forms
ln(Q) = a – bP + cY Q = a – bln(P) + c lnY ln(Q) = a - blnP + clnP2 + dlnY Q = a + bP +cY +dPY

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

Sum Up Price Elasticity P = % change quantity % change in P
Three ways to compute Q2-Q1 p = Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 = Q P P Q = lnQ lnP

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

L9 - More on D&S Responsiveness Elasticities
Q

Direct Purchases

Direct Purchases vs Cradle to Grave

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 = Pk = 20 Pl = 40 Pnr = 10 Qd = Pcd *10+0.1*200 = Pcd Qs = 30 + Pcs – 1* * *10 = Pcs Invert, Shift, Solve for equilibrium, Price controls

Last Time Price Elasticity How responsive Qs or Qd is to price
flatter is more responsive P = % change quantity % change in P S1 P Slr Dlr D1 Q Q

Last Time Price Elasticity P = % change quantity % change in P
Three ways to compute Q2-Q1 p = Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 = Q P P Q = lnQ lnP

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

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

PQ and Q P P D Q

Compute Price Increase
Q = spread over 8 weeks = Q P = Q/Q = = 0.208 P εp P = 0.208*2.5 = 0.521 P new = P + P = \$ \$0.521 = \$3.021

Compute Price Increase
Q = Q P = Q/Q = = 1.25 P εp P = 1.25*1.70 = 2.125 P new = P + P = \$ \$2.125 = \$3.825

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

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

Last Time Defined demand price elasticities Arc Q2-Q1 (Q1+ Q2)/2
P2 - P1 (P1+ P2)/2 Point (∂Q/∂P)(P/Q) = ∂lnQ/∂lnP Relationship of Revenue, Price and Elasticity

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

Last Time Simple Forecasting 1. ΔQ=εpΔP Q P 2. ΔP = εp ΔQ P Q

Forecast Using Income Elasticity
a. If εy 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 Qchina? about 1/3 of total world = 0.8*0.09*3 = billion tonnes 76

Price Change to Offset Coal Growth
Let εy China = 0.8, Y/Y=0.09 εp = -0.5 What P/P do we need to choke off coal growth

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 + -

Cross Price Iooty, Queiroz, and Roppa (2007)
Cross price elasticity of ethanol with respect to gasoline substitute or complement? εpg=+0.6 QEth = 100 Pg = \$2 per gallon Increases to \$3 Q/Q = εPg (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?

Create Function from Elasticity Q = PY Can add more variables
P =\$1.15 Q = Y = 5.40  = -0.8  = 1.4 Q = PY = ( )  = Q/(PY) = 8/( ) = 0.84 Q = 0.84P-0.8Y1.4

Elasticities to create demand equations linear
(Q = a + bP + cY) around the following values. Price Elasticity εp= -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

Create Demand from Elasticities Q = a + bP + cY
εp= εy= 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)* *5.4 = 3.2 Q = Pd Yd Could add another variable X Need values εx and X P =\$1.15 Q = Y = 5.40

Another Way With Constant Elasticity of Demand
Example: Q1= P1 Q2 =P2 Q2/Q1 = (P2/P1) if P growing at 2 then P2/P1 = 1.02P1/P1= Q2/Q1 = (1.02) only need ratio prices or growth rates no units to forecast Q2 multiply Q1 by your forecast of Q2/Q1 only works exactly for constant elasticity functions DELETE THIS ONE 83

Forecasts with Elasticity-2
^ Q2 = (0.98Ps1)b2(1.01Ppl)b3(1.02Pal) b4(1.015Y1)b5 Q Ps1b Pplb Palb Y1b5 = 0.98b21.01b31.02b41.015b5   = = Q1 = 111.9 Q2 = *Q1 = *111.9=

Sum Up: Why are Demand Elasticities important?
Why are they important? Forecast P → Q ΔQ/Q = εp(ΔP/P) Q → P ΔP/P = (ΔQ/Q)/εp Y → Q ΔQ/Q = εy(ΔY/Y) Pother → Q ΔQ/Q = εo(ΔPo/Po) P,Y, etc.  Q ΔQ1/Q1+ ΔQ2/Q2 Policy analysis P to offset Y increase Effect of carbon tax Create demand from elasticities linear and log

Chapter 3

Elasticities so Far 1. Measure of responsiveness
2. Where do they come from? a. compute from market data P Q (Q2-Q1) = (Q1+ Q2)/2 P2 - P1 (P1+ P2)/2 problem if other variables change beside P

Elasticities so Far 2. Where do they come from?
b. Estimate whole function market data Qd = f(Pd, Y, Ps, Pc, . . ., etc. ) εp = LnQ = Q P LnP P Q Function Forms linear: Q = a – bP εp= -b(P/Q) multiplicative: Q = aP-b εp= -b mixed: ln(Q) = a – bP εp= -bP mixed: Q = a – bln(P) εp= -b/Q Other

policy: price increase to offset income growth
Demand Elasticities εx = 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Q QP YQ PcrossQ

Price Change to Offset Coal Growth
Let εy China = 0.8, Y/Y=0.09 εp = -0.5 What P/P do we need to choke off coal growth

YQ and Q Y? P Po D' D Q

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  Po  more inflation, tighter monetary policy r up, GDP down Po  income transfer to OPEC

Elasticity Approximation - Linear
Q = 20-4P P = 3 Q = 8 εp = -4*3/8 = -1.5 P2= 4 Use elasticity dQ/Q = εp*dP/P = -1.5*1/3 = -0.5 Qnew = Q(1+dQ/Q) = 4 With function Qnew = *4 = 8(1-0.5) = 4

Elasticity Approximation - Log
Q = 10P-1 P = 2 Q = 5 εp= -1 P2 = 2.1 Use elasticity: dQ/Q = εp*dP/P = -1*0.1/2 = Qnew = Q(1+dQ/Q)= 5*(1-0.05) = 4.75 With function Qnew= 10(2.1)-1 = 4.76 Approximation gets worse the larger the price change

Cross Price Iooty, Queiroz, and Roppa (2007)
Cross price elasticity of ethanol with respect to gasoline substitute or complement? εpg=+0.6 QEth = 100 Pg = \$2 per gallon Increases to \$3 Q/Q = εPg (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?

Create Function from Elasticity Q = PY Can add more variables
P =\$1.15 Q = Y = 5.40  = -0.8  = 1.4 Q = PY = ( )  = Q/(PY) = 8/( ) = 0.84 Q = 0.84P-0.8Y1.4

Elasticities to create demand equations linear
(Q = a + bP + cY) around the following values. Price Elasticity εp= -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

Create Demand from Elasticities Q = a + bP + cY
εp= εy= 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)* *5.4 = 3.2 Q = Pd Yd Could add another variable X Need values εx and X P =\$1.15 Q = Y = 5.40

Another Way With Constant Elasticity of Demand
Example: Q1= P1 Q2 =P2 Q2/Q1 = (P2/P1) if P growing at 2 then P2/P1 = 1.02P1/P1= Q2/Q1 = (1.02) only need ratio prices or growth rates no units to forecast Q2 multiply Q1 by your forecast of Q2/Q1 only works exactly for constant elasticity functions DELETE THIS ONE 99

Forecasts with Elasticity-2
^ Q2 = (0.98Ps1)b2(1.01Ppl)b3(1.02Pal) b4(1.015Y1)b5 Q Ps1b Pplb Palb Y1b5 = 0.98b21.01b31.02b41.015b5   = = Q1 = 111.9 Q2 = *Q1 = *111.9=

Another Way With Constant Elasticity of Demand
Example: Q1= P1 Q2 =P2 Q2/Q1 = (P2/P1) if P growing at 2 then P2/P1 = 1.02P1/P1= Q2/Q1 = (1.02) only need ratio prices or growth rates no units to forecast Q2 multiply Q1 by your forecast of Q2/Q1 only works exactly for constant elasticity functions DELETE THIS ONE 101

Forecasts with Elasticity
^ Q2 = (0.98Ps1)b2(1.01Ppl)b3(1.02Pal) b4(1.015Y1)b5 Q Ps1b Pplb Palb Y1b5 = 0.98b21.01b31.02b41.015b5   = = Q1 = 111.9 Q2 = *Q1 = *111.9=

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+t P Ps Pe Pd Pd-t Q Qe

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

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

Coal Ad Valorem Tax 50 % of Price
50% of Ps (1+0.5)Ps = Pd ←(1+t%)Ps P Ps monopsony, monopoly, both Pd' tax Pe Ps' Pd Qe' Q Qe 106 106 106

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

50% of Pd (1-0.5)Pd = Ps P Ps ←(1-t%)Pd monopsony, monopoly, both Pd' tax Pe Ps' Pd Qe' Q Qe 108 108 108

Quantitative Model Tax supplier
Ps+6 Qd = 30 -2Pd Qs = -3 + Ps Solve for equilibrium 30 -2P=-3+P P = 11, Q = 8 Add tax of 6 to supply price Invert demand and supply Pd = Qd Ps = 3 + Qs Pd= Qd = Ps+t = 3 + Qs + 6 Solve Q = 4, Pd=13, Ps = 7 P 15 Ps 13 11 7 Pd 3 4 8 Q 30

Government Revenues Supplier tax of 6
110 Government Revenues Supplier tax of 6 Q = 4 t = 6 t*Q = 6*4=24 P Ps+6 15 Ps 13 11 7 Pd 3 4 6 Q 30

Who Pays the Tax Depends on Shape of Demand and Supply
111 Who Pays the Tax Depends on Shape of Demand and Supply Ps+t Perfectly Elastic Supply P P Ps+t Ps Pd' Pd' Ps Pe Ps'= Pe Ps' Pd Pd Qe' Qe Qe' Qe Q Q share the tax consumer pays

Incidence of Tax Depends Shape of Supply and Demand (Practice Four Extreme Cases)
Ps+t Pd-t Ps Pd Q Q D Ps+t Ps+t Ps P P Ps Pd Q Q

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

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/Q2 = – TC(Q)2/Q2 < 0 TC(Q)2/Q2 > 0 MC slopes up - increasing marginal cost

Social Welfare - Producer Surplus
Ps Pe Pd Q Qe Price Set by Marginal Producer and Consumer Ricardian Rent

Social Welfare - Consumer Surplus
Ps=MC Pe Pd=Marginal Benefit Q Qe

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/Q2 = – TC(Q)2/Q2 < 0 TC(Q)2/Q2 > 0 MC slopes up - increasing marginal cost

Consumer Surplus = ½(15-11)*8 =16 Producer Surplus = (1/2)(11-3)*8= 32
Social Welfare 15 P Qd = 30 -2Pd Qs = -3+ Ps Invert demand Pd = Qd Ps = 3 + Qs Solve for Equilibrium P = 11, Q = 8 Ps 11 Pd 3 Notice this maximizes social welfare4 Q 30 8 Consumer Surplus = ½(15-11)*8 =16 Producer Surplus = (1/2)(11-3)*8= 32 118

Welfare Cost of a Tax Ps+6 DWL = (1/2)(13-7)*(8-4) P 15
Pd = Qd Ps = 3 + Qs t = 6 Invert demand Ps + t = Pd 3 + Qs + 6 = Qd 1.5Q = 6 Q = 4 Ps= 7 Pd= 13 Pd=13 Ps Pe=11 Ps=7 Pd 3 4 8 Q 30 Tax revenues = 6*4 = 24 CS=16 PS=32

Welfare Cost of a Subsidy
DWL = (1/2)(10.4-8)*( )= 15 P Pd = Qd Ps = 3 + Qs Qe=8, Pe=11 sb= 3.6 Ps - sb = Pd 3 + Qs = Qd 1.5Q = 15.6 Q = 10.4 Ps= 13.4 Pd= 9.8 Ps=13.4 Ps-3.6 Ps 11 Pd= 9.8 Pd 3 Bring CS’ and PS’ tomorrow 10.4 8 Q 30 CS=16 PS=32 Subsidy Cost = 3.6*10.4= CS'=? PS'=? 120

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

Sum Up - P 15 Pd=13 Ps Pe=11 Ps=7 Pd 3 4 8 Q 30

More losses the more elastic are demand and supply
DWL and Elasticity Ps+t Ps+t P P S Ps D Pd Q Q More losses the more elastic are demand and supply

U.S. Tariff on Brazilian Ethanol (2008)
U.S. Ethanol Market P Sus tariff revenues Pw + tariff Pw Dus Q1 Q2 Q3 Q4 Q

Brazil Ethanol 2008 Exports of Q4-Q3
SBrazil Pw DBrazil Q3 Q4 Q

MR = MC P MC MR = MC Pm P(Q) ATC ACm Q Qm MR
Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm – oQmMC dQm 2.o.c. Is slope MR< slope of MC?

MR = MC P MR = MC Pm P(Q) ACm ATC MC Q Qm MR
Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm – oQmMC dQm 2.o.c. Is slope MR< slope of MC?

Social Optimum P = MC P Pd ATCo ATC Pso MC Q Qso Losses Pd = MC Choices: regulate or government own P = MC collect losses some other way P = ATC

Chapter 4

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

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

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

Incidence of Tax – Depends on Elasticity
Ps+t Depends on elasticity d = -0.5 s = 1 t = 1.5 dPd = εs = dPs εd P Ps Pd Pe Ps Pd Q Q' Qe

Incidence of Tax – Depends on Elasticity
Ps+t dPd = εs = dPs εd (1) dPd = (1/-0.5)dPs = -2dPs dPd-dPs = t (2) dPd-dPs = 1.5 Two equations two unknowns P Ps Pd dPd>0 Pe dPs<0 Ps Pd Q Q' Qe

Solve Two Equations for dPs, dPd
Ps+t P Ps (1) dPd = -2dPs (2) dPd-dPs = 1.5 Substitute (1) into (2) -dPs-dPs = 1.5 -3dPs = 1.5 dPs = -0.5 dPd = -2Ps = -2(-0.5) = 1 Pd dPd=1 Pe dPs=-0.5 Ps Pd Q Q' Qe

Consumer Surplus = ½(15-11)*8 =16
Social Welfare 15 P Qd = 30 -2Pd Qs = -3 + Ps P = 11, Q = 8 Invert demand Pd = Qd Ps = 3 + Qs Ps 11 Pd 3 Q 30 8 Consumer Surplus = ½(15-11)*8 =16

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/Q2 = – TC(Q)2/Q2 < 0 TC(Q)2/Q2 > 0 MC slopes up- increasing marginal cost

S = MC in competitive market

S = MC in competitive market
Producer Surplus

Social Welfare 15 P Qd = 30 -2Pd Qs = -3 + Ps P = 12, Q = 6
Invert demand Pd = Qd Ps = 3 + Qs Ps 11 Pd 3 Q 30 8 Consumer Surplus = ½(15-11)*8 =16 Producer Surplus = (1/2)(11-3)*8 = 32

Social Welfare - Consumer Surplus
Ps=MC Pe Pd=Marginal Benefit Q Qe

Social Welfare - Producer Surplus
Ps Pe Pd Q Qe Price Set by Marginal Producer and Consumer Hotelling Rent

Welfare loss from an ad valorem tax
Ps(1+t%) Ps P Pd Q'QePd(Q)dQ - Q'QePs(Q)dQ Ps Pe Pd Q' Qe Q

Government Revenues Ps(1+t%) P Ps Pd Pe Ps Pd Q' Qe Q

Change in Consumer and Producer Surplus
Ps(1+t%) P Ps Pd Pe Ps Pd Q' Qe Q

Welfare loss from unit subsidy tax
P Ps Ps-sb Ps Q'QePs(Q)dQ - Q'QePd(Q)dQ Pe Pd Pd Qe Q' Q

What if You Export Your Product?
Ps(1+t%) P Ps gain Pd Pe loss Ps Pd Q' Qe Q

What if your Demand is Perfectly Elastic
Ps(1+t%) P Ps Pd'= Pe Pd Ps' Qe Q' Q

Tariff is a Tax on Imports
Small Consumer and Producer Crude Price determined on world markets S P Pw D Q Qs Qd

Tariff on Crude Imports
Add tariff t S P Pw+t Pw D Qs Qd Q Qs' Qd'

Tariff on Crude Imports
Add tariff t S P Pw+t Pw D Qs Qd Q Qs' Qd'

Welfare loss from unit subsidy tax
cost to government benefit to producer P Ps Ps-sb Ps' Pe Pd' benefit to consumer Pd DWL loss Qe Q' Q

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 P P2 P1 D Q

Look at real income with two goods
X1 P1X1 + P2X2=Y Example 2X1 + 4X2=100 Graph X1 = 100/2 – (4/2)X2 X1 = 50 – 2X2 Raise P1 to 4 X1 = 100/4 – (4/4)X2 X1 = 25 – X2 50 Budget 25 25 X2

Sum Up Tariff from Last Time
U.S. Ethanol Market P Sus Loss in consumer surplus from tariff Pw + tariff Pw Dus Q1 Q2 Q3 Q4 Q

Sum Up Tariff from Last Time
U.S. Ethanol Market P Sus Gain in domestic producer surplus from tariff Pw + tariff Pw Dus Q1 Q2 Q3 Q4 Q

Sum Up Tariff from Last Time
U.S. Ethanol Market P Sus tariff revenues social loss Pw + tariff Pw Dus Q1 Q2 Q3 Q4 Q

Electricity - Decreasing Cost Industry
P D ATC Q Natural Monopoly

MR = MC P MC Pm P(Q) ATC ACm Q MR Qm MR = MC
Monopoly profit = TR – TC = Pm*Qm – AC*Qm = Pm*Qm – oQmMC dQm 2.o.c. Is slope MR< slope of MC?

Example (small village) – Monopoly Solution
P is US cents per kWh Q is kWh per year demand is P = Q total cost curve in cents is TC = 19Q Q2 AC = TC/Q = 19 – 0.25Q MC = TC/Q = 19 –0.50Q MR = Q Q = 7.467 P = 75 – 4(7.467)=45.132 P Monopoly Profits 75 45.132 19 AC =MC=19 –0.50Q MC D 7.467 18.75 MR Q

Example (small village) – Monopoly Solution
P is US cents per kWh = Q is kWh per year = 7,467 TC = 19Q Q2  = PQ – TC =45.132* (7.467) + 0.25(7.467)2= units Units PQ = cents/kWh*kWh = cents TC must be measured in cents What if TC measured in \$ TC\$*100¢ \$ P Monopoly Profits 75 45.132 19 AC MC D 7.467 18.75 MR Q

Social Optimum – Maximize Welfare
D P1 MC Q Q1 welfare (W) = sum of consumer plus producer surplus CS = area below demand and above price PS =area above marginal cost and below price W = 0QPd(X)dX - PQ + PQ - 0QMC(X)dX

What is 0QMCdQ P MC ` Q ∫0QMCdQ=TVC

What is Social Loss with Natural Monopoly
Decreasing Average Cost = Natural Monopoly Monopoly MR = MC Optimum P = MC P Social Loss Pm P(Q) ATC Po MC Q Qo Qm MR market failure

Lets Examine the Optimum
Pd ATCo ATC Pso MC Q Qso Losses Pd = MC Choices: regulate or government own P = MC collect losses some other way P = ATC

Example  piqi < expenses + s(RB)
0.08*1,966, *799,999 < ( )*1,966,667 + ( )*799,999 +0.105*750,000 197, ? 122, ,750= 201,083.32 < 201, Rates would be approved

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 = ? 14.864 2.608 0.068 0.136

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

Compounding p times a year
A(1+r/p)tp A = 100, t = 50, r = 10% p = p = p = continuous compounding p goes to  = ertA =

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 = p = p = continuous compounding p goes to  B = ertA A = B/ert = Be-rt = 100e-rt A = 100e-0.10*50 = \$0.674

Value a Stream of Income
D1 dollars at the end of 1 year D2 at the end of 2 years NPV = D D2 (1+r) (1+r)2 Example D1 = 50, D2 = 51, r = 0.10 NPV = = \$89.256 (1.1) (1.1)2 Could have changing interest rates NPV = D D2 (1+r1) (1+r2)2 \$89.256

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 What is the NPV or DCF of this power plant? – 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

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

Internal r (IRR) Solve for r that Makes NPV 0
= 0 (1+r) (1+r)2 Alternatively rearrange 100 = (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+r2) = r +59 100r2 +140r - 19 =0

100r2 +140r - 19 =0 Quadratic formula ar2 +br + c =0 -b  (b2 - 4ac)0.5 = -140 ( *100*140)0.5 2a *100 = 0.125 = 1.525 Excel alternative - Put stream of income in A1 to A3 =irr(a1.a3,guess) = irr(a1.a3, 0.05) = 12.5% seems to always take + root

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

Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class
residential customers (L) CL = QL industrial (H) CH = QH if produce both CLH = QL + 10QH but CL + CH = QL + 10QH sub-additive How to allocate 1500? one group not subsidize another don’t drive anyone out of market > CQLQH

Marginal Cost Pricing for Low Voltage
PL = 80 – 2QL CL = QL CLH = QL + 10QH MCL = 20 PL = MCL 80 – 2QL = 20 80-20 = 2QL QL = 30 PL = 20 Consumer surplus 0.5(80-20)30 = 900 80 PL 20 MCL 30 40 QL Standalone Fixed 1200

You Do Marginal Cost Pricing for High Voltage
PH CLH = QL + 10QH PH = 100 – 3QH MCH= QH = PH = Consumer surplus Standalone fixed PH MCH QH

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? PH = 100 – 3QH CLH = QL + 10QH CH = QH Stand fixed cost = Consumer surplus = P PH MCH QL

Pricing Across Time - Peak load pricing
one simple case – quantity independent of price in other period peak shifting more complicated problem P ck+co Dpk ck Dopk Q

Peak load pricing CS peak Social optimum Ppk = ck + co Popk = co Qpk
co+ck Qopk co Qopk' Qpk' Q CS offpeak

No peak switching Qpk = Ppk Qopk = 8 - 2Popk ck = 3 co = 2 Ppk = ck + co Popk = co Ppk = 10 - (1/5)Qpk Popk = 4 - (1/2)Qopk

Solve for Qpk and Qopk Social optimum
Ppk = 10 - (1/5)Qpk = ck + co = = 5 10 - (1/5)Qpk = 5 Qpk = 25 Popk = 4 - (1/2)Qopk = co = 2 4 - (1/2)Qopk = 2 Qopk = 4 P Qpk Co+Ck=5 Ppk Qopk Popk Co= 2 4 25 Q

Often Charge One Price If Charge One Price: P=5
Social Loss Qpk P=5 co + ck=5 Qopk co = 2 Q Qopk' Qpk'

If charge one price: P = 2 Social Loss P Qpk co+ck=5 Qopk co=2 P=2
also not covering capital cost

If charge one price: P=2.5 See if you can figure out losses
Losses in Peak Losses in Off Peak P Qpk Co+Ck=5 Qopk P=2.5 Co= 2 Qopk Qpk Q Qopk' Qpk' Peak Load Price if Losses Greater than Metering Cost

Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class
residential customers (L) CL = QL industrial (H) CH = QH if produce both CLH = QL + 10QH but CL + CH = QL + 10QH sub-additive > CLH

Fully Distributed Cost (FDC) Lump Sum to Each Consumer Class
residential customers (L) CL = QL industrial (H) CH = QH if produce both CLH = QL + 10QH but CL + CH = QL + 10QH sub-additive How to allocate 1500? one group not subsidize another don’t drive anyone out of market > CQLQH

You Do Marginal Cost Pricing for High Voltage
PH CH = QH CLH = QL + 10QH PH = 100 – 3QH MCH= QH = PH = Consumer surplus Standalone fixed PH MCH QH

You Do Marginal Cost Pricing for High Voltage
PH CH = QH CLH = QL + 10QH PH = 100 – 3QH MCH= – 3QH=10 QH = PH = Consumer surplus Standalone fixed 100 PH 10 MCH QH 30 33.33 30 10 1000 0.5(100-10)*30 =1350

Pricing Across Time - Peak load pricing
one simple case – quantity independent of price in other period peak shifting more complicated problem P ck+co Dpk ck Dopk Q

Two Curves Shift - Gasoline Market
Oil Prices Up Income Up P ↑ Q↓ P ↑ Q↑ ←S' S P S Pe' Pe monopsony, monopoly, both Pe' Pe D D →D' Q Q Qe' Qe Qe Qe' 193

Incidence of Subsidy – on Supply
Ps Ps-sb Ps' Pe Pd' Pd Qe Q' Q

Incidence of Subsidy – on Demand
P Ps Ps' Pe Pd' Pd+sb Pd Qe Q' Q

Pmax P S Pmax not binding monopsony, monopoly, both Pmax D Q Qd Qs 196

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

Marginal Cost Pricing PL PQL = 100 – 2QL PQL = 70 – 4QL
PQL = 100 – 2QL = MCL = 20 = 2QL QL = 40 PQH = 70 – 4QH = MCL = 30 70-30 = 4QH QH = 10 Haven't allocated fixed costs of 1700 PQL 100 20 MCL 50 QL 40

Marginal Cost Pricing PL PQL = 100 – 2QL CQLQH = 1700 + 20 QL + 30QH
CQL = QL Consumer surplus 0.5(100-20)40 = 1600 PQL 100 20 MCL 40 50 QL

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

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 = Q TC = 19Q Q2 AC = TC/Q = 19 – 0.25Q TC/Q = MC = 19 –0.50Q MR = Q = MC = 19 –0.50Q => Q = 7.467 P = 75 – 4(7.467)=45.132  = PQ – TC=45.132*7.467* -19(7.467) (7.467)2 = 209.1

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

Social Optimum P D P1 MC Q Q1
welfare (W) = sum of consumer plus producer surplus CS = area below demand and above price PS =area above marginal cost and below price W = 0QPd(Q)dQ - PQ + PQ - 0QMCdQ

Maximize CS = 0QPd(Q)dQ - 0QPs(Q)dQ
maximizing the area between D and MC f.o.c. W =  0QPd(Q)dQ - 0QMCdQ = 0 Q Q Q = Pd(Q) – MC = 0 2.o.c. 2CS = Pd(Q) – MC < 0 Q Q Q Pd(Q)< MC Q Q Slope of inverse demand less than slope of MC

Last Time Quiz - Cost Curves Sunk costs are part of total costs
TC = FC+VC P MC = TC/ TC ATC = ATC/Q FC=FCsun + FCnosunk FCsunk Q

Chapter 5

Generating Costs D' D

Price Regulation Transportation and Distribution
1. Rate of Return (piqi < 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=( )/115 = 0.07 productivity change some measure of output/input (O/I) (O/I)07=0.21 and (O/I) 08=0.22

Price Regulation Transportation and Distribution
X = ( )/0.21 = RPI-X = = P07 = \$0.10 P08<( )*0.10 = \$0.102 popular in UK 3. Light Handed New Zealand 4. Yardstick Scandinavia

Wholesale Market Q1=19 One sided bidding hour ahead, day ahead get bids – put in order one-sided P P1 Q1 Q

Wholesale Market System Marginal Price = SMP
Two sided bidding again get bids – put in order P \$0.07 P1 Q1 Q

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 = \$ Power Pool Price = PPP PPP = SMP + CC = = \$0.0775

How to allocate power at capacity Role of price signals Gaming the system
Dpk Dopk Qopk Q

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 = = \$0.0775

Demand up Supply down S' P S Pe D' D Qd' Q Qs' Qe Imports

Chapter 6

Typical Competitive Firm Cost Short Run Supply
Psr AVC1 AVC2 D Q Q1 Q2 Q Q Q1 +Q2 Si = MCi above AVC Market is horizontal sum

Last Time Reviewed- Long Run Supply With Entry and Exit
srMCi = Ssr P Slr D1 D3 D D2 Q Increasing Cost Industry

Long Run Supply With Entry and Exit
srMCi = Ssr P Slr D1 D3 D D2 Q Increasing Cost Industry

Long Run Supply With Entry and Exit
srMCi = Ssr P Slr D1 D3 D D2 Q Constant Cost Industry

Long Run Supply With Entry and Exit
srMCi = Ssr P Slr D1 D3 D D2 Q Constant Cost Industry

Inelastic Supply and Demand
Q Q

Multiplant Monopoly Marginal Cost – 2 countries
TC1 = 10 + Q1 + (1/2)Q12 TC2 = Q2 + Q22 MC1 = TC1/ Q1 = 1 + Q1 MC2 = TC2/ Q2 = 2 + 2Q2

MC for Monopolist- Horizontal Sum
Firm 1 MC1= 1 + 1Q1 Firm 2 MC2 = 2 + 2Q2 MC2 MC MC1 MC= 1 + 1Q 0 < Q < 1 Q = Q1+Q2 Q >1 MC1+2 2 explain why they should be the same across firms 1 Q1 Q2 Given MC sum the Q

MC Above Kink Firm 1 MC1= 1 + 1Q1 Firm 2 MC2 = 2 + 2Q2 Q = Q1+Q2,
Given MC sum the Q Q1 = -1 + MC1 Q2 = -1 + (1/2)MC2 Q1 + Q2 = -2 + (3/2)MC Q = -2 + (3/2)MC MC = 4/3 + 2/3Q MC1+2 2 explain why they should be the same across firms 1 Q1

Now Add Demand What Should Monopolist Do?
P=75-0.5Q MR = MC MR= 75-Q=4/3+2/3*Q Q=44.2 P=75-0.5Q = *44.2 = 52.9 MC=MR = =30.8 Q1 = -1 + MC = = 29.8 Q2 = -1 + (1/2)MC =-1 + (1/5)30.8 = 14.4 P MC1 52.9= Pm MC1+2 MCm explain why they should be the same across firms 2 D 1 1 Qm Q MR =44.2

Sum Up Competitive Market Short Run Supply
Competitive Market P = MC above AVC ΣMCi P P P MC1 MC2 MC3 Q2 Q3 Q1 Q MCi=fi(Qi) Invert Qi = fi-1(MCi) Horizontal Sum Q1+Q2+Q3= f1-1(MC1)+ f2-1(MC2) + f3-1(MC3) Set Q = Q1+Q2+Q3 and MCi=MCj

Competitive Market Long Run Supply With Entry and Exit Increasing, Constant, Decreasing Cost Industry P Slri Slrc Slrd Q

2 Order Conditions MR – MC = 0 MR – MC <0 P Q Q MR = 75-Q
MC = 4/3+2/3*Q MR = -1 Q MC = 2/3 MR – MC = -1 – (2/3) < 0 P MC1 Pm MC1+2 MCm explain why they should be same across firms D 1 Qm Q MR

Individual Producer's Profits
1 = P*Q1 - TC1 = P*Q Q1 - (1/2)Q1 = 52.9* – (1/2)29.82=1751 2 = P*Q2 – TC2 = P*Q2 – Q2 – Q22 = 52.9* *14.4 – (14.4)2=824

Competitive Model (P=MC)
Q1 Q2 Ppc MC1+2 explain why they should be same across firms D Qpc Q Be able to solve for Ppc, Qpc, Q1, Q2, 1, 1

Market Failure from Monopoly
Social Losses? redistributed from consumers to monopolist P Pm MC Ppc Efficiency Distribution explain why they should be same across firms D Qm Qpc Q MR

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

Where to Get Demand Qd = f(Pd, Y, Ps, Pc, . . ., etc. )
Collect data on Qd, Pd - all variables that change Fit a function using statistical techniques Simplified Two Variable Illustration Qt = b1 + b2Pt + et (truth) R.V. = et ~ 0, s2 P et êt Q

Can compute demand if know Eo
MRP = Factor Demand PEEo = Po MRP = Po marginal revenue product must slope down Po Can compute demand if know Eo PEEo Po1 Po2 Q O1 O2

Abdel Reviewed Competitive Short Run Supply
MC2 MC1 Psr AVC1 AVC2 D Q Q1 Q2 Q Q Q1 +Q2 Si = MCi above AVC Market is horizontal sum

Competitive Long Run Supply With Entry and Exit
srMCi = Ssr P Slr Slr Slr D D' Q

Market Failure from Monopoly
Social Losses? redistributed from consumers to monopolist P Pm MC Ppc Efficiency Distribution explain why httey should be same across firms D Qm Qpc Q MR

Price lower the farther from the market
Supplier Oil Price and Transport Cost Demand and Supplier Separated by Transport Cost tr S+ tr2 S+ tr1 P S Price lower the farther from the market Ps1 Ps2 D Q Q2 Q1

Location - Supplier Price and Arbitrage
>\$69? <\$69? \$1 D \$70 S1 \$1 \$69

Supplier Price and Arbitrage
>\$68? <\$68? \$2 D \$70 S1 \$1 \$69 Prices can only differ by transport and transaction cost

Income Redistribution If Producer Exports all of Product
Income distribution before tax Consumer Surplus P MC Pm info on government funding from oil Producer Surplus D Qm Q MR

Sum Up: Horizontal Sum MC Competitive Supply, MC for multi-plant Monopoly
explain why httey should be same across firms

Sum Up: Factor Demand = MRP = PQMPE Horizontal sum from individual to Market
explain why httey should be same across firms D Q

Sum Up: Add Demand to MC What Should Monopolist Do?
Put together Demand and MC MR = MC MC1 Pm MC1+2 MCm explain why they should be the same across firms 2 D 1 1 Qm Q MR

Next Horizontal Difference: Dominant Firm's Demand
World Demand Supply of fringe Qo = Qw – Qs call on OPEC horizontal difference Qs P call on OPEC Qw Q MRL MRU

What should OPEC Do? Add MC – Case 1 Qs MCf  Pf MCo P Po Fringe?
2 places call on OPEC Qw Q Qo MR

What should OPEC Do? Add MC – Case 2 MCf P MCo Po Qw Q Qo MR
call on OPEC Qw Q Qo MR

What should OPEC Do? Add MC – Case 3 MCf P MCo Po Qw Q Qo MR
call on OPEC Qw Q Qo MR

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

Numerical Example - 2 Country OPEC
Costs OPEC MC1 = Q1 MC2 = Q2 World Demand Qw = P Supply fringe Qf = P Qs P MC MRU Qw Q MRL

Numerical Example - 2 Country OPEC OPEC MC
Marginal costs MC1 = 2 + Q1 MC2 = Q2 Horizontal sum Q Invert (let MC1 = MC2 = MC) Q1 = -2 + MC Q2 = -1 +(1/2)MC add Q's, Q1+Q2 = Q = -3 + (3/2)MC invert back => MC = 2 + (2/3)Q P MC 2 Q

OPEC Demand - Find Kink MC fringe or supply fringe is MCf = P = 10 + Q
=> Qf = P Inverse Demand World P = Qw => Qw = P Kink P Qf = 0 = P => P = 10 Kink Q World demand = Q = (10) = 25 Qs P 10 Qw 25 Q

OPEC Demand Above kink P > 10 and Q < 25 Qw - Qf
Qo = P - (-10 + P) = P Below Kink P<10, Q > 25 Qo = Qw = P Qs P 10 Qw 25 Q

Marginal Revenue Above kink P > 10 or Q < 25: Qw - Qf
Qo = P Invert P = 40/1.5 -2/3Q MR = 40/ /3Q Below Kink Qo = P P = 60 -2Q MR = Q Qs P Qw Q

Solution - 3 choices Try above the kink MR = MC MR = 40/1.5 - 4/3Q
MC = 2 + (2/3)Q 40/ /3Q = 2 + (2/3)Q 74/3 = (6/3)Q 6Q = 74 Q = less than 25 P = 40/1.5-2/3(12.333) = Qs P MC 10 MRU Qw 25 Q MRL

Income Distribution Affect in Monopoly Market
Consumer Surplus P MC Pm info on government funding from oil Producer Surplus D Qm Q MR

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

Income Distribution in Monopoly Market Assume Producer Exports all of Product
Income distribution before tax Consumer Surplus P MC Pm info on government funding from oil Producer Surplus D Qm Q MR

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

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

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

Transfer to Producer Government
Tax in Monopoly Market Effect on Producers Producer Losses MC+t P Transfer to Producer Government Pm' MC Pm Qm' Qm Q MR

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

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 P 50 MC 36 = Pm info on government funding from oil 22 1 D 7= Qm Q MR

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 50 MC 36 = Pm info on government funding from oil 22 1 D 7= Qm Q MR

Tax in Monopoly Market Producer Tax of 7
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 MC+t 50 38 MC 36 38-7 19 1 6 Q MR

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

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 = 45.5 Tax from Producer (7-2)*6 = 30 DWL Producer = 15.5 P MC+t 50 MC 38 36 38-7 19 1 6 Q MR

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

Total Global Welfare Effects
Before CS = Now CS = 36 PS = PS + TR = Total= = 168 P MC Pm Total Losses = 16.5 D Qt Qm Q MR

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

Consumer Tax in Monopoly Market
Consumer Government Adds a Tax P MC Pt P P-T Qt MR Q MRt

What should OPEC Do? Add MC – Case 1 Qf MCf  Pf MCo P Po Fringe?
2 places Qf Qd-Qf call on OPEC Qw Qf Qo Q MR

What should OPEC Do? Add MC – Case 2 MCf P MCo Po Qw Q Qo MR
call on OPEC Qw Q Qo MR

What should OPEC Do? Add MC – Case 3 MCf P MCo Po Qw Q Qo MR
call on OPEC Qw Q Qo MR

Quiz - left of Kind Qf N.B. 1. read price off of OPEC demand not world demand 2. Price is the same for OPEC and the fringe MCo P Po call on OPEC Qw Qf Qo Q MR

Quiz OPEC at Kink Qf = 0 MCf P MCo Po call on OPEC Qw Q Qo MR

Quiz - OPEC to right of kink
Qf = 0 MCf P MCo Po call on OPEC Qw Q Qo MR

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

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)Qo 92 - (4/3)Qo= 2 + (2/3)Qo 90 = (6/3)Qo Qo = 45 MC Qs P 10 Qw 45 MR Q 123

What Else do We Know About the Market
Qo = 45 P = 92 - (2/3)Q = 92 - (2/3)45 = 62 Qf = P = = 52 MC Qs P 62 10 Qw 45 123 MR Q 52

OPEC Quotas MC1 MC2 MC MC = 2 + (2/3)Q = 2 + (2/3)45 = 32 OPEC Quotas
62 32 Qw 30 MR Q 15 45

Industry Profits OPEC π= P*Q - TC = P*Q - ATC*Q = P*Q - 0QMC(Q)dQ
Solve = 62*45 - 045(2 + (2/3)Q)dQ = (2Q + (1/3)Q2)|045 = [2*45 + (1/3) *0 + (1/3)02] = 2025 Individual OPEC countries or the fringe = P*Qi - 0QiMCi(Qi)dQ

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

Chapter 7

Price control versus Quantity control
D3 P P D3 D1 D1 D2 D2 Pc set P – Q more volatile, set Q, p more volatile Qcontract shortage Qcapacity Q Q

Opportunism - quasi rent
MC P ATC P1 AVC P2 quasi rents Q quasi-rent

Chapter 8

Policy - Negative Externalities on Supply?
Ssoc Spv P Psoc Ppv Dpv Qsoc Qpv Q

Drop subscripts and solve
Numerical example Qd = 90 - Pd Qs = 2Ps Externality X = 9 Solve for equilibrium Qd = 90 - Pd = Qs = 2Ps Drop subscripts and solve P = 30 Q = = 2*30 = 60

Numerical example Externality = 9 Invert Qd and Qs Pd = 90 - Qd Ps = Qs/2 Add negative externality to Ps Pd = Ps + X drop subscripts 90 - Q = Q/2 + 9 Q = 54, P = = 36 Ps = Qs/2 30 Pd = 90 - Qd 60

Numerical example- Social Costs
Pd = Ps + X = = 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 Ps = Qs/2 9 30 Pd = 90 - Qd 54 60

Social Loss - positive externality on supply
Spv Ssoc P Dpv Q

MB = MC MB,MC MC MB Xo X

Model Two Pollution - Optimal Level
\$ Benefits Costs MB of Pollution MC of Pollution G E A B C D X Pollution

Polluter has Property Rights? What are Social Losses?
\$ MB,MC MC of Pollution MB of Pollution F G E A B C D X Pollution

One Who Suffers has Property Rights? You Show Social Losses?
\$ MB,MC MB of Pollution MC of Pollution G F E A B C D X Pollution

Coase’s Law No Transaction Costs Suppose Dow has property rights
\$ MB,MC Dow MB of Pollution Exxon MC of Pollution F G Most benefit E A B C D X Pollution

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

Command and Control You can only emit C
\$ Benefits Costs MB of Pollution MC of Pollution at C no social loss F G E Q Pollution A B D C Pollution at C Polluter clean up cost

What Happens if Pollution Tax = T1
\$ Benefits Costs MB of Pollution MC of Pollution F G E T1 A B D C Q Pollution pollution

What Happens if Pollution Tax = T2
\$ Benefits Costs MB of Pollution MC of Pollution F T2 G E A B D C Q Pollution

Optimal Pollution Tax \$ Benefits Costs MB of Pollution MC of Pollution
taxes F G E T3 may be lots of money - corruption A B D C Q Pollution pollution

Polluter Had Property Rights Redistribution Affects With Tax
\$ Benefits Costs MB of Pollution MC of Pollution society gains back F G E How to use tax tax abatement A B D C Q Pollution polluter losses from tax

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

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

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

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

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

Which Policy Does Polluter Prefer

Model 3 Optimal level of Abatement CD Optimal Level – Pollution AC
\$ Benefits Costs MB of Pollution MC of Pollution F G E A B C D Q Pollution

Model 3 Abatement over to firms of CD
MC P2 Price of permits MC1 MC2 make new oh for P2 and P3 P3 A1’ A2’ A2 A1 needed abatement CD What happens at P2? P3?

Chapter 9

Public Good Quantitative Separate Players
MC MB MC = 6 MB1 = A1 MB2 = A2 MC= MB1 6 = A1 3A1 = = 24 A1 = 24/3 = 8 MC = MB2 6 = A2 2A2 = 14 A2 = 7 MB1=30-3A1 MC=6 A1 A1o =8 MC MB MB2= A2 MC=6 A2o =7 A2

Public Good Quantitative Gaming the System
MC MB MB1=30-3A1 Non-excludeable A1 wants A2 to produce? 7 A1 will produce 1 A2 wants A1 to produce? 8 A2 will produce 0 Each will want to free ride MC=6 A1 A1o =8 MC MB MB1+MB2 MB2 MC A2o=7 Aso A2

Public Good Quantitative Social Optimum
MC MB MB1=30-3A1 Since non-rivalrous benefits MB1 + MB2 MC = 6 MB1 = A1 MB2 = A2 MB = A MB = MC 50-5A = 6 5A = 44 A = 44/5 = 8.8 MC=6 A1 A1o =8 MC MB MB1+MB2 MB2 MC A2o=7 Aso =8.8 A2

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 = 5000 V = 5,000,000

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

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 (oi) = kilowatts per bulb X costs per kilowatt hour = (0.075)*0.10 = \$ per hour operating costs of/hour for fluorescent = (0.020)*0.10 = \$0.0020/hr

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) (1+r/12)n*12 Then K = \$X Si=1n*12 (1/(1+r/12) i) Solving for \$ = (K/X)/Si=1n*12 (1/(1+r/12)i)

Levelized Cost Fluorescent & Incandescent
Package of incandescents costs K = \$1.40 n=1 year, X = 100 hours per month \$i = Si=1n*12 (1/(1+r/12) i ) = (1.40/100)/ = \$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)/Σi=112*7(1/(1+0.1/12)i) = (12/100)/ = \$ compact fluorescent operating costs lower than capital costs

Total unit Cost Fluorescent & Incandescent
Adding capital and operating costs total incandescent costs \$i + oi = \$ \$ = \$0.0087 total compact fluorescent costs \$f + of = \$ \$ = \$0.0041 Total Formula unit cost = kilowatts*Pe + \$i = (K/X)/Si=1n*12 (1/(1+r/12)i)

Market Power Seller power MC P MC P Competition D D Q Q Competition
Monoply MR = MC Buyer Power

Graph the Decision Process MRP = P
PL & MRPL PL-1 PL-2 D = MRPL L1 L2 L

Marginal Factor Cost from Supply Market Power of Buyer
 = PEE(L) – PL(L)*L L = PEEL – (PL + dPLL)= 0 dL MRP - MFC =0 Example: L = PL supply P = L

Numerical Example of Marginal Factor Cost to Monopsonist
TC = PL L PL = L = ( L)L = 5L + 0.5L2 MFC= TCL= 5 + 2*0.5L = 5 + L

Chapter 10

Sum Up Factor Demand

Monopsony Outcome

Bilateral Monopoly Assume Both Want the Same Quantity
Reservation Prices Negotiation

Bilateral Monopoly Assume Both Want the Same Quantity

Reservation Prices Reservation Prices Negotiation

Graph the Decision Process MRP = P
PL & MRPL PL-1 PL-2 D = MRPL L1 L2 L

MRP = D (Marginal Benefit)
Need Buyer Marginal Cost MFCL PL & MRPL SL =Seller Marginal production cost PL-ms D = MRPL Lms L

Numerical Example – Monopsony Market
Sell Electricity PE = \$10 per megawatt Produce electricity from LNG (let Lng = L) E = 8L – 2L2 Buy LNG supply L = PL  PL = L Maximize profits PE*E - PLL  = 10(8L – 2L2) – (20+2L)L L = 80 – 10*4L - 20 – 4L = 0 80 – 10*4L = L MRP = MFC MFCL = 20+4L PL & MRPL PL = 20+2L PL-ms D = MRPL =80 – 10*4L Lms LNG L

Numerical Example – Monopsony Market
MRP = MFC 80 – 10*4L = L 44L = 60 L = 60/44 = 1.36 PL= L = *1.36 = 22.72 E = 8L – 2L2 = 8*1.36 – 2*1.362 = 8.156 MFCL = 20+4L PL & MRPL PL = 20+2L PL-ms =22.72 D = MRPL =80 – 10*4L Lms= = 8.156 LNG L

Chapter 11

Duopoly theory – Cournot model Two Players
Choose quantity to maximize profits given the other firms output Inverse demand function demand P = (q1 + q2) C1 = 5q1, C2 = 0.5q22 Profit functions 1 = ( (q1 + q2))q1- 5q1 2 = ( (q1 + q2))q q22

Duopoly theory – Cournot model First order conditions
Firm 1 1/q1=( (q1 + q2)) - 0.5q1 - 5 = 0 rearranged to = 95 - q q2 = 0 reaction function q1 = q2 Firm 2 1/q2 = ( (q1 + q2)) - 0.5q2 - q2 = 100- q q1 - 2q2 q2 = 100/2 - (0.5/2)q1 = q1

axes wrong switch q1 and q2

Equilibrium Solve Where Reaction Functions Cross
second equation into the first. q1 = ( q1) = 95 – q1 = q1 q q1 = 70 q1( ) = q1*0.875 = 70 q1 = 70/(0.875) = 80 Then q2 = *80 = 30 Price from P= ( ) = 45

What If Out of Equilibrium
q1 = q2 q2 = q1

Profits 1 = Pq1 - C1 = 45*(80) - 5*80 = 3200

P = 100 - 0.5(q1 + q2)C1 = 5q1, C2 = 0.5q22 MC1 = 5 MC2 = q2
Competitive Model P = MC P = (q1 + q2)C1 = 5q1, C2 = 0.5q22 MC1 = 5 MC2 = q2

Competitive Model P = MC
P = 5 = (q1+q2) q1+ q2 = 190 MC2 = P = 5 = q2 q2 = 5 q1 = = 185 1 = 185*5 - 5(185) = 0 no Ricardian rents normal rate of return 2 = 5* *(52)= 12.5 Ricardian rents

What if equal n-opolist
P = a –bnqi TC = c + dqi If act as competitors P = MC a –bnqi = d => qi = (a-d) P = a –bn(a-d) = d bn bn If act as duopolist i = (a-b[(n-1)qj+qi])qi – c – dqi = 0 i = aqi-b(n-1)qjqi+qi2 – c – dqi = 0

What if equal n-opolist
i/qi = a – b(n-1)qj+2bqi – d = 0 a – b(n+1)qi – d = 0 qi = (a-d) 2b(n+1) P= (a-b[a-d/2b(N+1)_= 0

What if sold gas on a monopoly market?
MR= MC 100 - (q1+q2) = 5 q1+ q2 = 95 P = (95) = 52.5

How much does each player produce?
MC2 = 5 = q2 q2 = 5 q1 = 95-q2 = 95-5 = 90 1 = 90* (52.5) = 2 = 5* *(52)= 250 Monopoly rents

Perfectly Price Discriminating Monopolist

Stackleberg solution q1
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 = ( (q1 + q2))q1 -5q1 but knows that firm 2’s reaction function is q2 = q1 1 = ( (q1 + ( q1))q1 - 5q1 1 = 100q q q q12 - 5q1 1/q1= 100 – q q1 – 5 = 0 0.75q1 = 70 => q1 = 70/0.75 = 93 1/3

Stackleberg solution q2
q2’s reaction function is the same as before q2 = q1 = *93.33 = 26.67 Stackleberg Cournot PC Monopoly q q P = profit 1= profit 2=

What If Both Try to be Leader
Firm 1 produces q1 = 93 1/3 expecting q2 = 26.67 Firm 2 maximizes 2 = ( (( q2) + q2))q q22 = 52.5q q22 2/q2 = q2 = 0 q2 = 35 expecting q1 = (35) = 77.5 P = 100 – 0.5*( ) = 15 1 = 933 2 = -88 Not a stable equilibrium

Bilateral Monopoly Model
Quantity agreed upon – Xc = 1 c1 reservation price of seller b reservation price of buyer Price between c1 and b

Add a Second Supplier with a Reservation (c2)
c1 <c2 < b possible rents, b-c1 divided between all players 1. 1 + 2 +3 = b - c1

Possible Rents at P If 1 sells 1 = p-c1 = rent supplier 1 If 2 sells
3 = b-p = rent buyer Find core no coalition can block

Core no coalition can block
1. 1 + 2 +3 = b - c1 Core = 5. i > 0 6. 1 + 2 > 0 7. 1 + 3 > b - c1 8. 2 + 3 > b - c2

Core no coalition can block
1. 1 + 2 +3 = b - c1 Core = 2. i > 1 + 2 > 0 4. 1 + 3 > b - c1 5. 2 + 3 > b - c2 1 & 3 => 2 = 0 (insight #1) 5 & 6 => 3 > b - c2 Substituting 6 into 1 8.  3 = b - c1 Rearranging 8 9. 1 = b - c1 - 3 Using 2 and 7 < 1 < b - c1 - (b - c2) < 1 < c2 - c1 (insight #2)

Case 2: c1 < c2 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

Limit Pricing Model

Chapter 12

first order conditions (foc)
International Energy Workshop collected forecasts

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

Demand Now

Demand Now and Next Year

Discount Next Year

Mathematical Solution Basic Model
Model Po = P1/(1+r) R = Qo + Q1 r = 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 – = 21.82 Po = 120-2*28.18 = P1 = 120-2*21.82 =

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

Basic Model – Increase Income Green for More Money

Increasing Income Period

Discount P1

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

Raise Interest Rate Green for More Interest

Discount Future More

Mathematical Solution Raise Interest
Model Po = P1/(1+r) R = Qo + Q1 r = 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 – = 23.33 Po = 120-2*26.67 = 66.67 P1 = *23.33 = 73.33

Model 4: Raise Reserves Green for More Reserves

Discount Next Period

Mathematical Solution Raise Interest
Model Po = P1/(1+r) R = Qo + Q1 r = 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 – = 35.45 Po = 120-2* = 40.91 P1 = 120-2*35.45 = 49.09

Add Constant Costs to the Model = 20

Red = Marginal Cost 20

Add Demand Next Period and Discount for Basic Model

P1 – MC!

Discount P1 - MC

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

Costs a Function of Current Production MCi= a + bQi
b > 0 = increasing cost industry b = 0 constant cost industry b< = decreasing cost industry MCo= a + bQo MC1= a + bQ1 Purple = Cost

Costs Increase with Production MCo

Po - MCo

MC1

PQ and MC1

P1 - MC1

Put Two Sides Together

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

Quantitative NPV Consumer Surplus =  Po dQo +  P1 dQ1 - NPV 
= 0.5*( )*28.57 + 0.5*( )*21.43/(1.1) =

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

MCo = 20 + Qo

Po-MCo Qo – ( 20 + Qo)

Next Period MC1 = 20 + Qo

Income increases

Change Interest Rate

Non-Scarce resources

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

Social Welfare

5. Model 3 + three cases for MC
a. MC = constant = 20 b. MC = function of current production MCo = Qo MC1 = Q1 technical progress in 2 that lowers costs c. MC = function of cumulative production MC1 = Qo 5a. P0 -MCo= P1 - MC1 (1+r)

Model with Costs

Po -MCo= P1 - MC1 (1+r) 200-2Qo -20= Q1 - 20 (1+0.05) substitute in the constraint 200-2Qo -20= (50-Qo) - 20 Qo/Q1 = 25.12/24.88 Po/P1 = /156.24 NPV Net Rev = NPV Cons Surp =

Compare to case 3 Qo/Q1= 25.37/24.63 Po/P1= 149.26 /156.74
Reduce current consumption higher costs delays consumption 5b. MC = function of current production MCo = Qo MC1 = Q1 technical progress in 2 that lowers costs

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

Back Stop Case 1: Po = Qo no Y grow r = 10% R = 50 MC = 0

Backstop @ 125 P1 = 125 Po = 125/(1.1)= 113.64 Q0 = 43.18
Q0 resource 43.18 Q0 bkstop 0.00 Q Q1 resource 6.82 Q1 bkstop 30.68 Resource price will gradually approach backstop price.

Backstop Analysis

Shortage Case P = 200 - 2Q MR = 200 - 4Q 200-4Qo = 100 - 4Q1 (1+.1)
200-4Qo = (50-Qo) Qo 26.19 Q P P Compared to PC Qo 28.57 Q P P Market imperfections 1. r social < r private 2. externalities 3. taxes 4. monopoly Static PC P = MC Dynamic PC Po-MCo = P1-MC1 (1+r) Static Monopoly MR = MC Dynamic Monopoly MRo-MCo = MR1-MC1 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 = Pt tdt (1.015)t St = Pt (1.02)CS/ St-1 St = supply of the competitive fringe CS = cumulative production competitive fringe CSt = CSt-1 + St Dt = Tdt - St Opec Demand Rt = Rt-1 - Dt N max W =  ((1+)t)-1(Pt - 250/R) * D t=1 indices given data decision variables constraints objective function sets parameters, tables, scalars variables equations model Solve (LP, NLP, MIP)

Shortage Other cases - n periods Po = P P P3 = Pn (1+r) (1+r) (1+r) (1+r) s.t. Q0 + Q1 + Q Qn = R Example: No income growth Case 1: P = Q Q = P Maximum price = 200 Q Reserves left Pn Pn-1= Pn/(1+r) Pn-2=Pn-1/(1+r) Pn Pn Pn Pn Pn Pn Pn Pn Pn Pn Pn Pn Pn Pn Pn Pn 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(Qo)Qo + P(Q1)Q1 (1+r) s.t. R = Qo + Q1  = P(Qo)Qo + P(Q1)Q1 + (R - Qo - Q1) Qo = P(Qo)+ dPo Qo -  = MRo - = 0 dQo Q1 = P(Q1) + dP1Q  = MR  = 0. (1+r) dQ1(1+r) (1+r) MRo = MR1

Monopoly

Price Control P = Pmax = P1 = \$112.00 Po = \$101.82 Q0 = 49.09
Q0 resource = 49.09 Q0 backstop = 0.00 Q1 = at \$101.82 but Q1 resource = 0.91 Then jumps to the backstop price P1 = \$150

Compare to competitive case

Calculus of Variation Chiang pick a time path that optimizes a function 0T 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

CV – Objective Function
=PodQo-MCodQo+P1dQ1 - MC1dQ1 + (R-Qo -Q1) (1+r) (1+r)  = R-Qo -Q1=0 Qo = P0 - MCo -  = 0 Q1 = P1 - MC1 -  = 0 (1+r) P0 -MCo=  P1 - MC1 = 

Backstop

Backstop Quantitative
Q = 60 – 0.5P Res = 60 P = 120 – 2Q r = 0.2 Backstop = \$42 P1 = 42 Po = 42/1.2 = 35 Qo = 60 – 0.5*35 = 42.50 Q1 = 60 – 0.5*42 = 39.00 Q1 + Qo = 81.50 Qo + Q1 – Res = backstop consumption = = 21.50

Chapter 13

Above Ground Costs - continuous
Suppose Ro = 100 and  = 0.10 (decline rate of ): Qo = 0.10*100 = 10 = aRo Q1 = e.10*t aRo = e-0.10*1 10 = Q2 = e.10*t aRo = e-0.10*2 10 = . . . Q20 = e.10*t aRo = e-0.10*2 10 = Q100 = e.10*t aRo = e-.10*2 10 =

Above Ground Costs - continuous
(decline rate of ): Qo = aRo K = \$Qte-rt dt = \$aRoe-t e-rt dt \$ = (K/(Roa))/(oe(--r)tdt) = denominator = [(e(--r)t/(--r)]|o = [(e(--r)/(--r) - (e(--r)0/(--r)] = [(0)(-1)/(--r) = 1/(+r) Solving \$ = (K/Ro)(a+r)/a = (K/Qo)(a+r) K/Qo is referred to as capacity cost

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

Nuclear Policy

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

Chapter 14

Whole Blending Problem
max  = \$0.08*X1 + \$0.09*X2 s.t. 0.4X X2 < 100,000 straight run 0.8X X2 < 140,000 cracked Graph in X1 X2 space constraint 1 constraint 2 X1 = 0 => X2 = 175,000 X1 = 0, X2 = 245,000 X2 = 0 => X1 = 250,000 X2 = 0, X1 = 175,000

Graph Constraints constraint 1 constraint 2
X1 = 0 => X2 = 175,000 X1 = 0, X2 = 245,000 X2 = 0 => X1 = 250,000 X2 = 0, X1 = 175,000

Objective Function = 0.08X1 + 0.09X2 X2 = /0.09 - (0.08/0.09)X1
Find highest line on constraint -slope dX2/dX1 =

For this Shaped Constraint Set Always on Corners
Check profits A, B, C (A) = 0.08X1+.09X2 = 0.08*(0) (175,000) = 15750 (C) = 0.08X1+.09X2 = 0.08*(175,000) (0) = 14000 (B) need to find what X1 and X2 are.

Solve simultaneously (1) 0.4X1 + 0.57143X2 = 100,000
Solve 1 for X1 (3) X1 = (100, X2)/0.4 Plug (3) into (2) (4) (100, X2)/ X2 = 140,000 (5) X2=105,000 (6) X1 = (100, *105000)/0.4 = 100,000

Solve simultaneously X2 = 105,000 X1 = 100,000 = 0.08X1 + 0.09X2
(B) = 0.08(100000) +0.09(105000) = 17,450 (A) = 0.08*(0) (175,000) = 15750 (C) = 0.08*(175,000) (0) = 14000

How to Blend X2 = 105,000 X1 = 100,000 straight run
0.40(100,000) (105,000) = 100,000 cracked 0.70(100,000) (105,000) = 140,000 u1 = 40,000 to grade 1 = 60,000 to grade 2 u2 = 80,000 to grade 1 = 60,000 to grade 2

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

Transport Costs

Math Formulation of Problem
Objective: Minimize TTC = * = Cij*Aij. supply shipments jAij < Yi for all i refinery satisfy crude oil needs iAij = Xj for all j Set up in Excel Solver

Simple Example 444 max  = \$0.08*X1 + \$0.09*X2
s.t. 0.4X X2 < 100,000 straight run 0.8X X2 < 140,000 cracked

Blending Model Profit Function
two processes grade 1 X1 = 2.5 min (u1 , u2/2) grade 2 X2 = 1.75 (u1,u2) u1 = straight run (100,000) u2 = cracked gasoline (140,000) 1 = \$0.08/gal X1 2 = \$0.09 / gal X2  = \$0.08*X1 + \$0.09*X2

What are technical constraints – u1
2.5 gallon of grade X1 requires 1 gallon of u1 gallon of u1 per gallon of X1 u1/X1 = 1/2.5 = 0.4 1.75 gallon of grade X2 gallon of u1 per gallon of X2 u1/X2 = 1/1.75 = 0.57 0.4X X2 < 100,000 u1 constraint

Total requirements of u2 for X1 and X2
2.5 gallon of grade X1 requires 2 gallon of u2 gallon of u2 per gallon of X1 u2/X1 = 2/2.5 = 0.8 1.75 gal of grade 2 requires 1 gallon of u2 gallon of u2 per gallon of X2 u2/X2 = 1/1.75 = 0.57 0.8X X2 < 140,000 u2 constraint

Whole problem max  = \$0.08*X1 + \$0.09*X2
s.t. 0.4X X2 < 100,000 straight run 0.8X X2 < 140,000 cracked Graph in X1 X2 space Constraint Constraint 2 X1 = 0 => X2 = 175,000 X1 = 0, X2 = 245,000 X2 = 0 => X1 = 250,000 X2 = 0, X1 = 175,000

Constraint Set = 0.08X X2 X2 = / (0.08/0.09)X1

Finding highest line that touches constraint set with slope dX2/dX1 = -0.8888
Will be one of points A, B, C Check the profit at each point (A) = 0.08X1+.09X2 = 0.08*(0) (175,000) = 15,750 (C) = 0.08X1+.09X2 = 0.08*(175,000) (0) = 14,000 (B) need to find what X1 and X2 are

Solve simultaneously 0.4X1 + 0.57143X2 = 100,000
If know matrix algebra X1 = 100,000 X ,000 Invert and multiply , = 100,000 , ,000 (C) = 0.08(100,000) +0.09(105,000) = 17,450

How much u1and u2 to blend to get X1 = 100,000 and X2 = 105,000
straight run 0.40(100,000) (105,000) = 100,000 u1 cracked 0.80(100,000) (105,000) = 140, u2 u1 = 40,000 to grade 1 60,000 to grade 2 u2 = 80,000 to grade 1

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

Transport Costs

Math Formulation of Problem
Objective: Minimize TTC = * = Cij*Aij. supply shipments jAij < Yi for all i refinery satisfy crude oil needs iAij = Xj for all j Set up in Excel Solver

Chapter 15

Cash Market Ignore transaction and storage costs
Trader has a barrel of crude in transit Current spot price is St = \$18 Trader is paid the spot price upon delivery at ST ST Gain or Loss Value \$

Trader Wants to Hedge Suppose FT = \$18 to deliver oil at time T
Sells one futures At time T the contract is worth FT – ST Futures Cash 18 - ST T ST Contract Sold \$ Price volatility down Cost is the transaction costs Speculator takes on the risk Doesn’t have to be same product - one correlated

Hedging When Futures Price Different than Current Spot (Mia and Edwards 111-117)
Heating Oil Distributor   1-Oct         Cost of Crude   Spot price   Cost of carry/gal/month Contract for Delivery Dec ,000 at market price Delivery in months If sell at current spot on Dec 15, profits would be ( *0.008)*420,000 = 4,200 Expecting price to rise but not certain If price < 0.54 less profits , > 0.54 more profits

Short with Zero Basis Risk
Suppose Futures price is for Dec. 15 = \$0.56 Basis = Cash Price – Minus futures Price = – 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

Long Hedge Distributor Short Crude 1/1 2/15
Agreed to Deliver Crude Price 0.55 Spot price Cost of carry/gal/month Deliver Crude March ,000 Delivery in month If buy at current spot, hold and sell at contract rate profits: ( *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

Convenience Yield < Storage plus Interest Rate
example r = 1%,  = 1%,  = 1%, St = 20. FT = Ste(r+ -)T = Ste( )T T FT = \$ FT = \$ Further out is T the higher is FT Normal market - contango

Convenience Yield > Storage Plus Interest Rate
(r+ -)< 0 => r+ < Example: r = 1%,  = 1%,  = 3%, St = 20. T FT = \$ FT = \$ Further out is T the lower is FT Backwardation or inverted market

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

What Determines Energy Future Prices
Market 1 Market 2 S1 a S2 P1 PT d e PT b c P2 D1 f D2 Qb Qc Q1 Qd Qe Q2 imports exports

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

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

Hedging When Futures Price Different than Current Spot (Mia and Edwards 111-117)
Heating Oil Distributor   1-Oct         Cost of Crude   Spot price   Cost of carry/gal/month Contract for Delivery Dec ,000 at market price Delivery in months If sell at current spot on Dec 15, profits would be ( *0.008)*420,000 = 4,200 Expecting price to rise but not certain If price < 0.54 less profits , > 0.54 more profits

Chapter 16

Value of Call at Expiration

Value of Put at expiration

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

Value before expiration depends on following variables
Increase Call Put 2. Exercise Price Value ST K ST K

Value before expiration depends on following variables
Increase Call Put 5. Stock Risk Value ST K ST K

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

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

Buy a stock and bond portfolio equivalent to C

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 \$ = \$7.55

Solve for N anf Bt PortuT = N*U*St + R*Bt = cu = ST - K = 10
PortdT = N*D*St + R*Bt = cd.= 0 N = (cu - cd)/[(U - D)*St], Bt = [cu*D - cd*U]/[(U - D)*(-R)] N = (10 - 0)/[( )*100] = 0.5, Bt = [(10*0.9) - (0*1.1)]/[( )*(-1.06)] = buy (+) half a stock sell (-) \$42.45 worth of bonds Value of the portfolio is, as before, N*St + Bt = 0.5St = \$50 - \$42.45 = \$7.55.

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 = 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

P for general case 1.1 (p) + 0.9(1-p) = 1.06
(p)*USt + (1 - p)*DSt = (1 + r)*St = R*St Solving we get p = (R - D)/(U - D)

What is Value of Your Portfolio?
What if two periods to maturity 0.8 (1.1) S = 121, C = 21 0.8 (1.1)* (2)0.2*0.8 (1-.1)(1+.1)100 S = 99, C=0 (1-.1)*100 0.22(1-.1)(1-.1) S = 81, C= 0 Value today 0.8*0.8*21 + 2*(0.8*0.2) *0.2(0) = \$11.962 1.062

Finish - Value of 2 period American Call
What if two periods to maturity (1.1) S = 121, C = 21 0.8(1.1)* (2)0.2*0.8(1-.1)(1+.1)100 S = 100, C=0 0.2(1-.1)*100 0.22(1-.1)(1-.1) S = 81, C= 0 Value today 0.8*0.8*21 + 2*(0.8*0.2) *0.2(0) = \$11.962 1.062

Chapter 17

Chapter 18

Chapter 19

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 (106 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)

Write in Equations: B = 0.05 B + 0.01M + 0.09E + 1
M = M E + 2 E = 0.20 B M E + 10 M all constants to the right and convert to matrices B M = d E I A x d (I-A)x = d

Last Time I-O Regionalize – Consumption, Investment, Government, Net Exports
a11 a12 a13 a14 a21 a22 a23 a24 . . . a91 a92 a93 a94 Region X Sector Sector X Sector a11C a12I a13G a14(X-M) a21C a22I a23G a24(X-M) AU= a91C a92I a93G a94(X-M) Region X Sector C 0 0 0 0 I 0 0 0 0 G 0 (X-M)

Last Time Keeping Track of Pollution (1)
fij is the amount of pollutant i per unit of good j Total amount of pollution i is Pi = j(fijXj) Example 2 pollutants (P1,P2), 3 goods (X1, X2, X3) pollutant 1 pollutant 2 f1j f2j Xj X1 = coal X2 = gas X3 = oil

Last Time Keeping Track of Pollution (2)
pollutant 1 pollutant 2 f1j f2j Xj X1 = coal X2 = gas X3 = oil P = P1 P2 F X P = F'X

Last Time Keeping Track of Pollution (2)
f1j f2j Xj P = F'X P = 30 = 25* * *30 = 2380 2* * *30 = 340

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

If a and b both \$ output input energy other energy 0.20 0.20
Sum 1- sum = value added 1-0.6= =0.7

Three Sectors - \$ E M O d E 0.20 0.30 0.25 10 M 0.35 0.10 0.15 20
x = (I-A)-1d x = 81.45 75.56 129.98

E M O d E M O x = (I-A)-1d xr = 136.86 238.92 xr-x = = 76.82 = 61.31 = sum =

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 = Value = Price'X = [1, 4, 3] 61.31 108.94 =

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

More complicated - model control industry
new control industry Z produce 0.05 lbs of pollutant/\$ of energy current pollution = 0.05*X1 = 0.05* =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 9.8573

More complicated - model control industry
New industry Z X1 = 0.2X1+0.2X2+0.25X X Z +10 X2 = 0.0X1+0.1X2+0.15X X Z+20 Fill in for X3 to X4 X3 = X4 = Add regulation Z = 0.9*0.05*X1 Notice: Old A plus extra row and extra column x includes Z x = Agx + d 0.3X1+0.3X2+0.30X X Z+100 0.3X1+0.1X2+0.20X3+0.40X

Variables left, constants right
X X1-0.2X2-0.25X X Z =10 X X1-0.1X2-0.15X X Z =20 X X1-0.3X2-0.30X X Z=100 Compute for X4 X4 Z - 0.9*0.05*X =0 Let's write as [I-Ag]x = d - 0.3X1-0.1X2-0.20X X = 30

Write as [I-Ag]x = d X1 X2 X3 X4 Z 10 20 100 30 - = 0.045 0.0 0.0 0.0 0.0 Solve:(I-Ag)x = d  x = (I-Ag)-1d

Opportunity Cost of Pollution Regulation
Before After Opportunity Cost X X X X Z pollution before = 0.05*X1 =0.05* =9.8573 pollution after = 0.05* = 1.00 cost as percent of GDP =

Cradle to Grave x = Ax + d x1 = a11x1 + a12x2 + d1 x2 = a21x1 + a22x2 + d2 x = (I-A)-1d x1 = τ11d1 + τ12d2 x2 = τ21d1 + τ22d2 cradle to grave use of x1 to get 1 more d1 dx1/dd1 = τ11 cradle to grave use of xi to get 1 more dk= τik More on eolca

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

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

Solve Substitute for M in 1, 3
B = 0.05 B (0.02E+4) E (4) E = 0.20B (0.02E+4) E (5) Rearrange some terms and simplify B B E E = 1 E B E E = 10 Further combine terms 0.95 B E = 1.04 - 0.20B E = 10.28

Solve Solve 2 equations with 2 unknowns 0.95 B - 0.0902E = 1.04 (1)
From eq. (1) B = (0.0902E +1.04)/0.95 = E Substitute B into eq (2) -0.20(0.0940E ) E = 10.28 Solve for E E E = 10.28 0.8255E = →E =

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

Let's Rewrite Technical Coefficients as Input per unit Output Matrix
From Slide basic materials (B) per B (tons) (B/B) 0.01 B per manufacturing (M) (tons) (B/M) 0.09 B into E (106 BTU) (E) (B/E) Inputs down/outputs across B M E B M E 0.05 0.01 0.09

Cradle to Grave, Wells to Wheels
A= B = 2.30 tons M = 4.25 tons E = X 106 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

Continue Solution A= 0.05 0.01 0.09 B = 2.30 tons
M = 4.25 tons E = X 106 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 = = 0.350

Cradle to Grave, Wells to Wheels
A= B = 2.30 tons M = 4.25 tons E = X 106 BTU Why can't diagonal elements > 1? Cradle to grave use of energy to produce mfg Direct: 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

Take a Look Back x = (I-A)-1d
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)-1d (kX1) (kXk)(kX1) f-19m.xlsx CA2 - hand in sheet with names and answers

Take a Look Back (b) When does solution exist When does solution make economic sense Disaggregate models (i regions, j products) aij = 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) a1,1 a1,2 a1,3 Coal Oil 0 a27,1 a27,2 a27,3 0 0 Ngas

Take a Look Back (c) A times FF = FF_Reg a1,1 a1,2 a1,3 Coal Oil 0 a27 a27,3 a27,3 0 0 Ngas FF_Reg = consumption of each fuel by region = a1,1 Coal a1,2Oil a1,3 Ngas a2,1 Coal a2,2Oil a2,3 Ngas … a27,1 Coal a27,2Oil a27,3 Ngas

Take a Look Back (d) fij is the amount of pollutant i per unit of good j i = 5 pollutants - O3, PM, CO, Nox, Sox j = 3 products electricity (E), metals (M), Pulp&Paper (PP)

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

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

Direct Inputs from One Industry to Another
x = (I-A)-1d = 2.30 tons (B) 4.25 tons (M) 12.66 X 106 (E) Direct E into each industry f-19m.xlsx, IO!A20:A22 E*B = aEB*B E*M = E*E B M E = 0.20*2.3 = = 0.07*4.25 = = ?*?

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

Solve In Excel

Multiply (I-A)-1d 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

Input Output in \$ aij = \$ 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 = 0.45 from other industries/\$ The remainder = 0.55 is called value added Value added for M = = 0.15 Value added for E = ?

More complicated - model control industry
new control industry Z produce 0.05 lbs of pollutant/\$ of energy current pollution = 0.05*X1 = 0.05* =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 9.8573

More complicated - model control industry
New industry Z X1 = 0.2X1+0.2X2+0.25X X Z +10 X2 = 0.0X1+0.1X2+0.15X X Z+20 Fill in for X3 to X4 X3 = X4 = Add regulation Z = 0.45*X1 Notice: Old A plus extra row and extra column x includes Z x = Agx + d 0.3X1+0.3X2+0.30X X Z+100 0.3X1+0.1X2+0.20X3+0.40X

Variables left, constants right
X X1-0.2X2-0.25X X Z =10 X X1-0.1X2-0.15X X Z =20 X X1-0.3X2-0.30X X Z=100 Compute for X4 X4 Z - 0.9*0.05*X =0 Let's write as [I-Ag]x = d - 0.3X1-0.1X2-0.20X X = 30

Write as [I-Ag]x = d Z - 0.045*X1 =0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
X1 X2 X3 X4 Z 10 20 100 30 - = ? ? ? ? ? 0.045 0.0 0.0 0.0 0.0 ? 0.0 Old A Ag Solve:(I-Ag)x = d  (I-Ag)-1(I-Ag)x = (I-Ag)-1d x = (I-Ag)-1d

Clean Up More Compliucated Model with Control Industry
(I-Ag)x = d X1 = 10 X X X Z (I-Ag) with augmented row and column

X with pollution control industry
x = (I-Ag)-1d = pollution after = 0.05* = 0.997 0.05* = 0.997 199.32 129.80 325.41 279.77 8.97

Opportunity Cost of Pollution Regulation
Before After Opportunity Cost X X X X Z pollution before = .05*X1 =0.05* =9.025 pollution after = 0.05* = 0.996 cost in extra output industry = 9.742 sometimes as a percent of GDP = 9.742/GDP

Chapter 20