1. 1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews 12-706/19-702 / 73-359 Lecture 7 - Microecon Recap

2. 12-706 and 73-3592 Discussion - “willingness to pay”  Survey of students of WTP for beer  How much for 1 beer? 2 beers? Etc.  Does similar form hold for all goods?  What types of goods different?  Economists also refer to this as demand

3. 12-706 and 73-3593 (Individual) Demand Curves  Downward Sloping is a result of diminishing marginal utility of each additional unit (also consider as WTP)  Presumes that at some point you have enough to make you happy and do not value additional units Price Quantity P* 0 1 2 3 4 Q* A B Actually an inverse demand curve (where P = f(Q) instead).

4. 12-706 and 73-3594 Market Demand Price P* 0 1 2 3 4 Q A B  If above graphs show two (groups of) consumer demands, what is social demand curve? P* 0 1 2 3 4 5 Q A B

5. 12-706 and 73-3595 Market Demand  Found by calculating the horizontal sum of individual demand curves  Market demand then measures ‘total consumer surplus of entire market’ P* 0 1 2 3 4 5 6 7 8 9 Q

6. 12-706 and 73-3596 Social WTP (i.e. market demand) Price Quantity P* 0 1 2 3 4 Q* A B  ‘Aggregate’ demand function: how all potential consumers in society value the good or service (i.e., someone willing to pay every price…)  This is the kind of demand curves we care about

7. 12-706 and 73-3597 First: Elasticities of Demand  Measurement of how “responsive” demand is to some change in price or income.  Slope of demand curve =  p/  q.  Elasticity of demand, , is defined to be the percent change in quantity divided by the percent change in price.

8. 12-706 and 73-3598 Elasticities of Demand Elastic demand:  > 1. If P inc. by 1%, demand dec. by more than 1%. Unit elasticity:  = 1. If P inc. by 1%, demand dec. by 1%. Inelastic demand:  < 1 If P inc. by 1%, demand dec. by less than 1%. Q P Q P

9. 12-706 and 73-3599 Elasticities of Demand Q P Q P Perfectly Inelastic Perfectly Elastic A change in price causes Demand to go to zero (no easy examples) Necessities, demand is Completely insensitive To price

10. 12-706 and 73-35910 Elasticity - Some Formulas  Point elasticity = dq/dp * (p/q)  For linear curve, q = (p-a)/b so dq/dp = 1/b  Linear curve point elasticity =(1/b) *p/q = (1/b)*(a+bq)/q =(a/bq) + 1

11. 12-706 and 73-35911 Sorta Timely Analysis zHow sensitive is gasoline demand to price changes? zHistorically, we have seen relatively little change in demand. Recently? zNew AAA report: higher gasoline prices have caused a 3 percent reduction in demand from a year ago.  What was  p?  q?  ? zWhat does that tell us about gasoline?

12. 12-706 and 73-35912 Maglev System Example  Maglev - downtown, tech center, UPMC, CMU  20,000 riders per day forecast by developers.  Let’s assume:  price elasticity -0.3;  linear demand;  20,000 riders @ average fare of $ 1.20.  Estimate Total Willingness to Pay.

13. 12-706 and 73-35913 Example calculations  We have one point on demand curve:  1.2 = a + b*(20,000)  We know an elasticity value:  elasticity for linear curve = 1 + a/bq  -0.3 = 1 + a/b*(20,000)  Solve with two simultaneous equations:  a = 5.2  b = -0.0002 or 2.0 x 10^-4

14. 12-706 and 73-35914 Types of Costs - from 3-03 zPrivate - paid by consumers zSocial - paid by all of society zOpportunity - cost of foregone options zFixed - do not vary with usage zVariable - vary directly with usage zExternal - imposed by users on non-users ye.g. traffic, pollution, health risks yPrivate decisions usually ignore external

15. 12-706 and 73-35915 Making Cost Functions zFundamental to analysis and policies zThree stages: y Technical knowledge of alternatives y Apply input (material) prices to options y Relate price to cost zObvious need for engineering/economics zMain point: consider cost of all parties zIncluded: labor, materials, hazard costs

16. 12-706 and 73-35916 Functional Forms  TC(q) = F+ VC(q)  Use TC eq’n to generate unit costs  Average Total: ATC = TC/q  Variable: AVC = VC/q  Marginal: MC =  [TC]/  q =  TC  q  but  F/  q = 0, so MC =  [VC]/  q

17. 12-706 and 73-35917 Short Run vs. Long Run Cost  Short term / short run - some costs fixed  In long run, “all costs variable”  Difference is in ‘degree of control of plans’  Generally say we are ‘constrained in the short run but not the long run’  So TC(q) < = SRTC(q)

18. 12-706 and 73-35918 Firm Production Functions MC Q P What do marginal, Average cost curves Tell us? AVC Variable cost shows Non-fixed components Of producing the good Marginal costs show us Cost of producing one Additional good Where would firm produce?

19. 12-706 and 73-35919 BCA Part 2: Cost Welfare Economics Continued The upper segment of a firm’s marginal cost curve corresponds to the firm’s SR supply curve. Again, diminishing returns occur. Quantity Price Supply=MC At any given price, determines how much output to produce to maximize profit AVC

20. 12-706 and 73-35920 Supply/Marginal Cost Notes Quantity Price Supply=MC At any given price, determines how much output to produce to maximize profit P* Q1 Q* Q2 Demand: WTP for each additional unit Supply: cost incurred for each additional unit

21. 12-706 and 73-35921 Supply/Marginal Cost Notes Quantity Price Supply=MC Area under MC is TVC - why? P* Q1 Q* Q2 Recall: We always want to be considering opportunity costs (total asset value to society) and not accounting costs

22. 12-706 and 73-35922 Unifying Cost and Supply  Economists learn “Supply and Demand”  Equilibrium (meeting point): where S = D  In our case, substitute ‘cost’ for supply  Why cost? Need to trade-off Demand  Using MC is a standard method  Recall this is a perfectly competitive world!

23. 12-706 and 73-35923 Example  Demand Function: p = 4 - 3q  Supply function: p = 1.5q  Assume equilibrium, what is p,q?  In eq: S=D; 4-3q=1.5q ; 4.5q=4 ; q=8/9  P=1.5q=(3/2)*(8/9)= 4/3

24. 12-706 and 73-35924 Pricing Strategies  Highway pricing  If price set equal to AC (which is assumed to be TC/q then at q, total costs covered  p ~ AVC: manages usage of highway  p = f(fares, fees, travel times, discomfort)  Price increase=> less users (BCA)  MC pricing: more users, higher price  What about social/external costs?  Might want to set p=MSC

25. 12-706 and 73-35925 Estimating Linear Demand Functions zAs above, sometimes we don’t know demand zFocus on demand (care more about CS) but can use similar methods to estimate costs (supply) zOrdinary least squares regression used yminimize the sum of squared deviations between estimated line and p,q observations: p = a + bq + e yStandard algorithms to compute parameter estimates - spreadsheets, Minitab, S, etc. yEstimates of uncertainty of estimates are obtained (based upon assumption of identically normally distributed error terms). zCan have multiple linear terms

26. 12-706 and 73-35926 Log-linear Function zq = a(p) b (hh) c ….. zConditions: a positive, b negative, c positive,... zIf q = a(p) b : Elasticity interesting = (dq/dp)*(p/q) = abp (b-1) *(p/q) = b*(ap b /ap b ) = b. yConstant elasticity at all points. zEasiest way to estimate: linearize and use ordinary least squares regression (see Chap 12) yE.g., ln q = ln a + b ln(p) + c ln(hh)..

27. 12-706 and 73-35927 Log-linear Function  q = a*p b and taking log of each side gives: ln q = ln a + b ln p which can be re-written as q’ = a’ + b p’, linear in the parameters and amenable to OLS regression.  This violates error term assumptions of OLS regression.  Alternative is maximum likelihood - select parameters to max. chance of seeing obs.

28. 12-706 and 73-35928 Maglev Log-Linear Function  q = ap b - From above, b = -0.3, so if p = 1.2 and q = 20,000; so 20,000 = a*(1.2) -0.3 ; a = 21,124.  If p becomes 1.0 then q = 21,124*(1) -0.3 = 21,124.  Linear model - 21,000  Remaining revenue, TWtP values similar but NOT EQUAL.

29. 12-706 and 73-35929 Demand Example (cont)  Maglev Demand Function:  p = 5.2 - 0.0002*q  Revenue: 1.2*20,000 = $ 24,000 per day  TWtP = Revenue + Consumer Surplus  TWtP = pq + 1/2*(a-p)q = 1.2*20,000 + 0.5*(5.2-1.2)*20,000 = 24,000 + 40,000 = $ 64,000 per day.

30. 12-706 and 73-35930 Change in Fare to $ 1.00  From demand curve: 1.0 = 5.2 - 0.0002q, so q becomes 21,000.  Using elasticity: 16.7% fare change (1.2-1/1.2), so q would change by -0.3*16.7 = 5.001% to 21,002 (slightly different value)  Change to Revenue = 1*21,000 - 1.2*20,000 = 21,000 - 24,000 = -3,000.  Change CS = 0.5*(0.2)*(20,000+21,000)= 4,100  Change to TWtP = (21,000-20,000)*1 + (1.2-1)*(21,000- 20,000)/2 = 1,100.