1. 1 Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: 12-706 and 73-359 Lecture 4 - 9/13/2004

2. 12-706 and 73-3592 Qualitative CBA  If can’t quantify all costs and benefits  Quantify as many as possible  Make assumptions  Estimate order of magnitude value of others  Make rough Net Benefits estimate

3. 12-706 and 73-3593 Welfare Economics Concepts  Perfect Competition  Homogeneous goods.  No agent affects prices.  Perfect information.  No transaction costs /entry issues  No transportation costs.  No externalities:  Private benefits = social benefits.  Private costs = social costs.

4. 12-706 and 73-3594 (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).

5. 12-706 and 73-3595 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

6. 12-706 and 73-3596 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

7. 12-706 and 73-3597 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

8. 12-706 and 73-3598 Total/Gross/User Benefits Price Quantity P* 0 1 2 3 4 Q* A B  Benefits received are related to WTP - and approximated by the shaded rectangles  Approximated by whole area under demand: triangle AP*B + rectangle 0P*BQ* P1

9. 12-706 and 73-3599 Benefits with WTP Price Quantity P* 0 1 2 3 4 Q* A B  Total/Gross/User Benefits = area under curve or willingness to pay for all people = Social WTP = their benefit from consuming = sum of all WTP values  Receive benefits from consuming this much regardless of how much they pay to get it

10. 12-706 and 73-35910 Net Benefits Price Quantity P* 0 1 2 3 4 Q* A B A B  Amount ‘paid’ by society at Q* is P*, so total payment is B to receive (A+B) total benefit  Net benefits = (A+B) - B = A = consumer surplus (benefit received - price paid)

11. 12-706 and 73-35911 Consumer Surplus Changes Price Quantity P* 0 1 2 Q* Q1 A B P1 CS1  New graph - assume CS1 is original consumer surplus at P*, Q* and price reduced to P1  Changes in CS approximate WTP for policies

12. 12-706 and 73-35912 Consumer Surplus Changes Price Quantity P* 0 1 2 Q* Q1 A B P1 CS2  CS2 is new cons. surplus as price decreases to (P1, Q1); consumers gain from lower price  Change in CS = P*ABP1 -> net benefits  Area : trapezoid = (1/2)(height)(sum of bases)

13. 12-706 and 73-35913 Consumer Surplus Changes Price Quantity P* 0 1 2 Q* Q1 A B P1 CS2  Same thing in reverse. If original price is P1, then increase price moves back to CS1

14. 12-706 and 73-35914 Consumer Surplus Changes Price Quantity P* 0 1 2 Q* Q1 A B P1 CS1  If original price is P1, then increase price moves back to CS1 - Trapezoid is loss in CS, negative net benefit

15. 12-706 and 73-35915 Further Analysis  Assume price increase is because of tax  Tax is P2-P* per unit, tax revenue =(P2-P*)Q2  Tax revenue is transfer from consumers to gov’t  To society overall, no effect  Pay taxes to gov’t, get same amount back  But we only get yellow part.. Price Quantity P2 0 1 2 Q2 Q* A B P* CS1 C Old NB: CS2 New NB: CS1 Change:P2ABP*

16. 12-706 and 73-35916 Deadweight Loss  Yellow paid to gov’t as tax  Green is pure cost (no offsetting benefit)  Called deadweight loss  Consumers buy less than they would w/o tax (exceeds some people’s WTP!) - loss of CS  There will always be DWL when tax imposed Price Quantity P2 0 1 2 Q* Q1 A B P* CS1

17. 12-706 and 73-35917 Net Social Benefit Accounting  Change in CS: P 2 ABP* (loss)  Government Spending: P 2 ACP* (gain)  Gain because society gets it back  Net Benefit: Triangle ABC (loss)  Because we don’t get all of CS loss back  OR.. NSB= (-P 2 ABP*)+ P 2 ACP* = -ABC

18. 12-706 and 73-35918 Commentary  It is trivial to do this math when demand curves, preferences, etc. are known. Without this information we have big problems.  Unfortunately, most of the ‘hard problems’ out there have unknown demand functions.  We need advanced methods to find demand

19. 12-706 and 73-35919 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.  = (p  q) / (q  p)

20. 12-706 and 73-35920 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

21. 12-706 and 73-35921 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

22. 12-706 and 73-35922 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

23. 12-706 and 73-35923 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 at average fare of $ 1.20. Estimate Total Willingness to Pay.

24. 12-706 and 73-35924 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

25. 12-706 and 73-35925 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 + (a-p)q/2 = 1.2*20,000 + (5.2- 1.2)*20,000/2 = 24,000 + 40,000 = $ 64,000 per day.

26. 12-706 and 73-35926 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.

27. 12-706 and 73-35927 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

28. 12-706 and 73-35928 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)..

29. 12-706 and 73-35929 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.

30. 12-706 and 73-35930 Maglev Log-Linear Function  q = a*p 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.