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Civil Systems Planning Benefit/Cost Analysis Scott Matthews Courses: 12-706 and 73-359 Lecture 4 - 9/13/2004 1 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 12-706 and 73-359 2 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. 12-706 and 73-359 3 (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 A Actually an inverse demand curve (where P = f(Q) instead). B P* 0 1 2 3 4 12-706 and 73-359 Q* Quantity 4 Market Demand Price A A B B P* P* 0 1 2 3 4 Q 0 1 2 3 4 5 Q If above graphs show two (groups of) consumer demands, what is social demand curve? 12-706 and 73-359 5 Market Demand P* 0 1 2 3 4 5 6 7 8 9 Q Found by calculating the horizontal sum of individual demand curves Market demand then measures ‘total and 73-359 market’ consumer surplus12-706 of entire 6 Social WTP (i.e. market demand) Price A B P* 0 1 2 3 4 Q* Quantity ‘Aggregate’ demand function: how all potential consumers in society value the good or service (i.e., someone willing to pay every price…) 12-706 and 73-359 This is the kind of demand curves we care about 7 Total/Gross/User Benefits Price A P1 B P* 0 1 2 3 4 Q* Quantity Benefits received are related to WTP - and approximated by the shaded rectangles Approximated by whole area under demand: 12-706 and 73-359 triangle AP*B + rectangle 0P*BQ* 8 Benefits with WTP Price A B P* 0 1 2 3 4 Q* Quantity 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 12-706 and 73-359 regardless of how much they pay to get it 9 Net Benefits Price A A B P* B 0 1 2 3 4 Q* Quantity 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 12-706 and 73-359 surplus (benefit received - price paid) 10 Consumer Surplus Changes Price CS1 A P* B P1 0 1 2 Q* Q1 Quantity New graph - assume CS1 is original consumer surplus at P*, Q* and price reduced to P1 Changes in CS approximate WTP for policies 12-706 and 73-359 11 Consumer Surplus Changes Price A CS2 P* B P1 0 1 2 Q* Q1 Quantity CS2 is new cons. surplus as price decreases to (P1, Q1); consumers gain from lower price Change in CS = P*ABP1 -> net benefits and 73-359 Area : trapezoid =12-706 (1/2)(height)(sum of bases) 12 Consumer Surplus Changes Price A CS2 P* B P1 0 1 2 Q* Q1 Quantity Same thing in reverse. If original price is P1, then increase price moves back to CS1 12-706 and 73-359 13 Consumer Surplus Changes Price A CS1 P* B P1 0 1 2 Q* Q1 Quantity If original price is P1, then increase price moves back to CS1 - Trapezoid is loss in CS, negative net benefit 12-706 and 73-359 14 Further Analysis Price A CS1 P2 P* 0 1 2 C B Q2 Q* Old NB: CS2 New NB: CS1 Change:P2ABP* Quantity 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 12-706 and 73-359 But we only get yellow part.. 15 Deadweight Loss Price A CS1 P2 B P* 0 1 2 Q* Q1 Quantity 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 always12-706 be and DWL when tax imposed 73-359 16 Net Social Benefit Accounting Change in CS: P2ABP* (loss) Government Spending: P2ACP* (gain) Gain because society gets it back Net Benefit: Triangle ABC (loss) Because we don’t get all of CS loss back OR.. NSB= (-P2ABP*)+ P2ACP* = -ABC 12-706 and 73-359 17 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 12-706 and 73-359 18 First: Elasticities of Demand Measurement of how “responsive” demand is to some change in price or income. Slope of demand curve = Dp/Dq. Elasticity of demand, e, is defined to be the percent change in quantity divided by the percent change in price. e = (p Dq) / (q Dp) 12-706 and 73-359 19 Elasticities of Demand Elastic demand: e > 1. If P inc. by 1%, demand dec. by more than 1%. Unit elasticity: e = 1. If P inc. by 1%, demand dec. by 1%. Inelastic demand: e < 1 If P inc. by 1%, demand dec. by less than 1%. P P Q Q 12-706 and 73-359 20 Elasticities of Demand P Necessities, demand is Completely insensitive To price Perfectly Inelastic P Q Perfectly Elastic A change in price causes Demand to go to zero (no easy examples) Q 12-706 and 73-359 21 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 12-706 and 73-359 22 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. 12-706 and 73-359 23 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 12-706 and 73-359 24 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 + (5.21.2)*20,000/2 = 24,000 + 40,000 = $ 64,000 per day. 12-706 and 73-359 25 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 + (1.21)*(21,000-20,000)/2 = 1,100. 12-706 and 73-359 26 Estimating Linear Demand Functions As above, sometimes we don’t know demand Focus on demand (care more about CS) but can use similar methods to estimate costs (supply) Ordinary least squares regression used minimize the sum of squared deviations between estimated line and p,q observations: p = a + bq + e Standard algorithms to compute parameter estimates - spreadsheets, Minitab, S, etc. Estimates of uncertainty of estimates are obtained (based upon assumption of identically normally distributed error terms). 12-706 and 73-359 Can have multiple linear terms 27 Log-linear Function q = a(p)b(hh)c….. Conditions: a positive, b negative, c positive,... If q = a(p)b : Elasticity interesting = (dq/dp)*(p/q) = abp(b-1)*(p/q) = b*(apb/apb) = b. Constant elasticity at all points. Easiest way to estimate: linearize and use ordinary least squares regression (see Chap 12) E.g., ln q = ln a + b ln(p) + c ln(hh) .. 12-706 and 73-359 28 Log-linear Function q = a*pb 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. 12-706 and 73-359 29 Maglev Log-Linear Function q = a*pb - 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. 12-706 and 73-359 30