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Civil Systems Planning Benefit/Cost Analysis

Civil Systems Planning Benefit/Cost Analysis. Scott Matthews Courses: 12-706 and 73-359 Lecture 4 - 9/13/2004. Qualitative CBA. If can’t quantify all costs and benefits Quantify as many as possible Make assumptions Estimate order of magnitude value of others

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Civil Systems Planning Benefit/Cost Analysis

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

  2. 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

  3. Welfare EconomicsConcepts • 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

  4. Price A B P* 0 1 2 3 4 Q* Quantity (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 Actually an inverse demand curve (where P = f(Q) instead). 12-706 and 73-359

  5. 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

  6. 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 consumer surplus of entire market’ 12-706 and 73-359

  7. Price A B P* 0 1 2 3 4 Q* Quantity Social WTP (i.e. market demand) • ‘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 12-706 and 73-359

  8. Price A B P* 0 1 2 3 4 Q* Quantity Total/Gross/User Benefits P1 • Benefits received are related to WTP - and approximated by the shaded rectangles • Approximated by whole area under demand: triangle AP*B + rectangle 0P*BQ* 12-706 and 73-359

  9. Price A B P* 0 1 2 3 4 Q* Quantity Benefits with WTP • 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 12-706 and 73-359

  10. Price A B P* 0 1 2 3 4 Q* Quantity Net Benefits 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) 12-706 and 73-359

  11. 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

  12. 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 • Area : trapezoid = (1/2)(height)(sum of bases) 12-706 and 73-359

  13. 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

  14. 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

  15. Further Analysis Price Old NB: CS2 New NB: CS1 Change:P2ABP* A CS1 P2 B C P* 0 1 2 Q2 Q* 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 • But we only get yellow part.. 12-706 and 73-359

  16. 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 always be DWL when tax imposed 12-706 and 73-359

  17. 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

  18. 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

  19. 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

  20. P P Q Q 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%. 12-706 and 73-359

  21. P P Q Q Elasticities of Demand Necessities, demand is Completely insensitive To price Perfectly Inelastic Perfectly Elastic A change in price causes Demand to go to zero (no easy examples) 12-706 and 73-359

  22. 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

  23. 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

  24. 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

  25. 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. 12-706 and 73-359

  26. 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. 12-706 and 73-359

  27. 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). • Can have multiple linear terms 12-706 and 73-359

  28. 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

  29. 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

  30. 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

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