It is not an essay assignment, but a problem set of 'Public Economics' course. I have attached the problem set that needs to be solved (file name: ps1.pdf) and the course materials (rest of files) that were used in class.
Public Economics — Problem Set #1 Due: October 3rd at 2:40pm (submit through the courseworks) 1. Consider a reform that changed welfare benefits in New Jersey and suppose that at the same time there was no change in New York (note: even though there was no reform in NY, that does not mean that nothing in New York has changed — for example, the composition of the group of welfare recipients may have changed over time). The data for welfare recipients (per month) in the two states looks as follows: New Jersey New York Average hours of work Average benefit Average hours of work Average benefit Before the reform 60 1000 55 1000 After the reform 70 600 60 800 (a) Explain what assumption(s) you need to make to rely on this data in order to estimate the effect of the reform (b) What is the difference-in-difference estimate of the effect of the reform on hours of work of welfare recipients? (c) What is the corresponding estimate on the effect on welfare benefits? (d) What is the implied elasticity of hours of work to the level of welfare benefits (ie., percentage change in hours of work per a one percent change in the level of benefits)? Note: the previous parts tell you what changes are, but the elasticity has to be evaluated at some reference point. It is obvious what the initial point is when one evaluates the elasticity theoretically — that’s where you take the derivative — but with real-life data one could use a lot of different points — before or after the reform, in New Jersey or New York or anything in between. One common choice is to take the mid-point between before and after reform values for the treated group. 2. The demand for smartphones is given by D(p) = 400− p+ p T 5 , where p is the price of a smartphone and p T is the price of a tablet (a substitute for smartphones). The supply is given by S(p) = 4 · p. The price of tablets is fixed and set at pT = 500. Suppose that the government imposes two taxes on phones: a $30 tax to be paid by the consumers and $70 tax that producers have to pay. (a) What is the economic incidence of this policy? (b) What is the excess burden here? (c) How would economic incidence change if government imposed instead a $70 tax on consumers and a $30 tax on producers (d) Imagine that the tax on producers increases to $120, while the tax on consumers remains unchanged at $30. How does the excess burden change? Divide the change in excess burden into components coming from the surplus of each of the parties involved. (e) Which component of the change in excess burden is the largest? Explain why. 3. The demand for food purchased in grocery stores is given by DG = 100 − PG + 12PT where PG is the price (index) of food in supermarkets and PT is the price of take-outs. Correspondingly, the demand for take-outs is DT = 100 − PT + 12PG. The supply functions are given by SG = 1 2PG and ST = 1 2PT respectively. The government imposes the tax of 40 on take-out food. Determine how the incidence of this tax is split between consumers and producers of the two types of food. Note: you have to find prices for both goods that yield an equilibrium in both markets at the same time. 4. The excess burden is an increasing function of a tax. True, false or uncertain: the excess burden of a (small?) subsidy is therefore lower than the excess burden of a tax.1 5. Suppose that the marginal private cost of providing higher education for n students is given by MPC(n) = n and that the marginal private benefit schedule is given by MPB(n) = 200 − n (ie, benefits decline with the number of students, presumably because additional students are less qualified and derive lower return from being educated). Imagine though that people with college education are more likely to vote and volunteer. Assume (on faith) that these behaviors benefit everyone. The additional social benefit from these activities is valued at 20 per person with college education. (a) Plot a graph showing private marginal benefit, private marginal cost and social marginal benefit. (b) Find the price and quantity that correspond to the private competitive equilibrium (i.e., with no intervention of any kind). 1A topic for additional consideration (no extra credit): denote the total excess burden as EB(t) where t can be positive (tax) or negative (subsidy). What happens at t = 0? Is EB(t) discontinuous or non-differentiable at t = 0? To be formal, this of course requires additional structure (though not much of it really), so let’s just assume for simplicity that we are dealing with linear demand/supply curves in partial equilibrium. (c) Find the socially efficient quantity and the deadweight loss from being at the private competitive equilibrium instead. (d) What value of a monetary subsidy to education would implement the efficient solution? 6. There are 4 firms in the industry that have the total costs of eliminating pollution given by P 2/4, P 2/3, P 2/2 and P 2 respectively. (a) Suppose that we want to reduce aggregate pollution in a way that minimizes the overall cost. Derive the marginal cost of doing so as a function of the overall reduction in pollution P ∗ (b) Suppose we want to reduce the overall pollution by 100 units. How much should each of the firms reduce pollution by in order to minimize the overall cost of doing so? (c) Suppose that we require each firm to reduce pollution by 30 units. Firms are allowed to trade obligations to lower their pollution reduction requirements. What will be the competitive market price of a unit of pollution reduction and how many units will be traded? (d) Suppose we do not allow firms to trade in part (c). What would be the deadweight loss compared to the solution in part (c) 2 Economic Tools ▶ Utility function: a mathematical representation of preferences ▶ Assumption: individuals have well-defined “rational” preferences and attempt to achieve the highest level of well-being ▶ Indifference curves Consumption Le is ur e ● B ● A ● C Utility ▶ Marginal utility U(Z ,Y ) = 50 ln(Y ) + 30 ln(Z ) (ln(X ))′ = 1 X MUZ (Z ,Y ) = 0 + 30 · 1 Z = 30 Z MUY (Z ,Y ) = 50 · 1 Y + 0 = 50 Y If one consumes (Z ,Y ) = (3, 2), the marginal utility of good Z is MUZ (3, 2) = 30 3 = 10 while the marginal utility of good Y is MUY (3, 2) = 50 2 = 25. ▶ The marginal rate of substitution (MRS) — the slope of the indifference curve. MRS of good Z to good Y: MRS = −MUZ MUY = − 30/Z 50/Y = −3 5 Y Z Good Z G oo d Y ● A ● B ∆∆Z ∆∆Y Slope: ∆∆Y ∆∆Z ≈≈ MRS u((A)) == u((B)) u((B)) ≈≈ u((A)) ++ MUZ∆∆Z ++ MUY∆∆Y MUZ∆∆Z ++ MUY∆∆Y == 0 ∆∆Y ∆∆Z == −− MUZ MUY Budget constraint ▶ Optimization is subject to (budget) constraints Price of apples (A) is pA. Price of bananas (B) is pB . Income is Y . The budget constraint is: pAA+ pBB = Y If price of apples was 5, price of bananas was 7 and income was 35, the budget constraint would be 5A+ 7B = 35 ▶ Equivalently: pBB = Y − pAA ⇒ B = Y pB − pA pB A ▶ The slope of the budget constraint is − pApB . Characterization of the optimum The budget constraint needs to be tangent to the indifference curve at the optimum. Two conditions: 1. The slopes of the budget constraint and the indifference curve need to be the same: −MUA MUB = MRS = −pA pB 2. The optimum is on the budget constraint pAA+ pBB = Y Good Z G oo d Y ● Example ▶ U(Y ,Z ) = 13 ln(Y ) + 2 3 ln(Z ). PY = 10, PZ = 20, Y = 120 ▶ Method 1: MRS = − 1 3 1 Y 2 3 1 Z = −12 ZY . The slope of the budget line is −1020 = −12 . We need to solve: −1 2 Z Y = −1 2 10Y + 20Z = 120 Solution: Z = Y = 4. ▶ Method 2: The budget constraint is 10Y + 20Z = 120 hence Y = 12− 2Z . We want to pick the point with the highest utility on the budget constraint, hence we want to maximize 1 3 ln(12− 2Z ) + 2 3 ln(Z ) That requires −23 112−2Z + 23 1Z = 0 ⇒ 12− 2Z = Z , hence Z = 4 and Y = 12− 2Z = 4. Nonlinear budget constraints ▶ Why care? Because they are pervasive in the tax/welfare context. ▶ Examples: Earned Income Tax Credit (we’ll talk more about it) provides a marginal subsidy if earnings are not too large and then slowly takes it away. Many related provisions in welfare programs. Tax exemptions — no tax (labor valuable, leisure costly) up to certain income level, tax afterwards. Progressive taxation — price of labor depends on your income. Health insurance subsidies — the amount depends on the level of income. Tax exemption Leisure C on su m pt io n ● A ● B ● C Income and substitution effects Good Z ● A ● C ● B Total effect: A to C Subsitution effect: A to B Income effect: B to C Elasticity (of demand) ▶ Demand at given price p is D(p) ▶ It could be individual demand or aggregate demand, we can derive it based on utility maximization or based on observation or assume ▶ Slope: D ′(p) — how much demand changes with a dollar change in price ▶ A common way is to instead measure the slope by the elasticity: the percentage change in the demand in response to a 1% change in price ▶ ε = % change in demand % change in price = ∆D(p)/D(p) ∆p/p = p D(p) D ′(p) ▶ Another (equivalent) definition: ε = ∆ ln(D(p)) ∆ ln(p) ▶ Aside: you can see it by substituting x = ln(p) so that d ln(D(p)) d ln(p) = d ln(D(e ln(x))) dx and work through the derivative with respect to x . Or, by simply noting that ∆ ln(x) is approximately a percentage change in x . Equilibrium and efficiency Quantity P ric e Demand Supply P ● Q Quantity P ric e Demand Supply ● ● Q Total surplus Deadweight loss Pareto efficiency ▶ An allocation at which the only way to make one person better off is to make another person worse off is called Pareto efficient ▶ If an allocation is not Pareto efficient, there must exist a Pareto improvement. ▶ At an (interior) Pareto efficient allocation MRSs for all individuals are the same. The First Theorem of Welfare Economics ▶ assume (1) perfect competition; (2) existence of markets for all