ECON 311 Fall Quarter, 2009 NAME: Prof. Hamilton FINAL EXAM 200 points 1. (30 points). Consider two ways for the government to collect tax revenue: a sales tax of on food, or an income tax. If you currently buy both food and clothes, which tax would you prefer if your total tax payment would be the same under either tax? Support your answer with a graph. If you currently buy both food and clothes, you will always be better off with the income tax. Under the sales tax, there is both an income effect (you purchase less clothes and food, because you have less after-tax income) and a substitution effect (you buy less clothes to avoid paying the tax, as clothes are now relatively more expensive). Under the income tax, there is only an income effect. To see why you are better off, consider the following figure: AOG A 0 tax revenue A S U 0 A I U I U S BC S BC I BC 0 F S F I F 0 Food First consider a sales tax on food. The total movement (income plus substitution effect) is from A = (F 0, A 0 ) to B = (F S, A S ). How much money has the government taken? At point B, the representative consumer now consumes C S units of clothes. The revenue taken by the government is summarized by the difference between the amount of AOG you used to be able to buy on your old budget line at C S, less what you can buy now on your new after-tax budget. Your utility under a sales tax on clothes is U S. An income tax lowers your income the same amount regardless of the market basket chosen. If government collects the same amount of tax revenue from you as under the sales tax, the IRS could increase the income tax, which would create a parallel shift in your budget constraint through point B. You will now change your consumption to C = (F I, A I ). Compared to the outcome under a sales tax, you will now consume more clothes and less other goods, but be on the higher indifference curve, U I. The sales tax has both an income effect and a substitution effect, while an income tax has only an income effect and is preferable because it causes less of a distortion. The income tax is better by the amount: U I - U S > 0.
2. (20 points). Write out (or describe in words) the condition which must hold at a cost-minimizing input mix for a firm that employs capital and labor in production. Show the cost-minimizing outcome on a diagram. Briefly provide intuition on why the condition you state must hold. K K* A B Q 0 K B 0 L* L B IC L At a cost-minimizing input mix, the firm s isoquant must be tangent to the isocost curve. The cheapest way to produce Q0 units of output is shown as point A in the above diagram, with K* units of capital and L* units of labor. The tangency between isoquant and isocost implies that the marginal rate of technical substitution (MRTS) must be equal to the ratio of factor prices, or MP K /r = MK L /w. This condition states that the marginal product per dollar must be the same for a dollar invested in capital or labor. If this condition did not hold, the firm could lower cost by shifting dollars away from the input with the lower marginal product per dollar and towards the input with the higher marginal product per dollar.
3. (30 points) (a) Depict an industry in long-run competitive equilibrium on a 2-panel diagram. In one panel, show the profit-maximizing output level and profit for an individual firm. In the other panel, show market supply and market demand. $/Q MARKET S 1 S 0 $/q Individual firm SR profit MC AC P 1 P 1 d = P 1 P 0 P 0 d = P 0 D Q E q 0 q 1 Quantity (millions) quantity (units) (b) Suppose your diagram depicts the situation of an individual firm in the U.S. selling in a global market. Now suppose an earthquake hits firms in China that compete with the U.S. firm in the global market and temporarily removes their output from global supply. Show on your diagram: (i) the effect of the supply disruption on the global market price, (ii) the short-run profit-maximizing output level of the U.S. firm, and (iii) the short-run profit level of the U.S. firm. Initial profit in the long run equilibrium is zero. After the supply disruption, market supply shifts back from S 0 to S 1, raising the market price along the global demand curve from P 0 to P 1. The individual firm responds to the higher global price by increasing short-run output from q 0 to q 1 until the new global price equals marginal cost. Marginal cost exceeds average cost at this price, so profit, which is shown as the shaded region in the right panel of the figure, is positive. 4. (40 points). An individual derives utility from consumption goods, X, and leisure time, N, to maximize the daily utility level, u = XN. Outside income per day is $m and the price of consumption goods is $p. The individual can earn money each day by working at a wage rate of $w per hour, but working of course takes away from leisure time (which implies it causes disutility). The individual s utility maximization problem is constrained by two things: (i) the budget constraint, px = m + wl, and (ii) the time constraint, L+ N = 24. (a) Write a single equation for the budget constraint in terms of N, X, and m and interpret. The utility is constrained by two things: the budget constraint and the time constraint ; px = m + wl, and L + N = 24. This can be written in terms of N, X and prices by substituting in for L = 24 N. Doing so gives: px = m + w(24 N) or 24w + m = px + wn. The left-hand side is potential income, the amount of income the individual could have if she devoted all her time to work. The right-hand side is her total expenditure on goods and leisure, where the price of leisure time is the wage that is foregone when lounging (opportunity cost).
(b) Derive the demands for goods and leisure (X * and N * ) CP: Max. XN s.t px + wn = 24w + m MU X = N; MU N = X MRS = price ratio => MU X / MU N = p/w or N/X = p/w X = N(w/p) Plug this into the budget constraint: 24w +m - wn - wn = 0 => N* = (24w+m) / 2w X* = (24w+m) / 2p (c) Solve for the individual s labor supply function, L * = 24 N *. Graph the labor supply function and verify that labor supply slopes upward in the wage rate. Labor supply: L* = 24 - N* = 24 - (24w+m) / 2w = 24-12 - m/ 2w = 12 - m/2w Notice: (1) If w m/24 (the wage is sufficiently small relative to the individual s outside income), then L* = 0 (since negative labor is not possible) (2) dl* / dw = m / 2w 2 > 0 (L-supply slopes upwards) w L 0 L 1 m 0 /24 Labor (d) What is the effect of an increase in outside income (m) on labor supply? Show the effect of a decrease in m (for instance due to graduation) on the individual s labor supply. dl* / dm = -1 / 2w < 0, so a decrease in outside income (i.e., from m 0 to m 1 < m 0 ) increases labor supply. An increase in labor supply is represented by a rightward shift in the labor supply curve, as more labor is now supplied at every wage. This is shown as a movement from L 0 to L 1. 5. (40 points). Consumer utility for calculators (x) and all other goods (y) is given by the utility function
2 uxy (, ) = 100x 1.5x + y, where p is the price of calculators and the price of all other goods is $1. (a) Compute consumer demand for calculators. 2 CP: Max. uxy (, ) = 100x 1.5x + y s.t px + $1y = I MU x = 100-3x; MU y = 1 MRS = price ratio => MU x / MU y = p/1, or (100-3x)/1 = p/1 100-3x = p (inverse demand) Demand for calculators is: x = (100 - p) / 3 The production function for a firm in the business of calculator assembly is given by x = 5K 0.5 L 0.5 (b) If both inputs are variable and w = $20 and r = $5, derive the cost function. Firm s Problem (FP): min. C = wl + rk s.t. x = 5K L 0.5-0.5-0.5 0.5 MP L = 2.5K L ; MP K = 2.5K L ; MRTS = factor price ratio => MP L / MP K = w/r or K/L = w/r K = L(w/r) = 4L 0.5 0.5 Plug this into the production function: x = 5(4L) L = 10L => L* = x / 10 And, since K* = 4L* K* = 4L* = 4x / 10 The long-run cost function is: c(x) = 20L* + 5K* = 2x + 2x = 4x 0.5 0.5 (c) Using the demand and cost information above, derive the competitive equilibrium quantity and
price in the calculator market. Show the competitive equilibrium on a diagram. Q/$ DWL under Monopoly Pm Pc P*= 4 Qm Q* Q In competitive equilibrium, the firm maximizes: π = px - 4x => FOC: p* = 4 (supply is a horizontal line at p* = 4). The equilibrium quantity is where demand = supply: p = 100-3x = 4 => 3x = 96 => ( x* = 32, p* = 4) (d) If the calculator business is controlled by a monopoly firm, calculate the monopoly price and quantity of calculators in the market. Show the monopoly equilibrium on your diagram above. Shade the region that represents deadweight loss under monopoly. In a monopoly market, monopoly profit is: π = p(x)x - 4x = (100-3x)x - 4x FOC: 100-6x - 4 = 0, or 6x = 96 => x m = 16 The monopoly price is a point on the demand curve: p = 100-3(16) => p = $52 m The deadweight loss is the shaded triangle, which has the area: DWL = ½(32-16)(52-4) = $384. 6. (20 points). Suppose Nokona and Rawlings compete in a Cournot duopoly market for baseball gloves with the cost functions c(q N ) = 5q N + q 2 N for Nokona and c(q R ) = 5q R + q 2 R for Rawlings.
Player s view Nokona and Rawling gloves to be perfect substitutes and inverse demand for baseball gloves is P = 85 Q, where Q = q N + q R is the total quantity of gloves produced by both firms. (a) Calculate the reaction functions, qnr and q R R. Show the equilibrium quantity of gloves produced by each firm as the intersection of two reaction functions on a graph. N okona s Profit: π = [85 q q ]q 5q - q FOC: π N ' = 85 2q N q R 5-2q N = 0 N N R N N N 2 Reaction fu nction: qn R = 20 ¼q R Similarly: q R R = 20 ¼q N q N 80 q R R 20 Cournot equilibrium 16 q N R 16 20 80 q R (b) Find the equilibrium quantity of gloves produced in a Cournot duopoly. What is the market price of a baseball glove in a Cournot duopoly? Cournot equilibrium is the intersection of the reaction functions. q N = 20 ¼[20 - ¼q N ] = 15 + (15/16)q N => q N * = 16 = q B * Market price: P* = 85 Q* = 85 (q N * + q R *) = 85-32 => P* = $53 7. (20 points). Billy John Pigskin of Mule Shoe, Texas has a von Neumann-Morgenstern utility function of uc () = c. Billy John weighs 400 pounds and can outrun jackrabbits and pizza delivery trucks. Billy John is beginning his senior year of college football. If he is not seriously
injured, he will receive a $1,000,000 contract for playing professional football. If an injury ends his football career, he will receive a $10,000 contract as a refuse removal facilitator. There is a 10% chance that Billy John will be injured badly enough to end his career. If Billy John can buy an insurance policy that pays him $1,000,000 if he suffers a career ending injury while in college at a price $p, what is the largest price that Billy John would be willing to pay for such an insurance policy? Derive Billy John s maximum willingness to pay and show this value on a diagram. U(w) U(w) 910 WTP = $171,900 ρ (risk premium) $10K CE = $828.1K E(w) $1m Wealth Suppose BJ elects not to buy insurance. In this case his expected utility is: No insurance: E(u) = 0.90( 1,000,000) + 0.10( 10,000) = 900 + 10 = 910 utils. If BJ elects to buy insurance, he will have $1,000,000 - p for sure, so that his expected utility is: With insurance: E(u) = 1,000,000 p utils BJ should buy insurance if 1,000,000 p > 910 The maximum WTP for insurance is the value of p that satisfies 1,000,000 p = 910 Squaring both sides: 1,000,000 p = 828,100 p* = 1,000,000-828,100 = $171,900 BJ would pay a maximum insurance payment of $171,900 THE END. Have a great winter break!