ECON 311 Winter Quarter, 2010 NAME: KEY Prof. Hamilton

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1 ECON 311 Winter Quarter, 2010 NAME: KEY Prof. Hamilton FINAL EXAM 200 points 1. (30 points). A firm produces rubber gaskets using labor, L, and capital, K, according to a production function Q = f(l,k). The wage rate for labor is w and the rental rate for capital is r. (a) Write out (or describe in words) the condition which must hold at a cost-minimizing input mix. Briefly provide intuition on why this must be so. Show this outcome on a diagram. The condition that must hold at a cost-minimizing input mix is that the MRTS = price ratio, MP MP L K w =, r which implies that the marginal productivity per dollar is equated across all factors of production. If this were not the case, the firm could reduce cost by exchanging units of the factor with lower marginal product per dollar for units of the factor with higher marginal product per dollar. The firm s cost-minimizing input mix is shown as the intersection between the isocost line C 1 and the isoquant Q 1. The cost-minimizing input mix is shown as the values L c and K c. K K c K * Q 1 L c L * C 1 L (b) Suppose the EPA finds that capital equipment used to produce rubber gaskets causes water pollution that harms human health, while labor does not pollute. EPA economists estimate that the true social cost of employing a unit of capital (inclusive of pollution damages), z, is twice as high as the rental rate on capital; or z = 2r. What condition must hold at a social cost-minimizing input mix that accounts for water pollution costs? Show the socially optimal input mix on your diagram and draw the socially optimal expansion path as the firm s output level expands. The socially optimal outcome is the tangency between the dotted line and the isoquant. The firm s cost under the environmental policy is higher (the lower dotted line, which reflects the original cost level under the higher capital price is no longer in contact with the isoquant); however, the new cost level is less than twice as high, because the firm responds to the higher capital price by substituting away from capital and using more labor.

2 2. (40 points). A Cal Poly student consumes 10 fish tacos and 10 slices of pizza per week. The current market price for fish tacos in San Luis Obispo is $1 per taco and the current market price for pizza slices is $1 per slice. Suppose the price of fish tacos rises to $4 per taco when Cabo San Luis burns down. (a) (15 points) If the student s parents contribute an extra $30 per week for the student s food budget in response to the rise in fish taco prices, is the student better off, worse off, or equally well off under the new arrangement? Provide a graph to support your argument. The student cannot be worse off. Her income is adjusted upwards by the parental contribution so that she can still afford to buy 10 fish tacos and 10 pizza slices with the extra $30. It turns out that the student will be better off, and the key to seeing this is to think of substitution effects. Fish tacos are now relatively more expensive than pizza slices, so the student will substitute away from buying fish tacos and towards greater consumption of pizza slices. To see this, consider the following diagram. In the graph, X is fish tacos and Y is pizza slices. Notice that relative prices change after the price of fish tacos rises. The student s old budget line was B 1, their original bundle was point A (10 fish tacos and 10 pizza slices). The student s original utility level was U 1. The new budget line (B 2 ) the dotted line-- is adjusted to allow the student to continue buying bundle A, so it also goes through point A. Notice that relative prices have changed, however, so the slope of B 2 is different than the slope of B 1. Therefore, the tangency between MRS and the (new) price ratio no longer occurs at point A, and the student will adjust by shifting consumption towards the (relatively) lower priced good Y, hence obtaining the higher indifference curve U 2 the dashed curve. Y A U 2 B 2 B 1 U 1 X

3 (b) (20 points). If the student s utility function is given by U = xy, where x is fish tacos per week and y is pizza slices per week, confirm your answer to part (a) by calculating the additional amount of money the student would need each week after the rise in fish taco prices to maintain her original utility level. This problem can be solved by finding either indirect utility or the expenditure function. To find the expenditure function, fix a target utility level of U* = XY, and minimize the expenditure E = PxX + PyY required to obtain U* The Langrangian is: L = PxX + PyY + λ( U* - XY) FOC: L/ X = Px - λy = 0 (1) L/ Y = Py - λx = 0 (2) L/ λ = U* - XY = 0 (3) Next, we solve these three equations simultaneously. Equate (1) and (2): P X / Y = P Y / X Y / X = (P X / P Y ) This is the familiar tangency condition that MRS = ratio of prices Solve for Y: Y = X(P X / P Y ) (4) We can now Substitute equation (4) into the budget constraint given in equation (3) to find the compensated demand for x and y: U* - XY = 0 U* - X 2 (Px / Py) = 0 X 2 = U*Py / Px X c = (U*Py / Px) 0.5 Y = X(P X / P Y ) Y c = (U*Px / Py) 0.5 The expenditure function is found by plugging the compensated demands back into the objective function, which gives the minimum expenditure necessary to achieve a particular utility level as E* = PxX c + PyY c = Px(UPy / Px) Py(UPx / Py) 0.5 = (UPxPy) (UPx Py) 0.5 = 2(UPxPy) 0.5 To answer the question, we first recover the original number of utils, U*: Since X c = Y c = 10 and Px = Py = $1, we can plug these values into the compensated demand for either good to get: X c = 10 = (U*) 0.5 => U* = 100 (alternatively, U* = X c Y c = 10*10 = 100) The expenditure level necessary to attain U*=100 at the new fish taco price is E 1 * = 2(UPxPy) 0.5 = 2(100*1*4) 0.5 = $40. The original expenditure level necessary to attain U*=100 at the old prices was: E 0 * = 2(UPxPy) 0.5 = 2(100*1*1) 0.5 = $20. (Alternatively, 10 tacos at $1 each + 10 slices at $1 each = $20) The student only needs $20 more per week food allowance to maintain U* = 100. Receiving compensation of $30 per week, the student is better off. 2(c). (5 points). The welfare measure you calculated in 2(b) is called COMPENSATING VARIATION

4 3. (30 points). A competitive firm has the cost function c(q) = 5q + 2q (a) Calculate the optimal production level (q * ) and profit level for the firm if the current market price is p = $65 per unit. Profit: π = pq c(q) = 65q 5q 2q => FOC: p c = 0 (or q = 0) => 4q = 60 => q* = 15 The firm s equilibrium profit level is: π* = 65(15) 5(15) 2(15) = $250 (b) Draw a graph that shows marginal cost, average cost, the market price and profit for the firm. $/x 65 MC AC 45 profit Q mes = 10 Q* = 15 Quantity (c) Calculate the long-run competitive equilibrium price level. In the long run competitive equilibrium, P = MC and P = AC (zero profit). The point where MC = AC is the minimum efficient scale of the firm, q MES. Since AC = c(q)/q = 5 + 2q + 200/q q MES solves: MC = 5 + 4q = 5 + 2q + 200/q => 2q = 200/q => q 2 = 100 => q MES = 10 The market price solves P=MC=AC, or P LR = 5 + 4(10) = $45

5 4. (30 points) A firm produces two different goods x and y with cost function c(x, y) = 0.5x 2 + y 2 - xy + 6. The firm is perfectly competitive and faces prices p x for good x and p y for good y. (a) (5 points) Does c(x, y) exhibit economies of scope? Define economies of scope in the context of your answer. Economies of scope exist when producing two goods together is cheaper than producing each good separately. For example, if goods x and y were produced separately, the cost would be: C(x,0) = 0.5x 2 + 6; C(0, y) = y 2 + 6; The total cost of producing the goods independently would be C(x,0) + C(0,y) = 0.5x 2 + y Economies of scope: C(x, y) < C(x,0) + C(0,y), since 0< xy. (b) (20 points) Derive a supply for goods x and y. 2 2 Total profit from sales of both x and y is π = Px+ Py 0.5x y + xy 6 FOC: x y π = P x+ y = 0 (1) x x π = P 2y+ x= 0 (2) y y Solving simultaneously, write (1) as x = P x + y and plug this into (2): P y -2y + P x + y = 0. y* = P x + P y Now use this in (1) or (2) to recover x*: x* = P x + y*, or x* = 2P x + P y (c) (5 points) Supply for good x in part (b) depends on both p x and p y. Does an increase in p y increase or decrease supply of good x? Explain how your answer to this question relates to your answer to part (a) on economies of scope. An increase in P y increases the firm s supply of good x. The reason for this is that an increase in the price of good y causes the firm to supply more of good y. Because the cost function involves economies of scope, there are synergies in producing both goods together. Indeed, the marginal cost of producing good x is MC x = x y, so that producing more of good y lowers the marginal cost of producing good x. This means that more of good x will now be produced at any P x.

6 5. (40 points). A monopoly firm produces in an industry facing input prices for capital and labor given by r = $5 and w = $5, respectively. The firm s production function is: Q= 2K L (a) (20 points) Derive the firm s cost function, c(q). To minimize cost in the long-run, MRTS = ratio of input prices MP MPK = 0.5K L and MPL = 0.5K L K 0.5K L, so that the MRTS = = = MPL 0.5K L Equating this with the ratio of input prices: MPK r L 5 = = => MPL w K 5 K = L Use of this in the production function gives: Q = 2(L) 0.25 L 0.25 = 2L 0.5 => L 0.5 = Q/2 So L* = Q 2 /4 L K Substituting back in for L gives: K* = L* => K* = Q 2 /4 The long-run cost function is c(q) = wl* + rk* = 5(Q 2 /4) + 5(Q 2 /4) = 5Q 2 /2 => c(q) = 2.5Q 2 (b) If (inverse) demand for the monopolist s product is given by p = 150 5Q, calculate the monopoly output level and price. Monopoly profit is: π = (150 5Q)Q 2.5Q 2 FOC: dπ/dq = Q - 5Q = 0 => 15Q = 150 => Q m = 10 Monopoly price: p m = 150-5(10) = $100

7 (c) Provide a diagram that shows the monopoly output level and price. If the monopolist was a pricetaking competitive firm, calculate the market output level and price and show the values on your diagram. Shade the region the represents excess burden (or deadweight loss) under monopoly. Competitive firm: π = pq - 2.5Q 2 => FOC: p - 5Q = 0 => p = 5Q Market equilibrium (S=D): p = 150-5Q = 5Q => 10Q = 150 => Q* = 15 P* = 5(15) = $75 Q/$ DWL under Monopoly MC = 5Q P m = $100 P c =$75 Q m =10 Q* =15 Q 6. (30 points). Mike Tyson has a logarithmic utility-of-wealth function, u = lnw and initial wealth of W 0 = $5 million. He considers gambling at the dog track on Brutus, who he estimates has a 75% change of winning the race. (a) Calculate the wager, X, that maximizes Mike s expected utility if he stands to win $X if Brutus wins the race and lose -$X if Brutus does not win the race. Expected utility (with all values expressed in millions of $s) is Eux ( ( )) = 0.25ln(5 x) ln(5 + x) The size of the gamble (x*) that maximizes expected utility solves: de( u( x)) 1 1 FOC : = = 0 dx (5 x) (5 + x) Solving for x*: 3(5-x) = 5+x => x* = $2.5 million.

8 (b) Calculate the certainty equivalent (W c ) of a gamble involving a $5 million wager on Brutus. Show the expected value of the gamble, the certainty equivalent, and the risk premium on a diagram. E(W) = 0.25(5-4) (5 + 4) = 0.25(1m) +.75(9m) = $7.0 million E(u(W)) = 0.25ln(1m) ln(9m) = = W c : ln(w c ) = => $5,196,152 Risk Premium: ρ = = $1,803,847 U(w) U(w) ρ = $1.804m $1m W c = $5.196m E(W)=$7m $9m Wealth THE END. Have a great Spring Break! KEY