1. Suppose a production process is described by a Cobb-Douglas production function f(v 1, v 2 ) = v 1 1/2 v 2 3/2.

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1 1. Suppose a production process is described by a Cobb-Douglas production function f(v 1, v 2 ) = v 1 1/2 v 2 3/2. a. Write an expression for the marginal product of v 1. Does the marginal product of v 1 increase, decrease, remain constant for small increases in v 1, holding v 2 fixed? b. Write an expression for the marginal product of v 2. Does the marginal product of v 2 increase, decrease, remain constant for small increases in v 2, holding v 1 fixed? c. Does an increase in the amount of v 2 used in production increase, leave unchanged, or decrease the marginal product of v 1? d. Write an expression for the marginal rate of technical substitution (MRTS). Does this technology exhibit a diminishing MRTS? e. What kind of returns to scale does this production function exhibit? 2. You manage a crew of 160 workers who could be assigned to make either of two products. Product requires 2 workers per unit of output. Product requires 4 workers per unit of output. a. Write an equation to express the combinations of products and that could be produced using exactly 160 workers. On the diagram below, use blue ink to shade in the area depicting the combinations of and that could be produced with 160 workers. (ssume that it is also possible for some workers to do nothing at all.) b. Suppose now that every unit of product that is produced requires the use of 4 shovels as well as 2 workers and that every unit of products requires 2 shovels and 4 workers. On the graph you have just modified, use red ink to shade in the area depicting combinations of and that could be produced with 180 shovels if there were no worries about the labor supply. Write down an equation for the set of combinations of and that require exactly 180 shovels. c. If you have 160 workers and 180 shovels, what is the largest amount of product that you could produce? If you produce this amount, you will not use your entire supply of one of the inputs. Which one? How many will be left unused? 3. Hydraulics Ltd. has designed a pipeline that provides a throughput of 70,000 gallons of water per 24- hour period. If the diameter of the pipeline were increased by 1 inch, throughput would increase by 4,000 gallons per day. lternatively, throughput could be increased by 6,000 gallons per day using the original pipe diameter with pumps that had 100 more horsepower. a. Estimate the marginal rate of technical substitution between pump horsepower and pipe diameter. b. Provide a verbal interpretation of the MRTS: If pipe diameter were reduced by 1 inch, how much would horsepower have to be increased in order to keep output (or throughput ) constant? c. ssuming the cost of additional pump size is $600 per horsepower and the cost of larger diameter pipe is $200,000 per inch, does the original design exhibit the property required for optimal input combinations? If so, why? If not, how would you change it and why?

2 4. firm uses a single input to produce a commodity according to the production function f(v) = v 1/2. The commodity sells for $100 per unit. The production input costs $50 per unit. a. Write out the firm s profit function. What is the profit maximizing level of output? How many units of the input factor will the firm use at this level of output? How much profit does it make when it maximizes profits? b. Suppose that the firm is taxed $20 per unit of its output and the price of its input is subsidized by $10. Revise the firm s profit function to reflect these new realities. What is the profit maximizing level of output? How many units of the input factor will the firm use at this level of output? How much profit does it make when it maximizes profits? c. Suppose that instead of the unit taxes and subsidies, the firm is taxed at 50% of its profits. Write out the firm s after-tax profit function. What is the profit maximizing level of output? How many units of the input factor will the firm use at this level of output? How much profit does it make when it maximizes profits? 5. profit-maximizing firm produces one output, x, and uses one input, v, to produce it. The price per unit of the input factor is denoted by w and the price of the output is denoted by p. You observe the firm s behavior over three production periods, and note the following: Period x v w p In the diagram below, draw an isoprofit line for each of the three periods, showing combinations of input and output that would yield the same profits that period as the combinations actually chosen. Write out the equations for these three lines: Period 1: Period 2: Period 3: Shade in the region on the graph that represents the feasible production area as far as one can deduce from the available evidence.

3 6. production process is described by a Cobb-Douglas production function f(v 1, v 2 ) = v 1 1/2 v 2 1/2. The price of its output is 4. The price of the first input factor is w 1 and the price of the second input factor is w 2. a. Write the two equations that say that the value of the marginal product of each factor is equal to its wage. b. The two equations in part (a) imply that if w 1 = 2w 2, then the optimal input ratio, v 1 /v 2, is. 7. Suppose that for a given production process and at the current levels of input usage, the marginal product of labor is 8 and the marginal product of capital is 2. If the wage rate is $4 and the price of capital is $2, is the firm using the cost minimizing capital and labor mix? If not, how should it change the input mix? 8. Suppose that the production function for a competitive firm is Q = K 0.5 L 0.5. The firm sells its output at a price of $10, and can hire labor at a wage of $5. Capital is fixed at 25 units. What is the profitmaximizing quantity of labor? 9. Curiously, two old friends, rchie and arney, have found themselves as the only two consumers in a little exchange economy which has only two commodities cheese and wine. rchie's initial endowment of these two commodities is (C, W ) = (3,2), and arney's initial endowment is (C, W ) = (1,6). They have identical utility functions of the form U i (C i,w i ) = C i *W i, where the subscripts denote the individual, using for rchie and for arney. a. Modify the Edgeworth box below to illustrate this initial allocation, and to illustrate one indifference curve for each of them, passing through this initial allocation. Use blue ink for rchie's indifference curve and red ink for arney's. b. t any Pareto optimal allocation where both consume some of each good, their marginal rates of substitution must be equal. Write an expression that states this condition in terms of the consumption of each good by each person. c. On the diagram, show the locus of points that are Pareto efficient (the contract curve). d. In this example, at any Pareto efficient allocation, where both persons consume both goods, the slope of rchie's indifference curve will be. Therefore, since we know that competitive equilibrium must be Pareto efficient, we know that at a competitive equilibrium, p p C W, e. What are rchie's and arney's consumption bundles in a competitive equilibrium?

4 Wine Cheese

5 10. Consider a pure exchange economy with two agents, and, and two goods, x and y, where the utility functions of the agents are nd the initial endowments of the two goods are U (x,y) = x 0.4 y 0.6 U (x,y) = x 0.6 y 0.4 (x,y) = (30, 20) (x,y) = (20, 30) a. Derive and describe the Pareto efficient equilibrium of this economy under these conditions. b. Construct an Edgeworth box that illustrates your answer in part (a). c. Derive and describe the Pareto efficient equilibrium of this economy if the only change to the above conditions is that s preferences are the same as s: U (x,y) = x 0.6 y 0.4 U (x,y) = x 0.6 y Charlotte and Wilber are two agents in a two-agent, two commodity pure exchange economy where apples and bananas are the two commodities. Charlotte loves apples and hates bananas. Her utility function is, where a is the number of apples she consumes and b in the number of bananas she consumes. Wilber likes both apples and bananas. His utility function is. Charlotte has an initial endowment of no apples and 8 bananas. Wilber has an initial endowment of 16 apples and 8 bananas. a. On the graph below, mark the initial endowment and label it E. Use red ink to draw the indifference curve for Charlotte that passes through this point. Use blue ink to draw the indifference curve for Wilbur that passes through this point. b. If Charlotte hates bananas and Wilber likes them, how many bananas can Charlotte be consuming at a Pareto optimal allocation? On the diagram, use black ink to mark the locus of Pareto optimal allocations of apples and bananas between Charlotte and Wilbur. c. We know that a competitive equilibrium allocation must be Pareto optimal and the total consumption of each good must equal the total supply, so we know that at a competitive equilibrium, Wilbur must be consuming how many bananas? If Wilbur is consuming this number of bananas, his marginal utility for bananas will be and his marginal utility of apples will be. If apples are the numeraire, then the only price of bananas at which he will want to consume exactly 16 bananas is. In competitive equilibrium for this Charlotte-Wilbur economy, Wilbur will consume bananas and apples, while Charlotte will consume bananas and apples.

6 d. This problem illustrates something about one of the two welfare theorems discussed on pages of Varian s Intermediate Microeconomics. What does it illustrate? 12. Suppose the utility possibility frontier for two individuals is given by U + 2U = 200. On the diagram below, plot the frontier. W U U = max U, U a. In order to maximize a Nietzschean social welfare function, ( ) { } on the utility possibility frontier shown in the diagram, one would set = U =. W U U = min U, U U and = b. If instead we use a Rawlsian criterion, ( ) { } function is maximized on the diagram where = c. Suppose that social welfare is given by ( ) would be maximized when =,, U and,, then the social welfare U. W U, U = U U U and =. In this case, social welfare U.

7 , y = x 2 + y 13. Roger and Gordon have identical utility functions, ( ) 2 10 units of y to be divided between them. U x. There are 10 units of x and a. Draw an Edgeworth box showing some of their indifference curves (Roger = blue, Gordon = red) and mark the Pareto Optimal allocations with black ink. What is it about their preferences that generates a peculiar outcome? b. What are the fair allocations in this case?

8 14. Carefully modify the diagram below so that it provides a clear representation of a general equilibrium for a two-agent (, ), two-good (x, y) economy with the following characteristics: The economy s PPF is concave and passes through the points (0, 16), (10, 12), (15, 10), (18, 5) and (20, 0). p y /p x = gent s utility function is U ( x, y) = x y. U x, y = x y gent s utility function is ( ) Your modified diagram should include the following: a. The PPF b. marginal rate of transformation line. c. community indifference curve. d. contract curve tracing out the Pareto set and reflecting the Cobb-Douglas preferences given above. e. consumption equilibrium that is fair in Varian s sense of the word (p. 639). f. Consumer indifference curves at equilibrium. g. Consumer terms of trade (or budget) line.