(0.50, 2.75) (0,3) Equivalent Variation Compensating Variation

Transcription

1 1. c(w 1, w 2, y) is the firm s cost function for processing y transactions when the wage of factor 1 is w 1 and the wage of factor 2 is w 2. Find the cost functions for the following firms: (10 Points) a. A firm with production function f (x 1, x 2 ) =min{4x 1, 5x 2 } b. A firm with production function f (x 1, x 2 ) = 7x 1 + 8x 2 c. A firm with production function f (x 1, x 2 ) =max{6x 1, 5x 2 } Ans: a. x 1 = y/4 and x 2 = y/5. Using c(w 1, w 2, y) = w 1 x 1 + w 2 x 2 as the cost function, we substitute for x 1 and x 2 to get w 1 (y/4) + w 2 (y/5) =[ (w 1 /4) + (w 2 /5)]y. b. The firm uses whichever input is less expensive, so min{w 1 /7,w 2 /8}y. c. The firm wants to minimize cost, so min{w 1 /6,w 2 /5}y. Page 1

2 2. Amazin.com has the short-run cost function c(y)= 0.001y + 10,000,000. Write down equations for: (10 Points) a. The firm s average variable cost function b. The firm s marginal cost function c. At what level of output is average variable cost minimized? d. Graph the short-run supply function for this firm, being careful to label the key points on the graph with the numbers specifying the exact prices and quantities at these points. Ans: a. The short run cost function is made up of fixed cost and variable cost. The fixed cost is 10,000,000 and the variable cost is 0.001y. To calculate average variable cost, divide the variable cost by the output, (0.001y)/y to get b. The marginal cost function is the first derivative of the cost function, 0.001y. c. To minimize average variable cost take the first derivative of the answer to part (a) and set it equal to zero and solve for y. The first derivative is 0, average variable cost is a constant, so there is no one value of y. Average variable cost is fixed for all values of y. d. The AVC is a horizontal line at y = 0.001, which is the same as the marginal cost. Page 2

3 3. Barney, an intelligent and strong bull, consumes only two goods, bull feed (made of ground corn and oats) and hay. His preferences are represented by the utility function U(x, y) = x (x 2 /2)+y, where x is his consumption of bull feed and y is his consumption of hay. Barney has been instructed in the mysteries of budgets and optimization and always maximizes his utility subject to his budget constraint. Barney has an income of $m that he is allowed to spend as he wishes on bull feed and hay. The price of hay is always $1, and the price of bull feed will be denoted by p, where 0 < p 1. (30 Points) a. Write Barney s inverse demand function for bull feed. b. If the price of bull feed is p and his income is m, how much hay does Barney choose? c. Find his utility level when the price of bull feed is p and his income is m. d. Suppose that Barney s daily income is $3 and that the price of feed is $.50. What bundle does he buy? e. What bundle would he buy if the price of bull feed rose to $1 and everything else remained the same? f. How much money would Barney be willing to pay to avoid having the price of bull feed rise to $1? g. Another way to state part (f) is what is the least extra income that, at the original prices, just restores Barney s original utility level? This amount is known as the (fill-in the blank). h. Suppose that the price of bull feed rose to $1. How much extra money would you have to pay Barney to make him as well-off as he was at the old prices? i. Another way to state part (h) is what is the least extra income that, at the new prices, just restores Barney s original utility level? This amount is known as the (fill-in the blank). j. Which is bigger, part (f) or part (h), or are they the same and why? k. At the price $.50 and income $3, how much (net) consumer s surplus is Barney getting? Page 3

4 (a) Write Barney s inverse demand function for bull feed. Barney always maximizes his utility subject to his budget constraint, so we can use the Lagrange multiplier method (or substitution) to optimize utility subject to the budget constraint. The utility function is U(x, y) = f(x)+y, where f(x) = x (x 2 /2), and the budget constraint is px + y = m. Then, the Lagrangian is L = f(x)+y - λ(px + y - m). First order conditions (FOC) are: (1) L/ x = f (x) - λp = 0 (2) L/ y = 1 - λ = 0 (3) L/ λ = px + y - m = 0 From equation (2), λ=1, and substituting λ=1 into equation (1) we get that p = f (x). Since f(x) = x (x 2 /2), then f (x) = 1 - x, and then p = 1 - x. We know that with a quasi linear utility, p = f (x), so p = 1 - x. (b) If the price of bull feed is p and his income is m, how much hay does Barney choose? The budget equation is px + y = m. The demand for bull feed is x. Solve part (a) for x, substitute into the budget constraint, and then solve for y. x = 1 p and p(1 p) + y = m, and y = m - p(1 p). (c) Plug these numbers into his utility function to find out the utility level that he enjoys at this price and this income. The utility function is U(x, y) = x (x 2 /2) + y, x = 1 p, and y = m - p(1 p). Substituting gives U(x, y=x (x 2 /2)+y=(1 p) ((1 p ) 2 /2)+ m - p(1 p) =(1 p )(1 (1 p)/2 - p) + m =[(1 p )/2](2 (1 p) - 2p) + m=(1 p ) 2 /2 + m (d) Suppose that Barney s daily income is $3 and that the price of feed is $.50. What bundle does he buy? Substitute these values into x = (1 p), and y = m - p(1 p) to get x = 0.50 and y = (0.50, 2.75) (e) What bundle would he buy if the price of bull feed rose to $1? Substitute these values into x=(1 p), and y=m-p(1 p) to get x=0 and y=3. (0,3) (f) How much money would Barney be willing to pay to avoid having the price of bull feed rise to $1? Barney s utility level at the new bundle, (0,3), is 3 and his utility level at the original bundle, (0.50,2.75), was Therefore, he should be willing to pay the difference, (g) Another way to state part (f) is what is the least extra income that, at the original prices, just restores Barney s original utility level? This amount is known as the Equivalent Variation (fill-in the blank). (h) Suppose that the price of bull feed rose to $1. How much extra money would you have to pay Barney to make him as well-off as he was at the old prices? Barney s utility level at the new bundle, (0,3), is 3 and his utility level at the original bundle, (0.50,2.75), was Therefore, we should take away the difference, (i) Another way to state part (h) is what is the least extra income that, at the new prices, just restores Barney s original utility level? This amount is known as the Compensating Variation_(fill-in the blank). (j) Which is bigger, part (f) or part (h), or are they the same and why? Page 4

5 Clearly, they are the same, but this is because we have quasilinear preferences. (k) At the price $.50 and income $3, how much (net) consumer s surplus is Barney getting? Consumer surplus is the difference in utility, so the difference in utility is Page 5

6 4. A firm has two variable factors and a production function, f(x 1, x 2 ) = (x 1 ) 1/4 (x 2 ) 5/4. The price of its output is 4. Factor 1 receives a wage of w 1 and factor 2 receives a wage of w 2. (30 Points) a. Does this production function have increasing, constant, or decreasing returns to scale? b. Write an equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1. c. Is the marginal product of factor 1 increasing, constant, or decreasing, as the amount of factor 1 is varied? d. Write an equation that says that the value of the marginal product of factor 2 is equal to the wage of factor 2. e. Is the marginal product of factor 2 increasing, constant, or decreasing, as the amount of factor 2 is varied? f. Solve two equations for the two unknowns, x 1 and x 2, to give the amounts of factors 1 and 2 that maximize the firm s profits as a function of w 1 and w 2. g. If the wage of factor 1 is 1, and the wage of factor 2 is 5, how many units of factor 1 will the firm demand? h. If the wage of factor 1 is 1, and the wage of factor 2 is 5, how many units of factor 2 will the firm demand? i. How much output will the firm produce? j. How much profit will the firm make? Page 6

7 a. Does this production function have increasing, constant, or decreasing returns to scale? Since this production function is Cobb Douglas and the exponents add to more than one, we have increasing returns to scale. b. Write an equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1. The marginal product of factor 1 is the partial derivative of the production function with respect to factor 1. f/ x 1 = 1/4(x 1 ) -3/4 (x 2 ) 5/4 The value of the marginal product is the price times the marginal product. 4[1/4(x 1 ) -3/4 (x 2 ) 5/4 ] = (x 1 ) -3/4 (x 2 ) 5/4 = w 1 c. Is the marginal product of factor 1 increasing, constant, or decreasing, as the amount of factor 1 is varied? We want to take the second partial derivative of the production function with respect to factor 1 2 f/ x 2 1 = -3/16(x 1 ) -7/4 (x 2 ) 5/4 which is always negative, so the marginal product of factor 1 is decreasing. d. Write an equation that says that the value of the marginal product of factor 2 is equal to the wage of factor 2. The marginal product of factor 2 is the partial derivative of the production function with respect to factor 2. f/ x 2 = 5/4(x 1 ) 1/4 (x 2 ) 1/4 The value of the marginal product is the price times the marginal product. 4[5/4(x 1 ) 1/4 (x 2 ) 1/4 ] = 5(x 1 ) 1/4 (x 2 ) 1/4 = w 2 e. Is the marginal product of factor 2 increasing, constant, or decreasing, as the amount of factor 2 is varied? We want to take the second partial derivative of the production function with respect to factor 2 2 f/ x 2 2 = 5/16(x 1 ) 1/4 (x 2 ) -3/4 which is always positive, so the marginal product of factor 2 is increasing. f. Solve two equations for the two unknowns, x 1 and x 2, to give the amounts of factors 1 and 2 that maximize the firm s profits as a function of w 1 and w 2. The two equations are the answers from part (b) and part (d). From part (d), 5(x 1 ) 1/4 (x 2 ) 1/4 = w 2, so (x 2 ) 1/4 = (w 2 /5)(x 1 ) -1/4. Rearranging the result from part (b) gives (x 2 ) 5/4 = w 1 (x 1 ) 3/4 or (x 2 ) 1/4 = (w 1 (x 1 ) 3/4 ) 1/5. Then, (w 2 /5)(x 1 ) -1/4 = (w 1 (x 1 ) 3/4 ) 1/5 and solving for x 1 gives x 1 =(w 5 2/(3125w 1 )) 1/2 and x 2 = (w 2 ) 3/2 (5w 1 ) 1/2 g. If the wage of factor 1 is 1, and the wage of factor 2 is 5, how many units of factor 1 will the firm demand? Substitute w 1 = 1 and w 2 = 5 into the result of part (f) to get x 1 = (w 5 2/(3125w 1 )) 1/2 = (5 5 /3125) 1/2 = (3125/3125) 1/2 = 1 Page 7

8 h. If the wage of factor 1 is 1, and the wage of factor 2 is 5, how many units of factor 2 will the firm demand? Substitute w 1 = 1 and w 2 = 5 into the result of part (f) to get x 2 = (w 2 ) 3/2 ](5w 1 ) 1/2 = (5) 3/2 (5) 1/2 = 5 i. How much output will the firm produce? Substitute the values for x 1 = 1 and x 2 = 5 from parts (g) and (h) into the production function to get y = (x 1 ) 1/4 (x 2 ) 5/4 = (1) 1/4 (5) 5/4 = 5(5 1/4 ). j. How much profit will the firm make? Profit is equal to revenue minus cost, so π = py - w 1 x 1 - w 2 x 2 = (4)(5(5 1/4 ).) (1)(1) (5)(5) = 20(5 1/4 )-1-25 = 20(1.5) 26 = 4. Page 8