Summer 2016 ECN 303 Problem Set #1

Transcription

1 Summer 2016 ECN 303 Problem Set #1 Due at the beginning of class on Monday, May 23. Give complete answers and show your work. The assignment will be graded on a credit/no credit basis. In order to receive credit for the assignment, you must demonstrate a good faith effort on each of the questions. I will post correct answers so that you may self-assess in preparing for Exam I. 1. a. Why must an isoquant be downward sloping when both labor and capital have positive marginal products? Explain and illustrate graphically. If the marginal product of labor is positive, then when we increase the level of labor (say from L 1 to L 2 in the graph below) holding everything else constant this will increase total output (Q must be > Q 1 at point B in the graph). To keep the level of output at the original level, we need to stay on the same isoquant. To do so, since the marginal product of capital is positive we would then need to reduce the amount of capital being used to K 2. So, to keep output constant, when the level of one input increases the level of the other input must decrease. This negative relationship between the inputs implies that the isoquant will have a negative slope, i.e., be downward sloping, and that the inputs are substitutable.

2 2 b. Why do isoquants not intersect? Explain and illustrate graphically. Suppose we draw isoquants for two levels of output Q 1 and Q 2 with Q2 Q1. In addition, suppose that these isoquants crossed at some point A as in the following diagram. Capital C B A Q 2 Q 1 Labor Point A implies that a given input combination can produce two different outputs. We define the production function, however, as showing the maximum amount of output that a given input combination can produce. Furthermore, because B is on Q 2 and C is on Q 1, input combination B produces more output than input combination C. This is not possible if the inputs have positive marginal products, however, since point C contains more of both inputs and therefore should achieve a higher level of output. Hence intersecting isoquants are neither consistent with the basic definition of the production function nor with the assumption that the inputs possess positive marginal products.

3 3 2. Consider the production data given in Table 5.2 of the textbook. Quantities of labor are given in the top row, quantities of capital are given in the left-most column, and output levels are recorded in the body of the table. a. Do labor and capital both have positive marginal products? Explain and provide numerical support for your answer. Labor and capital both have positive marginal products in the table. In order to show this pick any column and any row in the table and calculate the difference between one entry and the next to get the change in output due to a 1 unit change in the input (i.e., the marginal product). In the table below, I hold capital equal to 3 and calculate labor s marginal product for that amount of capital. Likewise I hold labor equal to 4 and calculate capital s marginal product for that amount of labor. Both input marginal products are clearly positive. K = 3 L = 4 L Q MP L K Q MP K

4 4 b. Do the inputs exhibit diminishing marginal returns? Explain and provide numerical support for your answer. Yes, both inputs exhibit diminishing marginal returns. In order to show this, hold one input constant and examine what happens to the marginal product of the other input as it is increased. In the table above, with capital fixed at 3 units, labor s marginal product decreases from 7 to 3 as labor is increased. Likewise, with labor fixed at 4 units, capital s marginal product decreases from 8 to 4 as capital usage is increased. c. In a well-labelled graph, draw the isoquant corresponding to an output level of Q = 35. Note that all four input combinations labelled in my graph can be found in the production function table given in the problem set-up. d. Is the isoquant drawn in part c convex to the origin of the graph? Why or why not? Why are isoquants typically convex? Explain. Yes, the isoquant in the graph of part c is convex to the origin of the graph. Isoquants for real world production processes will be convex as long as the inputs in question are subject to the law of diminishing marginal returns. The law of diminishing marginal returns states that the marginal product of an input decreases as more of that input is employed, holding other input usage levels constant. By definition, the absolute value of the slope of the isoquant evaluated at a particular input combination is MRTS = MP L MP K. Movement along the isoquant implies input substitution. Movement down

5 5 the isoquant for example means labor is being substituted for capital. By virtue of diminishing marginal returns, as more labor is combined with less capital, MP L necessarily decreases and MP K necessarily increases. Because its numerator decreases and its denominator increases, the MRTS must decrease. Hence, the slope of the isoquant decreases as labor is substituted for capital, which implies that the isoquant is convex to the origin. 3. Consider the following Cobb-Douglas production function: Q L K a. If L = 10 and K = 15, how much output is produced? Q = 500(10).6 (15).8 = 17, b. If L = 10 and K = 15, what is MP L? What is MP K? In a Cobb-Douglas production function, MP L = α AP L and MP K = β AP K. Given L= 10, K = 15, and Q = 17, , labor's marginal product is MP L =. 6 ( 17, ) = 1, and capital's marginal product is MP 10 K =. 8 ) = ( 17, c. Given L = 10 and K = 15, does the law of diminishing marginal returns hold for labor? Why or why not? We know from b that MP L = 1, at {L = 10 & K = 15}. So what happens to labor s marginal product as we increase labor and hold capital fixed? Labor s marginal product at {L = 11 & K = 15}, for example, is MP L =. 6 ( 18,394.1 ) = 1, which is a bit less than MP 11 L at {L = 10 & K = 15}, so the marginal product of labor is decreasing as labor is increased and capital is held fixed. [Note we can generalize this result as follows. Consider a general Cobb- Douglas production function: Q = AL α K β. Using calculus, the marginal product of labor is MP L = Q = L αalα 1 K β. The derivative of MP L with respect to labor usage tells us how the marginal product of labor changes as the amount of labor is increased, holding capital constant. If this derivative is negative, then, by implication, diminishing marginal returns prevails. So, taking the derivative of labor s marginal product: MP L = (α 1)αAL α 2 K β L This derivative must be less than 0 if 0 < α < 1. So it turns out that, given a Cobb-Douglas production function, diminishing marginal returns always holds for an input as long as its exponent in the production function is

6 6 between 0 and 1. Labor s exponent in this problem is 0.6, which is less than 1 and more than 0 and therefore implies that labor s marginal product decreases as labor is increased while capital is held constant.] d. Given L = 10 and K = 15, what is the marginal rate of technical substitution? We showed in class that, given a Cobb-Douglas production function, MRTS = ( α ) β (K ). Therefore, with 10 units of labor and 15 units of capital L the MRTS = (.6 ).8 (15 ) = An alternative approach to answering 10 this question would be to note that regardless of the type of production function, the MRTS = MP L by definition. Taking the values determined in MP K part b, MRTS = = e. Consider an alternative input combination where L = 20 and K = Is this input combination on the same isoquant as input combination {L = 10, K = 15}? Why or why not? Q = 500(20).6 ( ).8 = 17, Both the input combination in a and the input combination in e produce the same amount of output, hence both input combinations are on the same isoquant. f. What type of returns to scale does this production function exhibit? Why? Returns to scale are easily evaluated in a Cobb-Douglas production function. As we showed in class, all we need to do is add the input exponents together and compare with 1. If the sum exceeds 1, there are increasing returns to scale. If the sum equals 1, there are constant returns to scale. If the sum is less than 1, there are decreasing returns to scale. In this problem, α + β = = 1. 4 > 1. Therefore, production exhibits increasing returns to scale in this problem. g. Double the input amounts used in part a and calculate the resulting percentage change in output. What is the returns to scale elasticity here? First note that doubling the input amounts from part a implies Q = 500(20).6 (30).8 = 45, The returns to scale elasticity is defined as RTS ε = % Q % Inputs. % Q = ( ) 100 = Since we doubled the inputs, the percent change in the inputs is 100%. Hence the returns to scale elasticity is RTS ε = 163.9% = , which confirms the 100% assertion in part f that the production function exhibits increasing returns to scale.

7 7 4. Suppose that TC 0 = $2000, w = $90, and r = $60. a. Graph the isocost line corresponding to a total cost of $2,000. Be sure to label the axes and points of interest such as intercepts in the graph appropriately. b. The input price ratio, w/r, is a possible rate of input substitution. True or false? Explain. True. The input price ratio tells us the amount of capital that can replace 1 unit of labor with no change in total cost. Likewise the input price ratio tells us how much less capital must be used in order to afford employment of one more unit of labor with no change in total cost. In short, the input price ratio is the input substitution rate at which total cost is constant. In the example, 1.5 units of capital may be substituted for 1 unit of labor, and vice versa, with no change in total cost. c. If the price of labor and the price of capital both increase by 10%, how does that affect the isocost line corresponding to total cost of $2,000? Explain and illustrate graphically. If both input prices increase by 10%, then this causes the $2,000 isocost line to shift in toward the origin. The slope of the line remains the same because $99 = $90 = $66 $60

8 8 1. 5, thus the input substitution rate at which total cost remains the constant has not changed. This should seem reasonable because, while both input prices increased, the relative expense of the two inputs remained unchanged. The inward shift of the new isocost line reflects that less of both inputs can be purchased with two thousand dollars when both inputs become more expensive. d. If only the price of labor increases by 10%, how does that affect the isocost line for TC 0 = $2000? If only the price of labor increases by 10%, this reduces the horizontal intercept of the isocost line to 20.2 and increases the slope of the isocost line to w = $99 = These r $60 changes cause the isocost line to rotate down from its original position as in the graph below.

9 9 In this scenario, the higher price of labor means that less labor and capital can be afforded with $2000 and that the rate at which capital can be substituted for a unit of labor with no change in total cost has increased to A firm operates with a technology that is characterized by a standard set of negativelysloped, convex isoquants. At the current level of production, labor s marginal product is 20 and capital s marginal product is 10. A unit of labor costs w = $15 per hour while a unit of capital costs r = $12 per hour. Is the firm producing its current level of output at minimum cost? If yes, explain why. If no, show why not and indicate whether the firm should be using (i) more capital and less labor, or (ii) less capital and more labor. Long run cost minimization requires that inputs be employed such that MRTS = w r The input price ratio in the problem is w r = $15 = $12 In order to obtain the MRTS at the current input combination, recall the economic definition of the MRTS, i.e., MRTS = MP L MP K We are given the information that at the current production level MP L = 20 and MP K = 10. Hence MRTS = MP L = 20 MP K 10 = 2

10 10 Bringing together the MRTS and the input price ratio for comparison, 2 = MRTS > w r = This tells us that at the current input combination labor is twice as productive as capital yet only 25% more expensive than capital. Per class discussion, the firm should substitute labor for capital at the rate of the MRTS. For example, substitution of 1 unit of labor for 2 units of capital, by definition of the MRTS, maintains production at the current level but reduces cost by $9 (the extra unit of labor costs $15 but 2 fewer units of capital saves $24 (= r K = $12 ( 2)) in capital expense for a net cost reduction of $9). The firm should substitute labor for capital at the rate of the MRTS as long as MRTS > w r As more labor and less capital is employed, the MRTS decreases (due to diminishing marginal returns). Once the firm achieves the input mix at which MRTS = the firm will have minimized its cost of production. 6. The XYZ Corporation uses two homogeneous inputs labor and capital in the production of its product. A unit of labor costs XYZ $80 per day and a unit of capital costs them $60 per day. Presently, XYZ wishes to produce 200 units of output per day. Efficiency experts at the company have determined that the cost minimizing input combination for the company at this production level is L = 8 and K = 12. a. Provide a graph depicting the 200 unit isoquant and the isocost line that the firm is on at the cost minimizing input combination. Explicitly identify the cost minimizing combination of labor and capital in the graph. Calculate the total cost of production using the current input combination and label the actual numerical values of the isocost line intercepts in your graph. First, let s work out the total cost implied at this input combination. TC 0 = wl 0 + rk 0 = ($80 8) + ($60 12) = $1, 360. Next, determine the intercepts of the isocost line. Accordingly, TC 0 TC 0 w below. = $1360 $80 r = $1360 $60 = gives the vertical intercept and = 17 gives the horizontal intercept. The appropriate graph is given

11 11 b. What does the marginal rate of technical substitution equal at input combination {L = 8 and K = 12}? Explain. In this problem, we don t have enough information to calculate the MRTS directly, but we do know that if input combination A~{L = 8, K = 12} does minimize cost, then it must be the case that MRTS = w at input combination A. Given that r w = $80 and r = $60, w = $80 = Therefore, the MRTS must equal 1.33 at r $60 the input combination {L = 8, K = 12}. c. Suppose that workers at the company demand a 35% wage increase (raising daily, per unit labor cost to $108). What is the total cost of producing 200 units of output using {L = 8 and K = 12} now? Show graphically what happens to the isocost line going through {L = 8 and K = 12} after the change in the price of labor (be sure to provide the numerical values of the isocost line s new intercepts). Given the wage increase to w = $108, the initial input combination now costs out at TC 1 = w L 0 + rk 0 = ($108 8) + ($60 12) = $1, 584. By implication, the intercept values of the iso-cost line containing input combination A change to TC 1 r = $1584 $60 = and TC 1 w = $1584 $108 = The dotted line in the graph shows the isocost line passing through input combination A at input prices w = $108 and r = $60.

12 12 d. Illustrate graphically the substitution effect that the change in the price of labor has on the usage of labor and capital. What will ultimately happen to the total cost of production once optimal substitution has taken place? The increase in the price of labor breaks the equality between the MRTS and the input price ratio that prevailed in parts a and b. Specifically, = MRTS A < w r = $108 $60 = This says that labor s productivity is 33% greater than capital s productivity at the margin but labor s expense is 80% greater than that of capital. With the inequality running in this direction, the firm should substitute capital for labor at the rate of the MRTS. So, for example, if a unit of labor is replaced with 1.33 units of capital, production is unchanged at Q = 200, but total cost is reduced by -$28.20 (i.e., TC = w L + r K = ($108 ( 1)) + ($60 (1. 33)) = $108 + $ = ). As the firm substitutes capital for labor in this fashion, MP L increases and MP K decreases due to diminishing marginal returns all of which causes the MRTS to increase towards 1.8 in the process. The firm should continue to substitute capital for labor until the MRTS is driven back into equality with the input price ratio, as at point B in the graph below. Once point B is reached, the total cost of producing Q = 200 can be reduced no further. Using the vertical intercepts in the graph below we can deduce that $1360 = TC 0 < TC 2 < TC 1 = $1584. This tells us that some of the increased cost transmitted via the wage increase can be avoided by the firm via substitution away from labor and toward capital. Nevertheless, the original total cost value of $1360 can no longer be achieved, even with the optimal input adjustment. This implication should seem intuitively reasonable. That is, if the firm had been minimizing total cost in the first

13 13 place at the lower wage, then when the wage increases with no offsetting change in the price of capital, total cost must necessarily increase by some degree. Input adjustment/response to the price change allows the firm to avoid some but not all of the cost hit of the wage increase. The graph reveals the substitution effect for labor to be L = L 1 8 < 0 and the substitution effect for capital to be K = K 1 12 > 0.