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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain ECO 204 20082009 Ajaz Hussain HW 6 Solutions In lecture 7, we derived the optimal labor, capital and the long run average cost functions for imperfect substitutes, perfect substitutes and complements technologies for a company that has a target output q and procures labor and capital at prices PL and PK respectively. For your convenience these are summarized below: Imperfect Substitutes Technology: q = L K L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) C(q) = q1/( + ) PL /( + ) PK/( + ) [(/) + (/)]1/( + ) Perfect Substitutes Technology: q = L + K Case 1: If PL/PK < MRTS L = q/ K = 0 C(q) = PL q/ Case 2: If PL/PK > MRTS L = 0 1 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain K = q/ C(q) = PK q/ Case 3: If PL/PK = MRTS Any L and K satisfying: q = L + K C(q) = PLL + PK K Complements Technology: q = min(L, K) L = q/ K = q/ C(q) = q (PL/ + PK/) Question 1 Ajax Corporation has production function q = L1/3 K2/3 and has target output q. Currently, PL = $5 and PK = $10 a) What is Ajax's returns to scale: increasing, constant or decreasing? Answer: From lecture 6 and HW 5 we know that the production function q = L K has constant returns to scale (RTS = 1) whenever + = 1. In this case = 1/3 and = 2/3 implying that Ajax has constant returns to scale. b) From graphical analysis what must be true about the isocost and isoquant slopes? Answer: Because q = L1/3 K2/3 is an imperfect substitutes technology, the isoquants have a "convex shape" (see lecture 6 or 7). In lecture 7, we saw that at the optimal choice of inputs, the isoquant must be tangent to the isocost. The isoquant has slope (see lecture 6 and HW 5): MRTS = (/)(K/L) = (1/3)(2/3)(K/L) = (1/2)(K/L) while the isocost has slope PL/PK = $5/$10 = 1/2. c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. 2 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Answer: We need to calculate optimal L and K. With two unknowns, we need two equations. From lecture 7 we have the first equation from: q = L1/3 K2/3 (Equation 1) Next, at the optimal choice MRTS = PL/PK (see lecture 7). Therefore we have Equation 2: (1/2)(K/L) = 1/2 (Equation 2) (K/L) = 1 (K/L) = 1 K = L There are many ways to solve these two equations. Here is one suggested way. From equation 2, K = L. Substituting in equation 1: q = L1/3 K2/3 q = L1/3 L2/3 q = L Which implies that q = K. The optimal demands for inputs are: L = q K = q d) Now use the formulas above and check if you get the same answer. Answer: From lecture 7 we had the formulas: L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) 3 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Substitute , , PL and PK . L = q1/(1/3 + 2/3) [(1/2)(10/5)]2/3 L = q [(1/2)2]2/3 L = q K = q1/( + ) [(/)(PL/PK)]/( + ) K = q1/(1/3 + 2/3) [2(5/10)]1/2/(1/3 + 2/3) K = q [2(1/2)]1/2 K = q Which confirms our calculations in part (c). e) What is Ajax's long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above. Answer: With L and K as variable inputs, Ajax's cost of producing output q is: C(q) = PL L + PK K Substitute L = q and K = q: C(q) = PL q + PK q C(q) = (PL + PK)q C(q) = (5 + 10)q C(q) = 15 q Let's check if we get the same answer by the formula in lecture 7: C(q) = q1/( + ) PL /( + ) PK/( + ) [(/) + (/)]1/( + ) 4 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain You can substitute , , PL and PK above into a calculator or Excel model to show C(q) = 15q. If you want to use algebra to show that PL /( + ) PK/( + ) [(/) + (/)]1/( + ) "collapses" to 15, here is how: exploiting the fact that in this question + = 1: C(q) = q PL PK [(/) + (/)] C(q) = q PL PK [( / ) + ( /)] C(q) = q PL PK [ + + + )]/ C(q) = q (PL/) (PK/) [ + + + )] C(q) = q (5/(1/3))1/3 (10/(2/3)) 2/3 [(1/3) + (2/3)] C(q) = q (15)1/3 (15) 2/3 C(q) = 15 q f) What is the elasticity of Ajax's long run cost with respect to target output? Interpret this result given your answer in part (a). Answer: We want to calculate: % change in C(q)/ % change in q. You can use: (dC(q)/dq)(q/C(q)) or the following "logs" technique (as we did in consumer theory for constant elasticity demand functions): Now: C(q) = 15 q ln C(q) = ln (15 q) ln C(q) = ln 15 + ln q d ln C(q) / d ln q = 1 {dC(q)/C(q)}/ {dq/q} = 1 (see below for how this step is done) 5 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain % Change in C(q) / % Change in q = 1 Aside: When we did demand functions, we saw that: d ln x = dx/x. To see this again: Suppose: y = ln x dy/dx = 1/x dy = dx/x Since y = ln x we can express dy as d ln x and so: d ln x = dx/x Returning to: % Change in C(q) / % Change in q = 1 This says that if Ajax's target output increases by a certain percentage (say x%), his total cost will also rise by the same percentage (x %). This makes sense because Ajax has constant returns to scale. When desired output increases by x%, all inputs will also rise by the same percentage and therefore total costs will also increase by the same percentage. g) What is the long run average cost? Interpret this result given your answer in part (a). Answer: Average cost AC(q) is total cost/output or C(q)/q: AC(q) = C(q)/q = 15q/q = 15 The average cost is always constant and equal to $15. Again, this makes sense because Ajax has constant returns to scale. When desired output increases, all inputs will also rise by the same percentage and since total costs will also increase by the same percentage, it means the average cost stays constant. h) What is the long run marginal cost? Interpret this result given your answer in part (a). Answer: Marginal cost MC(q) is dC(q)/dq: 6 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain C(q) = 15 q dC(q)/dq = 15 The marginal cost is always constant and equal to $15. This makes sense because we saw that AC is constant and equal to $15. The only way the average can remain constant is if the marginal cost is constant and also equal to $15. You may remember this "rule" from ECO 100: When AC is falling, MC < AC When AC is flat, MC = AC When AC is rising, MC > AC Question 2 Tiesto Corporation has production function q = L1/5 K3/5 and has target output q. Currently, PL = $5 and PK = $10 a) What is Tiesto's returns to scale: increasing, constant or decreasing? Answer: From lecture 6 and HW 5 we know that the production function q = L K has decreasing returns to scale (RTS < 1) whenever + < 1. In this case = 1/5 and = 3/5 implying that Tiesto has decreasing returns to scale. b) From graphical analysis what must be true about the isocost and isoquant slopes? Answer: Because q = L1/5 K3/5 is an imperfect substitutes technology, the isoquants have a "convex shape". In lecture 7, we saw that at the optimal choice, the isoquant must be tangent to the isocost. The isoquant has slope (see lecture 6 and HW 5): MRTS = (/)(K/L) = (1/5)(3/5)(K/L) = (1/3)(K/L) while the isocost has slope PL/PK = $5/$10 = 1/2. c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. Answer: We need to calculate optimal L and K. With two unknowns, we need two equations. 7 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain From lecture 7 we have the first equation from: q = L1/5 K3/5 (Equation 1) Next, at the optimal choice MRTS = PL/PK (see lecture 7). Therefore: (1/3)(K/L) = 1/2 (Equation 2) (K/L) = 3/2 K = (3/2)L There are many ways to solve these two equations. Here is one suggested way. From equation 2, K = (3/2)L. Substituting in equation 1: q = L1/5 K3/5 q = L1/5 ((3/2)L)3/5 q = (3/2)3/5 L4/5 L = {(2/3)3/5}5/4 q 5/4 L = (2/3)3/4 q 5/4 Therefore from K = (3/2)L K = (3/2) (2/3)3/4 q 5/4 K = (3/2) (3/2)3/4 q5/4 K = (3/2)1/4 q5/4 The optimal demands for inputs are: L = (2/3)3/4 q 5/4 K = (3/2)1/4 q5/4 8 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain d) Now use the formulas above and check if you get the same answer. Answer: L = q1/( + ) [(/)(PK/PL)]/( + ) L = q1/(1/5 + 3/5) [((1/5)/(3/5))(10/5)](3/5)/((1/5) + (3/5)) L = q5/4 (2/3)3/4 K = q1/( + ) [(/)(PL/PK)]/( + ) K = q1/(1/5 + 3/5) [((3/5)/(1/5))(5/10)](1/5)/(1/5 + (3/5)) K = q5/4 (3/2)1/4 which confirms our answers in part (d). e) What is Tiesto's long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above. Answer: With L and K as variable inputs Tiesto's cost of producing output q is: C(q) = PL L + PK K Substitute L = q5/4 (2/3)3/4 and K = (3/2)1/4 q5/4 C(q) = PL q5/4 (2/3)3/4 + PK (3/2)1/4 q5/4 C(q) = q5/4 {PL (2/3)3/4 + PK (3/2)1/4 } C(q) = q5/4 {5 (2/3)3/4 + 10 (3/2)1/4 } Leave this expression as it is and check whether we get the same answer by the formula in lecture 7: C(q) = q1/( + ) PL /( + ) PK/( + ) [(/) + (/)]1/( + ) 9 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain C(q) = q1/( + ) {PL PK [(/) + (/)]}1/( + ) C(q) = q5/4 { PL1/5 PK3/5 [(1/3)3/5 + 31/5]}5/4 C(q) = q5/4 { 51/5 103/5 [(1/3)3/5 + 31/5]}5/4 Is this the same as C(q) = q5/4 {5 (2/3)3/4 + 10 (3/2)1/4 }? Yes. You can verify this by direct calculation using a calculator or the Excel model. Here is a long winded "proof" if you want to show it algebraically: We had: C(q) = q5/4 {5 (2/3)3/4 + 10 (3/2)1/4 } C(q) = q5/4 {54/5 (2/3)(3/4)(4/5) + 104/5(3/2)(1/4)(4/5) }5/4 C(q) = q5/4 {54/5 (2/3)3/5 + 104/5(3/2)1/5}5/4 C(q) = q5/4 {23/5 54/5 (1/3)3/5 + 24/5 54/5 31/5 (1/2)1/5 }5/4 C(q) = q5/4 {23/5 54/5 (1/3)3/5 + 23/5 54/5 31/5 }5/4 C(q) = q5/4 {23/5 54/5 [(1/3)3/5 + 31/5]}5/4 C(q) = q5/4 {23/5 53/5 51/5 [(1/3)3/5 + 31/5]}5/4 C(q) = q5/4 { 51/5 103/5 [(1/3)3/5 + 31/5]}5/4 which is exactly what we got from using the formula. f) What is the elasticity of Tiesto's long run cost with respect to target output? Interpret this result given your answer in part (a). Answer: We have C(q) = q5/4 { 51/5 103/5 [(1/3)3/5 + 31/5]}5/4 . Observe that this can be written as: C(q) = q5/4 Constant Applying the log technique: 10 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain ln C(q) = ln (q5/4 Constant) ln C(q) = ln q5/4 + ln Constant ln C(q) = (5/4) ln q + ln Constant d ln C(q)/ d ln q = 5/4 {dC(q)/C(q)}{dq/q} = 5/4 % Change in C(q) / (% Change in q) = 5/4 % Change in C(q) = (5/4) (% Change in q) If Tiesto increases his output by some percentage (say x %) then his long run total cost increases by (5/4)x %. The fact that his cost increases by more than x% is due to the fact that Tiesto has decreasing returns to scale (RTS < 1). As output increases by x%, all inputs must increase by more than x% so that his total costs increase by more than x% as well. g) What is the long run average cost? Interpret this result given your answer in part (a). Answer: We had C(q) = q5/4 { 51/5 103/5 [(1/3)3/5 + 31/5]}5/4 which was written as: C(q) = q5/4 Constant AC(q) = C(q)/q = q5/4 Constant/q AC(q) = q(5/4) 1 Constant AC(q) = q1/4 Constant The average cost increases with output. Again, this makes sense because Tiesto has decreasing returns to scale. When desired output increases, all inputs will rise by a greater percentage and since total costs will also increase by a greater percentage, it means the average cost will increase with output. 11 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain a) What is the long run marginal cost? Interpret this result given your answer in part (a). Answer: We had C(q) = q5/4 { 51/5 103/5 [(1/3)3/5 + 31/5]}5/4 which was written as: C(q) = q5/4 Constant MC(q) = dC(q)/q = (5/4) q(5/4) 1 Constant MC(q) = (5/4) q1/4 Constant The marginal cost increases with output. This makes sense because we saw that AC increases with output. The only way the average can increase is if the marginal cost also increases. In fact, observe that: MC(q) = (5/4) q1/4 Constant > q1/4 Constant = AC(q) Or that MC(q) > AC(q). Put simply, when AC(q) curve is rising, the MC(q) curve will be above the AC(q) curve. But you knew that already from ECO 100 . Question 3 Bob Sinclair Corporation has production function q = (1/5)L + (3/5)K and has target output q. Currently, PL = $5 and PK = $10. a) What is Bob Sinclair's returns to scale: increasing, constant or decreasing? Answer: From HW 5 we know that a perfect substitutes technology: Q = L + K + has: For < 0 increasing returns to scale (RTS > 1) For = 0 constant returns to scale (RTS = 1) For > 0 decreasing returns to scale (RTS < 1) In this case, = 0 and so Bob has constant returns to scale. b) From graphical analysis what must be true about the isocost and isoquant slopes? Answer: From lecture 7, we know that there can be three cases for perfect substitutes technology: If PL/PK < /, L = q/ and K = 0 12 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain If PL/PK > /, L = 0 and K = q/ If PL/PK < /, any L and K satisfying q = L + K In this question = 1/5 and = 3/5 / = (1/5)/(3/5) = 1/3. Now, PL/PK = $5/$10 = . Thus, PL/PK = 0.5 > 0.33 = /. c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. Answer: Since the isocost is steeper than the isocost: L = 0 and K = q/. K = q/ = q/(3/5) = (5/3)q Optimal demands are: L = 0 K = (5/3)q as long as PL/PK > / (what happens when PL/PK < /). d) Now use the formulas above and check if you get the same answer. Answer: From lecture 7, the formula is Case 1: If PL/PK < MRTS L = q/, K = 0, C(q) = PL q/ Case 2: If PL/PK > MRTS L = 0, K = q/, C(q) = PK q/ Case 3: If PL/PK = MRTS Any L and K satisfying: q = L + K, C(q) = PLL + PK K Since we have case 2, L = 0 and K = (5/3)q. e) What is Bob Sinclair's long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above. 13 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Answer: C(q) = PL L + PK K Substitute L = 0 and K = (5/3)q: C(q) = 5(0) + 10 (5/3)q C(q) = 10 (5/3)q C(q) = (50/3)q Let's verify this from the formula: for case 2 we have L = 0, K = q/, C(q) = PK q/ C(q) = 10 (q) / (3/5) C(q) = (50/3)q Which is what we got above. f) What is the elasticity of Bob Sinclair long run cost with respect to target output? Interpret this result given your answer in part (a). Answer: C(q) = (50/3)q ln C(q) = ln (50/3) + ln q d ln C(q)/d ln q = 1 % Change in C(q) /%Change in q = 1 This says that if Bob's target output increases by a certain percentage, his total cost will also rise by the same percentage. This makes sense because Bob has constant returns to scale. When desired output increases, all inputs will also rise by the same percentage and therefore total costs will also increase by the same percentage. g) What is the long run average cost? Interpret this result given your answer in part (a). Answer: C(q) = (50/3)q 14 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain AC(q) = C(q)/q = 50/3 The average cost is always constant and equal to $50/3. Again, this makes sense because Bob has constant returns to scale. When desired output increases, all inputs will also rise by the same percentage and since total costs will also increase by the same percentage, it means the average cost stays constant. i) What is the long run marginal cost? Interpret this result given your answer in part (a). Answer: C(q) = (50/3)q MC(q) = dC(q)/dq = 50/3 The marginal cost is always constant and equal to $50/3. This makes sense because we saw that AC is constant and equal to $50/3. The only way the average can remain constant is if the marginal cost is constant and also equal to $50/3. Question 4 Guetta Corporation has production function q = min((1/3)L , (2/3)K) and has target output q. Currently, PL = $5 and PK = $10 a) What is Guetta's returns to scale: increasing, constant or decreasing? Answer: Guetta's production function is q = min((1/3)L , (2/3)K), which from HW 5 always has constant returns to scale. b) From graphical analysis what must be true about Guetta's optimal choice of labor and capital? Answer: From lecture 7, we know the optimal choice of inputs must on the "ray". The equation of the ray gotten by equating the terms in the min function is: (1/3)L = (2/3)K c) Use your result in part (b) to solve for the optimal L and K without using the formulas above. Answer: We know that at the optimal choice (1/3)L = (2/3)K. Plugging this in q = min((1/3)L , (2/3)K) implies: 15 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain q = (1/3)L and q = (2/3)K These imply: L = 3q K = (3/2)q d) Now use the formulas above and check if you get the same answer. Answer: The formulas state: L = q/ L = q/(1/3) L =3q K = q/ K = q/(2/3) K = (3/2)q Which verifies our answers above. e) What is Guetta's long run cost function C(q) without using the formulas above? Next, verify your answer by using the formulas above. Answer: C(q) = PL L + PK K Substitute L =3q and K = (3/2)q: C(q) = PL 3q + PK (3/2)q C(q) = 5 (3q) + (10) (3/2)q 16 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain C(q) = 15 q + 15 q C(q) = 30q Let's check if we get the same answer from the formulas: C(q) = q (PL/ + PK/) C(q) = q (5/(1/3) + 10/(2/3)) C(q) = q (15 + 15) C(q) = 30q f) What is the elasticity of Guetta's long run cost with respect to target output? Interpret this result given your answer in part (a). Answer: We have C(q) = 30q. Now: C(q) = 30q ln C(q) = ln 30 + ln q d ln C(q) / d ln q= 1 % Change in C(q) / % Change in q = 1 % Change in C(q) = % Change in q This says that if Guetta's target output increases by a certain percentage, his total cost will also rise by the same percentage. This makes sense because Guetta has constant returns to scale. When desired output increases, all inputs will also rise by the same percentage and therefore total costs will also increase by the same percentage. g) What is the long run average cost? Interpret this result given your answer in part (a). Answer: C(q) = 30q AC(q) = C(q)/q = 30 The average cost is always constant and equal to $30. Again, this makes sense because 17 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Guetta has constant returns to scale. When desired output increases, all inputs will also rise by the same percentage and since total costs will also increase by the same percentage, it means the average cost stays constant. h) What is the long run marginal cost? Interpret this result given your answer in part (a). Answer: C(q) = 30q MC(q) = dC(q)/dq = 30 i) The marginal cost is always constant and equal to $30. This makes sense because we saw that AC is constant and equal to $30. The only way the average can remain constant is if the marginal cost is constant and also equal to $30. 18 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto- Toronto.

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