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Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 7 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. Question 1 Ajax Corporation has production function q = L1/3 K2/3 and has target output q. Currently, PL = $5 and PK = $10. Please note that PK can be either the price of leasing capital or using capital (aka "user cost of capital"). (a) Calculate the MRTS and the slope of the isocost curve. Answer: The isoquant has slope: MRTS = (/)(K/L) MRTS = (1/3)(2/3)(K/L) MRTS = (1/2)(K/L) Now, the isocost has slope: PL/PK = $5/$10 PL/PK = 1/2 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) Solve for the optimal L and K. You may want to check your answers with the "formulas" derived in lectures for q = L K the CMP yielded: L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) Answer: The CMP is: choose L and K to min cost of producing target output q: L = PL L + PK K [L1/3 K2/3 q] The FOCs are: L/L = 0 PL (1/3) L2/3 K2/3 = 0 PL = (1/3) L2/3 K2/3 L/K = 0 PK (2/3)L1/3 K1/3 = 0 PK = (2/3)L1/3 K1/3 L/ = 0 L1/3 K2/3 = q We need to calculate optimal L and K with two unknowns, we need two equations. The first equation comes from the constraint: q = L1/3 K2/3 ... (1) Dividing the first and second FOCs gives us equation 2: (1/2)(K/L) = 1/2 .. (2) This of course says: at the optimum, the slope of the isoquant is tangent to the slope of the isocost line. Simplifying equation (2): 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (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. Note how you'd get the same result had the production function been Q = min(L, K). This shows you that depending on the parameters, the CobbDouglas may yield the same demands as the complements technology. In summary, the optimal demands for inputs are: L = q K = q We can check if we get the same answer using the formulas from lectures. We had: L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) 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 Similarly: 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 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. Nice. Phew. (c) What is the elasticity of labor demand with respect to q, PL and PK? Interpret your answers. Answer: Demand for labor is: L = q This came from: L = q1/( + ) [(/)(PK/PL)]/( + ) L = q1/( + ) (/) /( + ) PK /( + )PL/( + ) L = q (1/2) 2/3 PK 2/3 PL2/3 Notice how this takes the "constant elasticity" form in fact you can either use logs technique or definition of elasticities to see that: Elasticity of L with respect to q = (% L) / (% q) = 1/( + ) = 1 Ceteris paribus, a 1% increase in target output q will lead to a 1% increase in labor Elasticity of L with respect to PL = (% L) / (% PL) = /( + ) = 2/3 Ceteris paribus, a 1% increase in wages PL will lead to a 2/3% decrease in labor Elasticity of L with respect to PK = (% L) / (% PK) = /( + ) = 2/3 Ceteris paribus, a 1% increase in capital lease rates PK will lead to a 2/3% increase in labor That demand for labor is independent of wages and lease rate is a special case due to the parameters chosen it is not true in general. 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (d) What is the change in labor demand with respect to q, PL and PK? Interpret your answers. Answer: Demand for labor is: L = q This came from: L = q1/( + ) [(/)(PK/PL)]/( + ) L = q1/( + ) (/) /( + ) PK /( + )PL/( + ) L = q (1/2) 2/3 PK 2/3 PL2/3 Now: dL/dq = 1 Ceteris paribus, a 1 unit increase in target output q will lead to a 1 unit increase in labor dL/dPL = (2/3) q (1/2) 2/3 PK 2/3 PL2/3 1 dL/dPL = 0.133 q Ceteris paribus, a $1 increase in wages PL will decrease labor by 0.133 units. dL/dPK = (2/3) q (1/2) 2/3 PK (2/3)1 PL2/3 dL/dPK = 0.067 q Ceteris paribus, a $1 increase in lease rates PK will increase labor by 0.067 units. That demand for labor is independent of wages and lease rate is a special case due to the parameters chosen it is not true in general. (e) What is Ajax's long run cost function C(q)? 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 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain C(q) = (PL + PK)q C(q) = (5 + 10)q C(q) = 15 q (f) What is the elasticity of Ajax's long run cost with respect to target output? Answer: You can see that C = 15q is a "constant elasticity" function in q where: Elasticity of C with respect to q = (% C) / (% q) = 1 Ceteris paribus, a 1% increase in target output q will lead to a 1% increase in total cost You can also show it this way: C(q) = 15 q log C(q) = log (15 q) log C(q) = log 15 + log q d log C(q) / d log q = 1 % Change in C(q) / % Change in q = 1 (g) What is the long run average cost? Answer: Average cost AC(q) is total cost/output or C(q)/q: AC(q) = 15q/q = 15 (h) What is the long run marginal cost? Answer: Marginal cost MC(q) is dC(q)/dq: C(q) = 15 q 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 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 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) Solve for the optimal demand for labor and capital. Answer: We know that there can be three cases for perfect substitutes technology: If PL/PK < /, L = q/ and K = 0 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 = /. Since the isocost is steeper than the isocost: L = 0 and K = q/. K = q/ = q/(3/5) = (5/3)q Optimal demands are: 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L = 0 K = (5/3)q as long as PL/PK > / You now note that if wages fall, for some wage rates the inequality PL/PK > / will continue holding; but if wage rates fall substantially, there will come a point where PL/PK < /, in which case the firm stop using capital and only use labor. You may want to calculate that "threshold" wage, where this "switch" from "only capital, no labor" to "no capital, only labor" happens. (b) What is Bob Sinclair's long run cost function C(q)? Answer: The cost of producing target output q is: 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 (c) What is the elasticity of Bob Sinclair long run cost with respect to target output Answer: C(q) = (50/3)q log C(q) = log (50/3) + log q d log C(q)/d log 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 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain rise by the same percentage. (d) What is the long run average cost? Answer: C(q) = (50/3)q AC(q) = C(q)/q = 50/3 The average cost is always constant and equal to $50/3. (e) What is the long run marginal cost? 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 3 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) Solve for the optimal labor and capital demands: Answer: We know the optimal choice of inputs must on the corners of the isoquants which lie on the ray with equation: (1/3)L = (2/3)K Plugging this in q = min((1/3)L , (2/3)K) implies: q = (1/3)L and q = (2/3)K These imply: 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L = 3q K = (3/2)q (b) What is Guetta's long run cost function C(q)? 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 C(q) = 15 q + 15 q C(q) = 30q Let's check if we get the same answer from the formula in lecture slides: C(q) = q (PL/ + PK/) C(q) = q (5/(1/3) + 10/(2/3)) C(q) = q (15 + 15) C(q) = 30q (c) What is the elasticity of Guetta's long run cost with respect to target output? Answer: We have C(q) = 30q. Now: C(q) = 30q log C(q) = log 30 + log q d log C(q) / d log q = 1 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain % 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. (d) What is the long run average cost? 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 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. (e) What is the long run marginal cost? Answer: C(q) = 30q MC(q) = dC(q)/dq = 30 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. About Questions 4 6 Suppose we analyze a company with 2 inputs capital (K) and labor (L). The production function is Q = f(K, L). We've discussed these production technologies: CobbDouglas Technology: Q = L K Perfect Substitutes Technology: Q = L + K 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Complements Inputs Technology: Q = min(L, K) Now you'll look at the CES production function: Q = [ L( 1)/ + K( 1)/ ] /( 1) It turns out that the isoquants of the CES production function approximate the isoquants of: CobbDouglas Technology: when lim 1 of Q = [ L( 1)/ + K( 1)/ ] /( 1) Perfect Substitutes Inputs Technology: when lim of Q = [ L( 1)/ + K( 1)/ ] /( 1) Complements Inputs Technology: Q = min(L, K): when lim 0 of Q = [ L( 1)/ + K( 1)/ ] /( 1) Question 4 Assume 0 and derive the MRTS = (dK/dL) = (dQ/dL)/(dQ/dK) of the CES production function. Answer: The CES production function is Q = [ L( 1)/ + K( 1)/ ] /( 1). To get the MRTS we need dQ/dL and dQ/dK. First, dQ/dL: dQ/dL = {/( 1)} [ L( 1)/ + K( 1)/ ] {/( 1) 1} {( 1)/} L{( 1)/ 1} dQ/dL = [ L( 1)/ + K( 1)/ ] 1/( 1) L 1/ Similarly: dQ/dK = {/( 1)} [ L( 1)/ + K( 1)/ ] {/( 1) 1} {( 1)/} K{( 1)/ 1} dQ/dK = [ L( 1)/ + K( 1)/ ] 1/( 1) K 1/ Hence: MRTS = (dQ/dL)/(dQ/dK) = L 1/ / K 1/ 12 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain MRTS = K 1/ / L 1/ = (/) (K/L) 1/ MRTS = (/) (K/L) 1/ Question 5 Consider a company using K and L as imperfect substitute: Q = L K. (a) Derive the MPK = dQ/dK and interpret it. Answer: From Q = L K the MPK is: dQ/dK = L K 1 The interpretation is that if the company uses one more machine, then it can produce an additional output of L K 1. (b) Derive the MPL = dQ/dL and interpret it. Answer: From Q = L K the MPK is: dQ/dL = L1 K The interpretation is that if the company uses one more labor, then it can produce an additional output of L1 K. (c) Derive the MRTS = dK/dL = (dQ/dL)/(dQ/dK) = MPL/MPK. Interpret this. Answer: The MRTS is MPL/MPK. Using the expressions above: MRTS = MPL/MPK MRTS = (dQ/dL)/(dQ/dK) MRTS = [ L1 K ]/[ L K 1] 13 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain MRTS = (/)(K/L) The slope of the imperfect substitute inputs isoquant is (/)(K/L). Since / is constant, observe that as L increases (/)(K/L) decreases, giving us the imperfect substitutes inputs isoquant curves. (d) Substitute = 1 in the MRTS of the CES production function: do you get MRTS of Q = L K? Answer: We just derived the MRTS = (/)(K/L). Will we get the same answer if we substitute = 1 in the MRTS of the CES production function? From above, the MRTS of the CES production function is: MRTS = (/) (K/L) 1/ If = 1 we get MRTS = (/)(K/L), thus confirming the result. (e) In consumer theory, we saw the utility function U = Q11/3 Q22/3 represents the same preferences as (say) U = Q12/3 Q24/3. By the same token, does the production function Q = L1/3 K2/3 represent the same technology as Q = L2/3 K4/3? Answer: No. Comparing U = Q11/3 Q22/3 and U = Q12/3 Q24/3, we see that the second utility function is the square of the first utility function. Recall how we showed that these utility functions will represent the same preferences. For example, if bundle A is preferred to bundle B, then each utility function will assign a higher value to bundle A. On the other hand, in Q = L1/3 K2/3 the variable Q is measure in units and literally represents the maximum output that be produced with K and L. As such, for a given K and L, Q = L1/3 K2/3 always produces less output than Q = L2/3 K4/3 and therefore do not represent the same technology. Question 6 Consider a company using K and L as perfect substitute: Q = L + K. (a) Derive the MPK = dQ/dK and interpret it. 14 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: From Q = L + K the MPK is: dQ/dK = The interpretation is that if the company uses one more machine, then it can produce additional output (b) Derive the MPL = dQ/dL and interpret it. Answer: From Q = L + K the MPL is: dQ/dL = The interpretation is that if the company uses one more unit of labor, then it can produce additional output (c) Derive the MRTS = dK/dL = (dQ/dL)/(dQ/dK) = MPL/MPK. Interpret this. Answer: The MRTS is MPL/MPK. Using the expressions above: MRTS = MPL/MPK MRTS = [dQ/dL]/[dQ/dL] MRTS = / The slope of the perfect substitute inputs isoquant is (/): a constant. Thus, the isoquant curves are straight lines. (d) Substitute = in the MRTS of the CES production function: do you get MRTS of Q = L + K? Answer: We just derived the MRTS = /. Will we get the same answer if we substitute = in the MRTS of the CES production function? 15 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain From above, the MRTS of the CES production function is: MRTS = (/) (K/L) 1/ As observe that 1/ 0 so that MRTS = (/)(K/L)0 = (/) thus confirming the result. Question 7 Consider a company using K and L as complements: Q = min(L, K). Solve for the optimal demand for labor and capital. Answer: The complements inputs production function cannot be differentiated at the corner. Hence, we appeal to the graphical analysis (that optimal choice has to be at a corner) and the fact that the company will not waste any inputs so that, at the optimum: L = K Which implies that: q = L = K L = q/ K = q/ Observe how the demand for L and K do not depend on wages. This is a different result from demands for good in the UMP there, demands did depend on prices. Question 8 (2008 Test 2 Question) Edison Chang has the production function Q = L K + L + K. Suppose Edison Chang is in the long run. Calculate Edison Chang's MRTS. Answer: Because Edison is in the long run, all inputs are variable. By definition: MRTS = MPL/MPK = (dQ/dL)/(dQ/dK) 16 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Q = L K + L + K dQ/dL = L1 K + dQ/dK = L K1 + MRTS = ( L1 K + )/( L K1 + ) Question 9 (2008 Test 2 Question) ETorre Enterprises used labor (L) and capital (K) to produce its target output q = 80 according to the production function q = L + K where = 2 and = 8. ETorre procures labor and leases capital in competitive markets. Currently, PL = $100 and PK = $10. (a) If ETorre is in the long run, how much labor and capital does it use? Show your calculations below. Answer: ETorre has a perfect substitutes production function. Optimal labor and capital depends on whether the isocost is steeper, tangent, or flatter than the isoquant. The isocost has slope: PL/PK = 100/10 = 10 The isoquant has slope: MRTS = MPL/MPK = / = 2/8 = 1/4. Given the isocost is steeper than the isoquant, ETorre will use zero labor. Thus: q = L + K q = K K = q/ K = 80/8 = 10 17 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L = 0 and K = 10. Incidentally, the y and x intercepts of the isoquant are 10 and 40 respectively. (b) Plot your answer to part (a) in on a graph with labor on xaxis and wages on yaxis. Now suppose PL decreases: graph the resulting demand curve for labor in the figure. Answer: To see how the demand curve was derived note from part (a) that when PL = $100, there is no demand for labor. This implies L = 0 for PL 100. As PL falls, there will be no demand for labor so long as the isocost is steeper than the isoquant: 18 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain When the isocost becomes tangent to the isoquant, any amount of labor from 0 to 40 can be hired. This will happen when: PL/PK = / = PL = PK PL = 10 = $2.50 As PL falls further, the isocost will become flatter than the isoquant. This lets ETorre to produce his target output at a lower cost. For any wage lower than $2.50, demand for labor will be 40. 19 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 10 (Summer 2008 Final Exam Question) Etorre manufactures Italian gelato using machines and workers as complements in a 2:1 ratio. It leases machines at a price of $20/machine and hires workers at a wage of $10/worker. If E torre sells gelato into a competitive market and currently has 0 profits, what is the price of gelato? Show all steps and calculations clearly. Answer: We need to find the price of gelato given that Etorre is earning zero profits. Now profits are: = pq (PKK + PLL) Thus, we need to solve for p and given PK, PL for q, K and L. Now, the production function is: q = min(K,2L) The CMP inputs are: K = q and 2L = q L = q/2. 20 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Thus profits become: = pq (PKK + PLL) = pq (PKq + PLq/2) = (p PK PL/2)q If profits are zero, then: = (p PK PL/2)q = 0 (p PK PL/2)q = 0 p = PK + PL/2 p = $20 + $10/2 = $25 Given Etorre's production technology and input prices, if he is earning 0 profits ("breakeven profits") the price of the output, gelato, must be $25 (that's one expensive gelato!). Question 11 Ever wonder how a particular brand of milk has the same "taste"? It's because the dairy and animal husbandry industry uses scientific methods to design animal specific diet. There are methods which change diets according to the age, health, and condition etc of animals. In this question you will use actual figures to solve a simple CMP. A 200 pound steer can be sustained on a diet with the following combinations of grass and grain (pounds/day): Pounds of Grass Pounds of Grain 50 80 56 70 60 65 68 60 80 54 88 52 Currently, grass is $0.10/lb and grain is $0.07/lb. Suppose a farmer currently uses a feed mix of 68 lbs grass and 60 lbs grain is this optimal? 21 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: In this real life no equations problem, we have to think a little differently. First notice how grass and grain are imperfect substitute inputs for the sustenance of a 200 pound steer. In fact, if you draw the isoquants, they will "look like" those of CobbDouglas technology. The answer to this problem is simple: simply compute the total cost of each combination: Pounds of Grass 50 56 60 68 80 88 Pounds of Grain 80 70 65 60 54 52 Total Cost = 0.1 Grass + 0.07 Grain 10.60 10.50 10.55 11.00 11.78 12.44 Thus, the farmer is not minimizing cost to do that, she should use the 56 70 combination. 22 ...
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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.
 Fall '08
 HUSSEIN
 Economics, Microeconomics

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