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 8 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 Consider a company using K and L as imperfect substitute: Q = L K. (a) Suppose = 1/3 = 2/3. What are the returns to scale for this technology? Answer: Consider the general form: Q = L K. To check when we have increasing, constant or decreasing returns to scale (RTS) let's double inputs and compare the output with double inputs versus double the output. Initially we have: Initial Q = f(K, L) Initial Q = L K When inputs double, the new output will be: New Q = f(2K, 2L) New Q = (2L) (2K) 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain New Q = 2(+)L K New Q = 2(+) f(K, L) New Q = 2(+)*Initial Output That is: output with doubled inputs = 2(+) Initial output. Clearly (try some values!): if + > 1, doubling inputs will more than double output (increasing RTS) o This implies as the firm produces more, AC will decrease (graph it!) if + = 1, doubling inputs will exactly double output (constant RTS) o This implies as the firm produces more, AC will be constant (graph it!) if + < 1, doubling inputs will less than double output (decreasing RTS) o This implies as the firm produces more, AC will increase (graph it!) In this question + = 1/3 + 2/3 = 1 and therefore the production function : Q = L1/3 K2/3 always has constant returns to scale. (b) For what values of and will this technology exhibit increasing returns to scale? Decreasing returns to scale? Answer: When + > 1 and + < 1 respectively. For example, Q = L2/3 K2/3 has increasing RTS while Q = L1/3 K1/3 has decreasing RTS. Question 2 Consider a company using K and L as perfect substitute: Q = L + K. (a) Suppose = 1/3 = 2/3. What are the returns to scale for this technology? Answer: Consider the general form: Q = L + K. To check when we have increasing, constant or decreasing returns to scale (RTS) let's double inputs and compare the output with double inputs versus double the output. 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Initially we have: Initial Q = f(K, L) Initial Q = L + K With doubled inputs the new output will be: New Q = f(2K, 2L) New Q = (2L) + (2K) New Q = 2 {L + K} New Q = 2 f(K, L) Thus, Q = L + K (with > 0 and > 0) always has constant RTS. (b) Now suppose Q = L + K + . For what values of , and will this technology exhibit increasing, constant and decreasing returns to scale? Answer: In this question, you can get Q = L + K from Q = L + K + by setting = 0 which we know is constant RTS. But what happens when 0? Consider: Initially we have: Initial Q = f(K, L) Initial Q = L + K + With double inputs the new output will be: New Q = f(2K, 2L) New Q = (2L) + (2K) + 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain New Q = 2{L + K} + New Q = 2 f(K,L) New Q = 2 Initial Output Now: the new output with double inputs > 2 initial output if < 0 (for then > 0). Thus, Q = L + K + will have increasing RTS if < 0 the new output with double inputs < 2 initial output if > 0 (for then < 0). Thus, Q = L + K + will have decreasing RTS if > 0 (and as we just proved) the new output with double inputs = 2 initial output if = 0. Q = L + K + will have constant RTS if = 0 As an example, suppose = 1, = 1 Case 1: < 0, say, = 1 When L = K = 1, initial output is: f(K, L) = 1 + 1 1 = 1. Twice this output is 2. Now double inputs: L = K = 2 and so new output is: f(K, L) = 2 + 2 1 = 3 Note how the output with double inputs > Double the initial output. Hence, we have RTS > 1. Case 2: > 0, say, = 1 When L = K = 1, initial output is: f(K, L) = 1 + 1 + 1 = 3. Twice this output is 6. Now double inputs: L = K = 2 and so new output is: f(K, L) = 2 + 2 + 1 = 5 Note how the output with double inputs < Double the initial output. Hence, we have RTS < 1. Question 3 When does the complements production function Q = min(L, K) have increasing, constant, and decreasing RTS? Answer: 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Initially we have: Initial Q = min(L, K) With double inputs the new output will be: New Q = f(2K, 2L) New Q = min(2L, 2K) Observe this is the same as: New Q = min(2L, 2K) New Q = 2 min(L, K) New Q = 2 Initial Output. Thus, the complementary outputs technology, Q = min(L, K), always has constant RTS. As a study question, investigate when the production technology Q = min(L, K) + for and > 0 will have increasing, constant or decreasing RTS. Question 4 Jenn uses a complements technology with long run production function q = min(L, K) . Currently, = 1/2, =1/2, PL = $10 and PK = $10. (a) Suppose demand for Jenn's product is given the equation P = 100 10q. What is Jenn's target output if she wants to maximize profits? Answer: Jenn wants to maximize profits and should produce where MR = MC. Now: P = 100 10q You can use the shortcut that if the demand curve is linear, then the MR curve will also be linear, with the same intercept and twice the slope: MR = 100 20q 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain You can also show this as follows: P = 100 10q R = Pq R = (100 10q)q R = 100q 10q2 MR = dR/dQ MR = 100 20q But how do we find Jenn's MC? You know that for the complements technology: C(q) = q (PL/ + PK/) If you have forgotten how we got this result, consider Jenn's CMP: choose L and K to minimize cost of producing target output q. At the optimum, the solution must be at the corner of the isoquants. These corners lie on the line: L = K Thus, with: q = min(L, K) we have: L = q L = q/ and K = q K = q/ Thus: 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain C = PL L + PK K C = q (PL/ + PK/) Notice there is no "fixed cost" component here there shouldn't be, since all inputs are variables. Moreover, the cost function is linear in q. Now, from the cost function: MC = dC/dq MC = (PL/ + PK/) Observe how this is a constant as it should be, since the cost function is linear. For the complements technology, the long run MC is always constant and equal to (PL/ + PK/). For Jenn: MC = ($10/[1/2] + $10/[1/2]) MC = ($20 + $20) MC = $40 Setting MR = MC: 100 20q = 40 20q = 60 q = 3 Jenn will have a target output of 3 units. The astute student will wonder: what would happen if one is asked to choose profit maximizing output for the CobDouglas technology? In that case, matters are much trickier because there the MC is not a constant and in fact depends on the amount of L and K. Put simply, to know L and K you need to know MC and to know MC you need to know L and K. The intrepid student can do it, but I suggest holding off on it. 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) Given the target output q in part (a), how many workers should Jenn hire? How much capital should Jenn lease? Answer: We know the optimal choice of labor and capital are: L = q/ and K = q/ Thus: L = 3/(1/2) = 6 K = 3/(1/2) = 6 Jenn should hire 6 workers and lease 6 machines. (c) What is the long run cost of producing Jenn's target output? Answer: You can calculate by direct algebra C(q) = PL L + PK K or use the formula: C = q (PL/ + PK/) Either way you'll get the same answer. I will use the formula: C(q) = q (10/[1/2] + 10/[1/2]) C(q) = q (20 + 20) C(q) = 40q This is a linear cost curve, as we showed above, and substituting q = 3 gives the actual cost: C(q) = $120. (d) Suppose Jenn's workers clamor for higher wages. The labor union negotiates a higher wage of PL = $20. With the expenditure in part (c), will Jenn be able to produce the target output? 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: When Jenn's workers clamor for higher wages, with current expenditure levels, her new iso cost line does not allow her to reach the target q = 3 isoquant: (e) How will Jenn react to higher wages? In particular, what is the percentage change in labor, capital and long run cost due to the higher wages? Answer: With higher wages, the only way Jenn can produce q = 3 is by increasing expenditure (budget). Observe that Jenn continues to use the same number of workers and machines (albeit at a higher cost). Here is why. Initially, Jenn's optimal L and K are: When wages rise, the isocost line moves inwards: 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain However, this puts Jenn at a lower isoquant, meaning that she cannot produce her target output. To "get back" on the target isoquant, she has no choice but to increase the budget cost of production which will shift the isocost line out and back to the target isoquant: That Jenn uses the same number of workers and machines will now be shown formally. First, note how the optimal demands for labor and capital: L = q/ and K = q/ are independent of prices. Now cost at the higher wages is: C(q) = q (PL/ + PK/) C(q) = 3 (20/[1/2] + 10/[1/2]) C(q) = 3 (40 + 20) C(q) = 3 (60) 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain C(q) = $180 which is higher than the initial cost of $120. Thus, when wages increase Jenn continues using 6 workers and 6 machines at a higher cost of $180. You can see the same points from the elasticity of L, K and C(q) with respect to PL. L = q/ % change in L / % change in PL = 0 and: K = q/ % change in K / % change in PL = 0 and: C(q) = q (PL/ + PK/) % change in C(q) / % change in PL = (dC(q)/dPL)(PL/C(q)) (q/)(PL/C(q)) (q/)(PL/[ q (PL/ + PK/)]) PL/[(PL/ + PK/)] For Jenn: % change in C(q) / % change in PL = PL/[(PL/ + PK/)] % change in C(q) / % change in PL = $10/[(1/2) ($10/(1/2)+ $10/(1/2))] % change in C(q) / % change in PL = $20/[$20 + $20] % change in C(q) / % change in PL = That is, for every 1% increase in wages, total cost increases by %. In this question, wages 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain increased by 100% ($10 to $20) so cost should increase by 50%. This is exactly what happened here cost rose from $120 to $180, a 50% increase. Question 5 InHart uses a perfect substitutes technology with long run production function q = L + K. Currently, =1, =1, PL = $10 and PK = $20. (a) Suppose demand for InHart's product is given the equation P = 100 10q. What is InHart's target output if he wants to maximize profits? Answer: As in Jenn's case, we need to set MR = MC. Now: MR = 100 20q. What about MC? Perhaps we can use the same trick as in Jenn's question where the MC is constant? But we know for the 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 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 12 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain For InHart, = 1, =1, PL = $10 and PK = $20. The isoquant slope is / = 1 and isocost slope is PL/PK = . The isocost being flatter than the isoquant shows that we are in Case 1 above. The cost function is: C = PL q/ MC = dC/dq MC = = PL/ which is a constant. For InHart: MC = 10/1 = 10 Setting: MR = MC: 100 20q = 10 20q = 90 q = 9/2 = 4.5 (b) Given the target output q in part (a), how many workers should InHart hire? How much capital should InHart lease? Answer: Since the MRTS < isocost slope, we have: Case 1: If PL/PK < MRTS L = q/ L = 9/2 K = 0 InHart should hire 4.5 workers (eek) and lease no machines. 13 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (c) What is the long run cost of producing InHart's target output? Answer: Since InHart only hires labor: C(q) = PL L C = PL q/ C = $10*9/2*1 C = $45. (d) Suppose the price of leasing capital is volatile. For what range of PK will InHart use the same number of workers and capital as in part (b)? Answer: So long as the isocost is flatter than the isoquant, InHart will lease no machines and employ 4.5 workers i.e. as long as: PL/PK < MRTS = 1 PL < PK $10 < PK Thus, as long as $10 PK InHart will employ 4.5 workers and 0 machines. Question 6 (2008 Test 2 Question) A company uses Labor (L), Capital (K) and Materials (M) as inputs to produce target output q with the production function q = L K M. (a) For what values of , and will this company have constant returns to scale? Show your calculations below. Answer: This production function will have constant returns to scale (RTS = 1) if: f(2L, 2K, 2M) = 2f(L, K, M). 14 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain That is, when output with double inputs is equal to double the output. Now: the output with L, K and M is: Initial q = f(L, K, M) Initial q = L K M. Next, the output with twice the L, K and M is: New q = f(2L, 2K, 2M) New q = (2L) (2K) (2M) New q = 2 2 2 L K M New q = 2 + + L K M New q = 2 + + (Initial q) The new output (with all inputs doubled) will be double the initial output when + + = 1. Question 7 (2008 Test 2 Question) A company uses Labor (L) and Capital (K) as inputs to produce target output q with the production function q = min(L, K) + . For what values of , and will this company have increasing returns to scale? Show your calculations below. Answer: First, the initial output with L, K is: Initial q = f(L, K) Initial q = min(L, K) + Next, the output with twice the L and K is: New q = f(2L, 2K) 15 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain New q = min(2 L, 2 K) + New q = 2 min(L, K) + We need to see when the output with twice the inputs is greater than double the initial output. Double the initial output is: Double Initial q = 2 min(L, K) + 2 New output > Double initial output when: 2 min(L, K) + > 2 min(L, K) + 2 Which can only happen if < 0. If you're having problems seeing this, look at this way: New output > Double initial output when: 2 min(L, K) + > 2 min(L, K) + 2 2 min(L, K) + + > 2 min(L, K) + 2 2 min(L, K) + 2 > 2 min(L, K) + 2 > 0 < 0 Thus: q = min(L, K) + has increasing returns to scale when < 0. Question 8 (20082009 Final Exam Question) Fuelled by easy access to drugs from the USA (United States of Addicts) the citizens of the Canadian town Crackotoa are fast becoming drug addicts. The mayor of Crackotoa, Ms. Cindy Bong, has started a drug treatment program in which police officers (P) and drug counselors (D) work as complements in a 1:1 ratio to treat each drug addict (A). That is, the treatment program uses police officers and drug counselors as complementary "inputs" to treat drug addicts as the "output". 16 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (a) The program currently has 1,000 police officers and 1,000 drug counselors. If Crackotoa's government doubles the budget for the drug treatment program, how many more police officers and drug counselors should Ms. Cindy Bong hire? Assume the price of police officers and drug counselors is constant. Answer: The trick here is to recognize that drug addicts, the "output", are being treated by police officers and drug counselors as complementary "inputs" in a 1:1 ratio. Denoting addicts by "A", police officers by "P" and drug counselors by "D", we have: Addicts = min(Police, Drug Counselors) Addicts = min(P, D) This has constant returns to scale: thus if the budget doubles, the number of police officers and drug counselors also doubles. You can also show this as follows: being complementary inputs, the optimal input entails hiring equal numbers of police officers and drug counselors: Police officers = Drug counselors P = D The number of addicts treated is equal to police officers (which is equal to the drug counselors): A = P = D. The cost function or the drug treatment "budget" to treat a target level of addicts is: C(A) = PPP + PDD Since P = D: C(A) = (PP + PD)P P = C(A)/(PP + PD) Since the cost of the program is equal to the budget, equating C(A) to Budget, we have: P = Budget/(PP + PD) = D Thus, if the budget doubles, then ceteris paribus, the number of police officers and drug counselors must also double. That is, 1,000 additional police officers and drug counselors should be hired. 17 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) Repeat part (a) for the case where the price of police officers and drug counselors also doubles. Answer: From part (a): P = Budget/(PP + PD) = D From this equation note that if the budget and the price of police officers and drug counselors double then the number of police officers and drug counselors remains constant since both the numerator and denominator double, leaving the ratio unaltered. Question 10 This question was motivated by the recent announcement by the Ontario province and the federal government to give Toronto close to $1 billion to (principally) extend the Sheppard subway line. In this question, you will investigate whether it's better to receive an unrestricted grant that be, for example, spent on any combination of mass transit and highways, versus a block of money which specifies a specific level of mass transit and highways. Suppose the "inputs" for a city's transportation level ("output") are "mass transit" (xaxis) and "highways" (yaxis). Assume mass transit and highways are imperfect substitutes for transportation in a city. (a) Draw the isoquants for transport as the output and mass transit and highways as inputs. Answer: The isoquant for mass transit and highways as imperfect substitutes is given below. (b) Suppose the Federal government gives cities block grant money for a specific amount of mass transit and highways to "produce" a target level of transport. Depict this on the isoquant figure. 18 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: The city has been given a specific amount of mass transit and highways to produce a target level of transport: (c) Now suppose a unit of mass transit costs Pm and a unit of highways costs Ph. Instead of giving cities block grants as in part (b), now suppose the Federal government gives cities an unrestricted block grant (budget) which they are free to spend on any combination of mass transit, highways. Draw a picture for two cases: Case (i): a city which benefits from the unrestricted block grant; Case (ii): a city which does not benefit from an unrestricted block grant. Answer: This question is like the free education question from consumer choice. Note that the government specifies the amount of mass transit and highway "inputs" with a specific bundle of inputs, the city will on some isoquant. If the Federal government lets cities choose how to "allocate" inputs, each city will minimize cost for a target level of output. It may be that the optimal choice will lead to higher transport output than before (remember, higher isoquants correspond to higher output): 19 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Here, the unrestricted grant is large enough so that the city is able to provide a greater level of transportation compared to the case when it was given a specific level of mass transit and highways. Incidentally, it is a complete coincidence that in the graph, mass transit goes up while highways remain the same it is a feature of the way I have drawn the graph. Here is the opposite case: the unrestricted grant may result in a lower level of transport than before: Cities of type (i) will favor unrestricted transport grants since it leads to more transportation. Similarly, cities of type (ii) will be against unrestricted grants and for restricted grants. The moral of this question is: unrestricted grants don't necessarily result in more output. We had the same kind of result in the free education example for consumers: free education, or an education coupon, does not necessarily result in more education or increased happiness. Question 11 This question is a review of the Coke example discussed in the lectures. In the 1970s, Coca Colathe number 1 global brandused sugar in its secret formula. Rising sugar prices and changing consumer tastes away from high calorie drinks to low calorie drinks, forced CocaCola to tinker with its formula. In particular, CocaCola considered switching from sugar to corn fructose syrup. In all questions, put fructose on the xaxis and sugar on the yaxis (that's the 20 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain opposite of what I did in the lectures). (a) CocaCola initially could not develop a new formula for Coke that used fructose as an input while maintaining the "taste" of Coke. What do Coke's isoquants look like? How much of sugar and fructose will Coke use in its products? Answer: Your understanding of consumer choice will help in this question of producer choice. Coke is unable to formulate its secret recipe to use fructose and maintain the taste of the product. This implies, no matter how much fructose is used, or no matter how cheap fructose is, Coke will only use sugar in its products. Think of a similar question in consumer choice where a person likes martinis but not sushi. The iso quants are: This implies that regardless of the price of sugar or fructose, Coke will only use sugar: (b) CocaCola figured out a way to mix sugar and fructose and maintain Coke's taste. What will Coke's isoquants look like? At that time, Coke used a 5050 combination of sugar and fructose in its productsshow this on your diagram. 21 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: If Coke manages to tweak the formula, then fructose and sugar can be mixed. One possible shape is: If Coke was using a 5050 mixture of sugar and fructose, the isoquants and isocost combinations could be: Note that sugarfructose now being imperfect substitutes does not preclude Coke from using only sugar: 22 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (c) Fructose prices continued to drop and CocaCola got better at tweaking its formula to incorporate fructose. By the later 70s, CocaCola was using only fructose in its products. Show this on an isoquant/isocost diagram. Answer: If Coke is only using fructose, then we should be draw a picture so that regardless of sugar/fructose prices, Coke only uses fructose (these are "quasilinear" isoquants): Question 12 In this question, you will explore the idea that if a company has an imperfect substitutes technology, it is able to alter the mix of inputs in response to input prices. For example, a company with a complements technology is "hostage" to input prices if input prices increase it is unable to switch to a relatively cheaper input. In contrast, a firm using imperfect substitutes technology can defray higher input prices by switching. Consider a firm with a CobbDouglas production function Q = L K. For your convenience, here are the optimal demands for labor and capital and the optimized cost function: L = q1/( + ) [(/)(PK/PL)]/( + ) K = q1/( + ) [(/)(PL/PK)]/( + ) C(q) = q1/( + ) PL /( + ) PK/( + ) [(/)/( + ) + (/)/( + )] From the demands for labor and capital, show that if wages increase, then to still produce target output q, this firm will use less labor and more capital. Answer: Look at the demand for labor: 23 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L = q1/( + ) [(/)(PK/PL)]/( + ) Observe how it is decreasing in wages: as PL then L . In fact the elasticity of L with respect o PL is: Elasticity of L with respect to PL = (% L) / (% PL ) = /( + ) so that a 1% increase in wages will result in /( + )% decrease in labor. But, this would result in lower output after all Q = L K so that L means Q . Since the firm must produce the target output q it must use more capital. How do we know this? Look at the demand for capital: K = q1/( + ) [(/)(PL/PK)]/( + ) Observe how it is increasing in wages: as PL then K . In fact the elasticity of K with respect to PL is: Elasticity of K with respect to PL = (% K) / (% PL ) = /( + ) so that a 1% increase in wages will result in /( + )% increase in Capital. What's happening here is that as wages, rise, the isocost line moves in: To get back on the target isoquant, the firm increases expenditure by just enough to get back on the target isoquant: 24 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain We can see, to maintain the same target output q, the firm must increase expenditure when wages rise from the equation: C(q) = q1/( + ) PL /( + ) PK/( + ) [(/)/( + ) + (/)/( + )] Ceteris paribus as PL note how C also . Optional: An increase in wages has made the firm move up the target isoquant less labor, more capital and the same target output. Can we "prove" this? For sure. Here is how: a 1% increase in PL has led to a /( + )% decrease in labor and a /( + )% increase in capital. What is the % effect on total output? We know it has to be 0% since the firms stays on the same isoquant. But can we prove this? Yes. Start with: Q = L K log Q = log (L K ) log Q = log L + log K log Q = log L + log K d log Q = d log L + d log K Intuitively this says: if log Q = log L + log K then: Change in log Q = change in log L + change in log K 25 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Now you know that d log x = dx/x (go over mathecon review). Thus: dQ/Q = dL/L + dK/K If you multiply both sides by 100 this equation says: % in Q = (% in L) + (% in K) We know from above that: % in L = /( + )% and % in K = /( + )%. Substituting these: % in Q = ( /( + )) + /( + ) % in Q = /( + )) + /( + ) % in Q = 0 That is, a % in L = /( + )% and a % in K = /( + )% leads to no change in target output, which means we must have moved on the same isoquant. Nice to have a confirmation of our graphical result. Question 13 Consider a firm with a CobbDouglas production function Q = L1/2 K1/2. Currently, PL = $5 and PK = $5 and the firm has target output q. (a) Solve for the optimal levels of labor and capital: Answer: The CMP is: choose L and K to min cost of producing target output q. The Lagrangian is: L = PL L + PK K [L K q] L = 5 L + 5 K [L1/2 K1/2 q] The FOCs are: L/L = 0 5 (1/2) L1/2 K1/2 = 0 26 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 5 = (1/2) L1/2 K1/2 L/K = 0 5 (1/2)L1/2 K1/2 = 0 5 = (1/2)L1/2 K1/2 L/ = 0 L1/2 K1/2 = q Dividing the 1st and 2nd FOCs and solving, we get: K = L Substitute this in L1/2 K1/2 = q to get: L = q and K = q (b) What is the cost function? Answer: Now: C = PL L + PK K C = 5L + 5K C = 10q This is a linear cost function with MC = AC = 10. (c) Calculate the Lagrange multiplier and interpret it. Answer: We can use any of the first two FOCs. Let's use the first: 5 = (1/2) L1/2 K1/2 27 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Now L = K = q, so that: 5 = (1/2) L1/2 K1/2 5 = (1/2) (q)1/2 (q)1/2 5 = (1/2) = 10 To interpret this, recall the Lagrangian, expressed in terms of parameters and variables is: L = PL L + PK K [L K q] L = PL L + PK K L K + q A facile interpretation is to ask the question "What is the impact on total cost, subject to the constraint that output produced must equal target output, if the target output rises". The answer from the envelope theorem is: dL/dq = dL/dq = 10 That is, each time the target output increases by 1 unit, the optimal cost will rise by $10. In this question you can see why this makes complete sense. Recall L = K = q and PL = PK = $5. If the firm produces another unit then from L = q and K = q it needs to hire another worker and lease another machine. The cost of another worker is her wage $5 and the cost leasing another machine is $5, so the total cost needed to produce another unit of output is $10. That is exactly what we have from = 10. Nice. 28 ...
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 Economics, Microeconomics, Returns to scale, S. Ajaz Hussain, Ajaz Hussain

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