ProblemSet3-AK - Economics 3203 Fall 2010 Problem Set # 3:...

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Economics 3203 – Fall 2010 Problem Set # 3: Labor Demand (2) ANSWER KEY 1. a. We can set MC = MR or MRP L = ME L in order to maximize profits. Both will give the same result. The second one requires more math but gets to the employment level directly. First method: MC = MR: Find MR: Total Revenue = P * Q = 60Q – 2Q 2 MR = dTR/dQ = 60 – 4Q Find MC: TC = W * L = 2*L We need to take the derivative of TC with respect to Q in order to find MC. This means we need TC in terms of Q instead of L. We have the production function given to us that can be solved for L and plugged into TC: Q = [2L-200] 1/2 ---> Q 2 = 2L – 200 ---> Q 2 + 200 = 2L ---> L = (Q 2 /2) + 100 So TC = 2*[(Q 2 /2) + 100] = Q 2 + 200 Then MC = dTC/dQ = 2Q MR = MC yields: 60 – 4Q = 2Q ---> Q=10 and P=$40. This level of Q can be plugged into the production function to find L = 150. Second Method: MRP = ME L (it’s longer, but it works) Note that the wage is given, so ME L = W = $2. MRP = ME L MP L * MR = 2 Find MR: Total Revenue = P * Q = 60Q – 2Q 2 MR = dTR/dQ = 60 – 4Q Find MP L : MP L = dQ/dL = (1/2)*(2L-200) -(1/2) * 2 (note that this used the chain rule – see your calc review sheet if this is unfamiliar) MP L =(2L-200) -(1/2) Now plug these back in to the profit-maximizing equation (MRP=ME L ) (2L-200) -(1/2) * (60 – 4Q) = 2 We still have an L and a Q here, so we need to substitute in for Q using the production function. (2L-200) -(1/2) * (60 – 4(2L-200) (1/2) ) = 2 1
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Multiply this out: 60*(2L-200) -(1/2) – 4(2L-200) (1/2) * (2L-200) -(1/2) = 2 The two ugly pieces of the second term cancel out. Then we just have: 60*(2L-200) -(1/2) – 4 = 2 60*(2L-200) -(1/2) = 6 Get L alone on one side, with the other numbers on the other side: (2L-200) -(1/2) = 6/60 ---> (2L-200) -(1/2) = .1 Square both sides: (2L-200) -1 = .01 Multiply both sides by (2L-200) to get rid of negative exponent: (2L-200) -1 *(2L-200) = .01*(2L-200) 1 = .02L – 2 Æ 3 = .02L Æ 3/.02 = L Æ L = 150 To find price, use the demand curve – but first we need Q to use that. Q = (2L-200) 1/2) Q = (300-200) (1/2) Æ Q = 100 (1/2) Æ Q = 10 Plug into demand curve: P = 60 – 2Q Æ P = 60 – 20 Æ P = $40 b. This price ceiling is below the equilibrium price, so it is binding (in other words, it affects the decisions of the firm because it restricts them from choosing their profit-maximizing price and quantity). The firm will not be allowed to sell its output for $40; the price will have to be less than or equal to $36. First consider P = $36, and see what Q maximizes profit if the firm assumes it can sell as much as it wants at
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ProblemSet3-AK - Economics 3203 Fall 2010 Problem Set # 3:...

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