# ECON201PSet7Answers - WELLESLEY COLLEGE DEPARTMENT OF ECONOMICS E CONOMICS 201-03 Answers to Problem Set#7 J OHNSON 1 P&R page 550#6 I fwe set up

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WELLESLEY COLLEGE DEPARTMENT OF ECONOMICS ECONOMICS 201-03 JOHNSON Answers to Problem Set #7 1. P&R, page 550, #6: Ifwe set up the profit maximization problem directly, we can solve for the L*(w,P) function directly: IT= P·\-w.L~ P(12L-L!)-wL ~:::~ = 12f'-.2.V L - W oL ~'? L -: t 2. f - vi L lt ~w,r)'" Now, proceed with the following: Suppose a firm's production function is given by Q = 12L . L , I for L = 0 to 6, where L is labor input per day and Q is output per day. Derive and draw the firm's demand for curve if the firm's output sells for \$10 in a competitive market. How many workers will the firm hire when the wage rate is \$30 per day? \$60 per day? (Hint: The marginal product of is 12 - 2L.) The demand for labor is given by the marginal revenue product oflabor. This is equal to the product ofmarginal revenue and the marginal product oflabor: MRP = (MR)(MP). L L In a competitive market, price is equal to marginal revenue, so MR = 10. We are given MP = 12 - 21, (the slope ofthe production function). L Wage 120 100 80 60 40 20 Labor 1.5 3.0 4.5 Figure 14.6 Therefore, the = (10)(12 - 21,). The firm's profit-maximizing quantity of labor L occurs where = w. If w = 30, then 30 = 120 - 20L at the optimum. Solving for L L yields 4.5 hours per day. Similarly, ifw = 60, solving for L yields 3 hours per day.

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2. The cost minimizing demands for L and K are homogeneous of degree ZERO in wand r. Why? Well, if we double w and r ,let's say, we do NOT change the slope of the isocost line and therefore, for a GIVEN q ,we will NOT change the cost minimizing demands for L and K. Same tangency point, given that the isocost slope does not change. However, the TC*(w,r,q) function is homogeneous of degree ONE in wand r. Why? Well, given that L* and K* don't change when, say, w and r double, it IS true that the TOTAL COST ofthat bundle ofinputs is now TWICE as expensive. Thus, even though the isocost slope doesn't change, the TC value (in \$) associated with that same level ofL*, K* and thus q has to double. Think of it this way: the original isocost line would shift IN because of the doubling of w and r. SO, to reach the original isoquant, the TC outlay must also double to shift the isocost curve back out. As for the unconditional (profit maximizing) functions, L*(w,r,P) and K*(w,r,P) are homogeneous of degree ZERO in w, r, and P. Why? Well, firms will hire L and K up to the point where the marginal product of the factor equals its real input price (either the real wage or the real rental rate ofcapital). If w , r , and P all double, say, then the real wage and the real rental price of capital stay the same, implying that the profit maximizing firm will not alter its demands for L and K, thus leaving output unchanged as well.
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## This note was uploaded on 06/15/2010 for the course ECON 201 taught by Professor Johnson during the Fall '08 term at Wellesley College.

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ECON201PSet7Answers - WELLESLEY COLLEGE DEPARTMENT OF ECONOMICS E CONOMICS 201-03 Answers to Problem Set#7 J OHNSON 1 P&R page 550#6 I fwe set up

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