Chapters%206%20-%208%20-%20answers

Chapters%206%20-%208%20-%20answers - Sample Problems -...

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Sample Problems - Answers 1. Assume that the following table represents the short run production function q = f(L): L Q AP MP 0 0 7 1 7 7 17 2 24 12 15 3 39 13 14 4 53 13.25 8 5 61 12.2 9 6 70 11.6667 7 7 77 11 6 8 83 10.375 5 9 88 9.77778 -1 10 87 8.7 a. Calculate the marginal product and average product of labor for each value of L. (see above) b. Draw a graph of the total product, marginal product and average product curves. Connect the points with smooth curves. (See below) -5 0 5 10 15 20 0 5 10 15 Labor MP and AP AP MP
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Assume that Firm A has the following production function: Q A = 100K A 0.5 L A 0.3 . Determine the returns to scale of Firm A’s production function. Now suppose that Firm B has the following production function, Q B =80K B + 45L B . Determine the returns to scale of Firm B’s production function. Answer 1: Multiply K A and L A by some positive constant, z, and call the resulting output Q A ’. Thus, Q A ’ = 100(zK A ) 0.5 (zL A ) 0.3 Q A ’ = 100(z 0.5 K A 0.5 )(z 0.3 L A 0.3 ) Q A ’ = z 0.8 100K A 0.5 L A 0.3 . Q A ’ = z 0.8 Q A . Because Q A ’<zQ A , the production exhibits diminishing returns to scale. Answer 2: Multiply K B and L B by some positive constant, z, and call the resulting output Q B ’. Thus Q B ’ = 80(zK B ) + 45(zL B ) Q B ’ = z(80K B + 45L B ) Q B ’ = zQ B . Because Q B ’=zQ B , the production exhibits constant returns to scale. 3. Assume in the long run that a firm uses 30 units of labor and 60 units of capital to produce output q* when the market wage is $40 and the rental rate is $60. Assume further that the marginal product of labor is 20 and the marginal product of capital is 40 when L is 30 and K is 60. You may assume that the isoquants for the production function are strictly convex. Is the firm maximizing long run profits? Explain your answer. If your answer is no, what actions should the firm take? Convex isoquants (usually) rules out corner solutions. The MRTS = -MP L /MP K , which should equal the negative ratio of input prices (w/r) at the profit maximizing solution. Therefore, does MP L /MP K =w/r? No. 20/40 does not equal $40/$60. The MRTS is less than the input price ratio in absolute value, therefore one dollar spent on capital will give more output than one dollar spent on labor. Thus, the firm should reduce its labor inputs and increase its capital inputs. 4. Describe a firm’s expansion path if the following conditions are true: a. Capital and Labor are perfect complements. Under perfect complements, the inputs are always purchased in fixed proportions. Therefore, the expansion path should be a straight line from the origin and through the corners of each successive isoquant. b.
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Chapters%206%20-%208%20-%20answers - Sample Problems -...

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