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Questions and Answers from Econ 210A Final: Fall 2008 I have gone to some trouble to explain the answers to all of these questions, because I think that there is much to be learned by working through them. Please let me know if you find mistakes or inadequate explanations. Ted Bergstrom Question 1) Firm A has production function f ( x 1 , x 2 ) = x 1 / 2 1 + x 1 / 2 2 2 k where k > 0. a) For what values of k is f a quasi-concave function? For what values of k is f a concave function? Explain your answers. Firm B has production function g ( x 1 , x 2 ) = ( x 2 1 + x 2 2 ) k/ 2 where k > 0. b) For what values of k is g a quasi-concave function? For what values of k is g a concave function? Explain your answers. Answers to Question 1 Answer to 1a: The easiest way to check for quasi-concavity of f is to remem- ber that a function is quasi-concave if and only if every monotonic increasing transformation of that function is quasi-concave. In particular, the function f ( x 1 , x 2 ) = x 1 / 2 1 + x 1 / 2 2 2 k is a monotonic increasing transformation of the function g ( x 1 , x 2 ) = x 1 / 2 1 + x 1 / 2 2 , But it is easy to verify that g is quasi-concave. You could do this by checking the bordered Hessian conditions, which are now pretty simple because there are no off-diagonal terms in the regular Hessian part of the bordered Hessian. You could make the problem even easier by checking that g is a concave function and then using the fact that a concave function must be quasi-concave. Or you could alternatively just note that the isoquants for g have diminishing marginal rate of substitution. When is the function f concave? We could check this by writing out the Hessian matrix and making sure that it is negative semi-definite. But there is an easier way. We see that the function f is homogeneous of degree k . We know that it is quasi-concave, and our textbook tells us that if a function is quasi- concave and homogeneous of degree 1 it is concave. Suppose that k < 1. Define 1

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the function h ( x ) = f ( x ) 1 /k . We know that h ( x ) is homogeneous of degree one and quasi-concave, so it is concave. But now f ( x ) = h ( x ) k where k < 1. This means that f is a concave function of a concave function and hence must be concave. What if f is homogeneous of degree k > 1? Then it will not be a concave function. There are increasing returns to scale. It is easy to check that f is not a concave function. One could prove this by showing a single example of two points that violate the condition for concavity For example let x = (0 , 0) and x 0 = (1 , 0). Then f ( x ) = 0, f ( x 0 ) = 1. Now for any t (0 , 1), tx 0 + (1 - t ) x = ( t, 0). We have (1 - t ) f ( x ) + tf ( x 0 ) = t . But f ((1 - t ) x + tx 0 ) = f ( t, 0) = t k . If k > 1 and 0 < t < 1, we see that t k < t . Therefore f ((1 - t ) x + tx 0 ) < (1 - t ) f ( x ) + tf ( x 0 ) = t , which means that f is not a concave function. Answer to 1b: This one should be really easy if you think about it. The isoquants for this production function have the equation x 2 1 + x 2 2 = Constant. What do they look like? Can this function be quasi-concave? No! This should be obvious once you look at an isoquant. To be a little more formal about this: Much as we remarked in Part 1a) the function f
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