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CU-Boulder ECON 6808 - Review Questions

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Econ 6808 Introduction to Quantitative Analysis review questions -set 2. III. Economic Curvature and VI. Economic Applications of Duality TheoryReview questions. 1. What does it mean to say that the function y = g(m1, m2) is concave? Give ananswer in both words and in mathematical notation. 2. Convince me that the cost function, c = c(y, w) where w = [w1, w2,...wN), is notstrictly convex in input prices. Assume that this cost function is increasing iny, increasing in w, and everywhere twice differentiable. As part of youranswer define strictly convex. 3. Consider a cost function, c = c(y, w) where w = [w1, w2,...wN). Assume that thiscost function is nondecreasing in y, nondecreasing in w, quasiconcave in w,and everywhere twice differentiable. What does it mean to say that this costfunction is nondecreasing in y? Use the definition of a cost function to prove (toconvince me) that a cost function must by nondecreasing in y. 4. Consider a cost function, c = c(y, w) where w = [w1, w2,...wN). Assume that thiscost function is nondecreasing in y, nondecreasing in w, quasiconcave in w,and everywhere twice differentiable. What does it mean to say that this costfunction is nondecreasing in w? Use the definition of a cost function to provethat a cost function must by nondecreasing in w. 5. Define Shephard's Lemma in terms of production theory. Prove it. 6. Define both in words, and in functional notation the competitive firm's costfunction and its conditional input demand functions. 7. Convince me that an individual, Wilma, who is maximizing her utility, u(x,y), subject to a budget constraint will behave as if her utility function isquasiconcave even if it is not. You might want to convince me using graphs ofindifference curves and budget lines to identify utility maximizing bundles.As part of your answer, define quasiconcave for this utility function, andexplain what it tells you about the shape of the indifference curves. Assumethat Wilma's utility function is increasing in x and y. The point of thisexercise is to demonstrate that it is not restrictive to assume that a utilityfunction that is increasing in its arguments is also quasiconcave. 8. Assume that Wilma's utility function, u(x, y) is increasing in x and y. Giventhis, what is gained by assuming that it is strictly quasiconcave rather than justquasiconcave? 9. Consider an individual's utility function, u(x) where x = [x1, x2,...xN], that hasall the standard properties (increasing in x, quasiconcave in x, and twicedifferentiable. One can think of this utility function as a production functionwhere utility is being produced using goods as inputs. Given this, define in both words and functional notation the consumer theoryanalog of the competitive firm's cost function. This function is called theexpenditure function where E is conventionally used as the dependent variable.Duality theory tells us that this expenditure function is dual to the direct utilityfunction, u(x), and also describes the individual's preferences. Then define both in words and in functional notation the consumer theoryanalog of a firm's conditional demand functions for inputs. In consumertheory, these are called the hicksian demand functions. Can one use the expenditure function to derive these hicksian demandfunction? How?Assume a simple functional form for a two-good utility function and deriveboth the expenditure function and the hicksian demand functions. 10. Intuitively explain, using a graphical analysis, how one can derive theproduction function, y = f(k, l) from the cost function c = c(y, r, w). 11. Choose some explicit cost function, and from it derive the correspondingproduction function.12. Define an individual’s compensating variation for a change from to(,)mp00in four ways: in words, in terms of an indifference relationship(,)mp11between two states of the world, in terms of the indirect utility function , andin terms of the expenditure function.13. Explain how one can derive a competitive firms cost function, c = c(y, w)where w = [w1, w2,...wN) from its production function y = f(x) where x = [x1,x2,...xN]. Now assume a simple functional form for a 2-input productionfunction and derive the firm's cost


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