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# Derive the degree of homogeneity

Consider a firm which produces a good, y, using two inputs or factors of production, x1and x2. The firm’s production function describes the mathematical relationship betweeninputs and output, and is given byy = A( x1, x2) = xqx2, a, BER+.(a) Derive the degree of homogeneity of the firm’s production function.(b) The setS = {(X1,x2) E RHxix2 = yo}is the set of combinations of (x1,x2) which produce output level yo. S is a level curve of and is referred to by economists as the isoquant associated with output level yo. Theisoquant implicitly defines x2 as a function of x1.i) Use implicit differentiation to derive the slope of the isoquantxix2 = yo-That is, derive "2.(Note that along a given isoquant Ay = 0).ii) Use implicit differentiation to derive *2 for the isoquantxix2 = yo.iii) What conclusions do you draw regarding the slope and curvature of the isoquant?Briefly explain.(c)i) Derive the Hessian matrix associated with the firm’s production functionA(x1 , X2 ) = xqx2.ii) State a sufficient condition(s) for (1) to be a strictly concave function.iii) Derive a restriction on the parameters a and B which ensures that (1) is a strictlyconcave function.

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