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Unformatted text preview: 0.1 Production functions with a single output 0.1.1 Homothetic and Homogeneous Production Functions Homothetic production functions have the property that f ( x ) = f ( y ) implies f ( x ) = f ( y ). Homogeneous production functions have the property that f ( x ) = k f ( x ) for some k . Homogeneity of degree one is constant returns to scale. Homoge- neous implies homothetic, but not conversely. Example f ( x 1 , x 2 ) = x 1 x 2 + 1 is homothetic, but not homogeneous. Draw a picture. Separable production function. Example f ( x 1 , x 2 , x 3 ) = F ( g ( x 1 , x 2 ) , x 3 ). Suppose factors 1 and 2 are skilled and unskilled labor and factor 3 is capital. What does separability mean? Additively separable production function. f ( x 1 , . . . , x n ) = F ( i g ( x i )) Abram Bergsons theorem, A function is additively separable and homothetic if and only if it is of one of the following two forms: F ( i a i x i ) or F ( i a i ln x i ). Corollary It is additively separable and homogeneous of degree k iff it is of the form X i a i x i ! k/ or of the form Y i x a i i where i a i = k . The first of these cases is known as a constant elasticity of substitution pro- duction function (ces) and the second as a Cobb-Douglas production function. Why is it called ces? The elasticity of substitution between two factors is defined to be the absolute value of the percent change in the ratio in which the factors are used that results from a one percent change in the ratio of the wages of the two factors. That is, it is the partial derivative of the log of the ratio of factor inputs with respect to the log of the ratio of the factor prices. (elasticityfactor inputs with respect to the log of the ratio of the factor prices....
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