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Unformatted text preview: Economics 313 Lecture 2, Spring 2008 (TA) Jayant V. Ganguli Suggested solutions for homework 4 I. 1. As discussed in the lectures and section, for this question we have to compare X ( &L;&K ) with &X ( L;K ) for arbitrary & & 1 ; L & ; and K & . a. Since X ( &L;&K ) = a L ( &L ) 2 + a K ( &K ) 2 = & 2 & a L L 2 + a K K 2 = & 2 X ( L;K ) > &X ( L;K ) = & & a L L 2 + a K K 2 , the production function exhibits increasing returns to scale. b. Here X ( L;K ) = min f a L L;a K K g . There are two possibilities. First, a L L & a K K , then X ( L;K ) = a K K . So, X ( &L;&K ) = min f a L ( &L ) ;a K ( &K ) g = a K ( &K ) = &X ( L;K ) . 1 Now consider the second of a L L < a K K , then we again get X ( &L;&K ) = &X ( L;K ) [exercise]. So, this function has constant returns to scale. c. Since X ( &L;&K ) = a L ( &L ) + a K ( &K ) = & [ a L L + a K K ] = &X ( L;K ) , the production function exhibits constant returns to scale. 2. To answer this we need to &nd the marginal products of the factors. a. MPP K ( L;K ) = @X ( L;K ) @K = 2 a K K . To check if MPP K is decreasing in K , consider @MPP K ( L;K ) @K = 2 a K > . Hence, MPP K is increasing, not decreasing so the function does not exhibit diminishing marginal returns with respect to K . A similar result applies to L [exercise]. b. Given the production function X ( L;K ) = 8 < : a L L if a L L a K K a K K if a L L > a K K ; we get that MPP K ( L;K ) = 8 > > > < > > > : if a L L < a K K a K if a L L > a K K [0 ;a K ] if a L L = a K K : (Why is this?) If a L L > a K K , then MPP K ( L;K ) = 0 . Now increase K by an in&nitesimal amount > to ( K + ) . Since the change is very small, we have a L L > a K ( K + ) so that MPP K ( L;K + ) = 0 (why?). This shows that X ( L;K ) does not exhibit diminishing marginal returns (with respect to capital). [As an exercise, show that MPP K is also constant for the case a L L < a K K .] A similar analysis for L shows that MPP L ( L;K ) is also not decreasing in L . c. MPP K ( L;K ) = @X ( L;K ) @K = a K . Further, @MPP K ( L;K ) @K = 0 , so the function does not exhibit diminishing marginal returns with respect to K . A similar analysis applies to L . 1 Why is min f a L ( &L ) ;a K ( &K ) g = a K ( &K )? 1 3. The production functions in (b) and (c) both have indi/erence curves that have &at portions or lines and so do not have strictly convex isoquants, see gure 18.2 on p 325 and gure 18.3 on p 326 of the text respectively. The production function in (a) does not have strictly convex isoquants, as shown in gure 1. To check this analytically, consider the MRTS (or TRS as in the text) for (a), MRTS ( L;K ) = @X @L @X @K = a L L a K K : Clearly, MRTS ( L;K ) will increase as ( L=K ) increases which means that the isoquant is not convex....
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This note was uploaded on 09/28/2008 for the course ECON 3130 taught by Professor Masson during the Spring '06 term at Cornell University (Engineering School).
- Spring '06