# Famous Figures and Diagrams in Economics

## Edited by Mark Blaug and Peter Lloyd

# Chapter 7: Homothetic Production and Utility Functions

## Rolf Färe and Shawna Grosskopf

## Extract

Rolf Färe and Shawna Grosskopf Let us begin by quoting Professor Dale W. Jorgenson, from the Forward to the 1981 reprint of Shephard’s Cost and Production Functions, first published in 1953: Shephard developed the notion of a homothetic production function and employed the idea in function and formulating the concept of homothetic separability. The critical importance of homothetic separability to duality in the theory of aggregation and index numbers was fully appreciated by Shephard. Although Shephard developed the notion of homotheticity and homothetic separability in 1953, it was not until his 1970 book Theory of Cost and Production Functions that we can find a figure illustrating homotheticity. This figure illustrates that a ‘homothetic production function (structure) may be obtained from that for unit output rate by radial expansion’ (Shephard, 1970, p. 34). This type of expansion is equivalent to stating that the production function has linear expansion paths. We note that the figure may not be as famous as the concept itself. To introduce it, we need some formalities. Using Shephard’s notation, inputs are denoted by x [ RN , the scalar output by u [ R1. The production func1 tion F (x) denotes the maximal feasible output that inputs x can produce. A production structure is said to be homothetic if we can write F (F (x)) , where F (x) is homogeneous of degree +1 (F (lx) 5 lF (x) , l . 0) and F is a monotone transformation of F. Denote the level set for output level u (where F21 (u)...

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