The Definition of Capital for Solow's Growth Accounting Formula

Abstract

Solow's growth formula [14] for ascertaining the contribution of factor input growth to total growth in output is a major contribution to our understanding of the determinants of growth, and was one of the reasons he was recently awarded the award of the Nobel Prize in Economics. The one detail that still requires explication is which of the many possible definitions of capital quantity should be used for growth accounting. The issue has been heavily debated since the sixties, but is still unresolved. A simple resolution based on the dependence of his formula on the law of one price will be offered. By the law of one price is meant that the price for a unit of capital services is the same regardless of which type or age of capital is rendering the services. Meeting this necessary condition forces the use of a particular definition of the quantity of capital services in growth accounting, namely that the quantity of capital services must be a weighted sum of the service provided by the different goods, with weights proportional to the prices of the services of the different capital goods. The proportionality of the weights to the prices must apply to goods of different ages as well as to goods of different types. With this definition of capital services, Solow's results can be obtained without assuming the existence of an aggregate production function. Solow's argument went as follows. If Q represents output and K and L represent capital and labor inputs in 'physical' units, then the aggregate production function can be written as: Q = F(K, L; t). He then goes on to assume neutral technical change where the production function takes the special form Q = A(t)f(K, L) and the multiplicative factor A(t) measures the cumulated effect of shifts over time. Differentiating the equation totally with respect to time and dividing by Q, one obtains Q'/Q = A'/A + A(af/aK)K'/Q + A(af/aL)L'/Q where ' indicates the time derivatives. Defining wk = (aQ/aK)K/Q and WL = (aQ/aL)L/Q, the relative shares of capital and labor, and substituting in the above equation gives: