IN this day of rationally designed econometric studies and super-input-output tables, it takes something more than the usual “willing suspension of disbelief” to talk seriously of the aggregate production function. But the aggregate production function is only a little less legitimate a concept than, say, the aggregate consumption function, and for some kinds of long-run macro-models it is almost as indispensable as the latter is for the short-run. As long as we insist on practicing macro-economics we shall need aggregate relationships.
Even so, there would hardly be any justification for returning to this old-fashioned topic if I had no novelty to suggest. The new wrinkle I want to describe is an elementary way of segregating variations in output per head due to technical change from those due to changes in the availability of capital per head. Naturally, every additional bit of information has its price. In this case the price consists of one new required time series, the share of labor or property in total income, and one new assumption, that factors are paid their marginal products. Since the former is probably more respectable than the other data I shall use, and since the latter is an assumption often made, the price may not be unreasonably high.
Before going on, let me be explicit that I would not try to justify what follows by calling on fancy theorems on aggregation and index numbers. 1 Either this kind of aggregate economics appeals or it doesn’t. Personally I belong to both schools. If it does, I think one can draw some crude but useful conclusions from the results.
I will first explain what I have in mind mathematically and then give a diagrammatic exposition. In this case the mathematics seems simpler. 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:
The variable t for time appears in F to allow for technical change. It will be seen that I am using the phrase “technical change” as a short-hand expression for any kind of shift in the production function. Thus slowdowns, speed-ups, improvements in the education of the labor force, and all sorts of things will appear as “technical change. ”
It is convenient to begin with the special case of neutral technical change. Shifts in the production function are defined as neutral if they leave marginal rates of substitution untouched but simply increase or decrease the output attainable from given inputs. In that case the production function takes the special form
and the multiplicative factor A (t) measures the cumulated effect of shifts over time. Differentiate (Ia) totally with respect to time and divide by Q and one obtains
where dots indicate time derivatives. Now defineand the relative shares of capital and labor, and substitute in the above equation (note that ∂Q/∂K= A ∂f/∂K, etc.) and there results:
* I owe a debt of gratitude to Dr. Louis Lefeber for statistical and other assistance, and to Professors Fellner, Leontief, and Schultz for stimulating suggestions.
1 Mrs. Robinson in particular has explored many of the profound difficulties that stand in the way of giving any precise meaning to the quantity of capital (“The Production Function and the Theory of Capital, ” Review of Economic Studies, Vol. 21, No. 2), and I have thrown up still further obstacles (ibid., Vol. 23, No. 2). Were the data available, it would be better to apply the analysis to some precisely defined production function with many precisely defined inputs. One can at least hope that the aggregate analysis gives some notion of the way a detailed analysis would lead.