Here is a way of seeing why greater input variety in production is akin to technical progress. Suppose that an output Y is produced by n different kinds of machines, X₁, X₂, . . . , X_n. No machine is necessary in production, and there are diminishing returns to each machine, but the whole process exhibits constant returns to scale. One way of capturing this is to use the production function:

where α lies between zero and 1.

(a) Show that if all machine inputs are increased by some factor λ, then output also increases by the same factor. This means that there is constant returns to scale in production.

(b) Show that if there are at least two different kinds of machine inputs, then there is diminishing returns to each input.

(c) Now suppose that each machine has a new blueprint for its manufacture, but once the blueprint is known, B units of any of these machines can be produced by using one unit of “capital” (which is measured in the same units as final output). Show that if a given amount of capital K is available, it should be used to make the same quantity of each machine so as to maximize the final output from production.

(d) Show that for a given variety of machines (i.e., given n), the production function can be rewritten as a function of aggregate “capital” K in the form

(e) Now using this expression, show that both an increase in the productivity of machine-production and the variety of available machines is akin to technical progress. Provide intuition for why an increase in machine variety is akin to an increase in overall productivity.

Transcribed Image Text:

Y = (X + X2+.+Xa)¹/a,