Question

please show all work and derivatives Consider a firm with long-run production function [ Q=f(L, K)={1}{2} \cdot L\frac{1}{2}} \cdot K{1}{2}} ] where ( Q ) is output, ( L ) is labor input, and ( K ) is capital input.

Answer 1

The question involves a long-run production function Q = f(L, K) where both labor and capital are variable, and the short-run scenario where capital is fixed, meaning Q = f(L). An example illustrates a secretarial firm's capacity to produce documents, constrained by fixed capital in the short run.

Step-by-step explanation:

The student's question pertains to the concept of a long-run production function in economics, which describes the output of a firm as a function of input factors labor (L) and capital (K). In the long run, both labor and capital are variable inputs, meaning that a firm can adjust the amount of these resources it uses in production. The given production function is Q = f(L, K) = ½ · L^{½} · K^{½}, which exhibits constant returns to scale. In the short run, however, capital is fixed, and therefore, the production function is reduced to Q = f(L) which means the output depends solely on the variable input of labor as capital cannot be adjusted.

For example, if a secretarial firm uses typists for labor (L) and personal computers as capital (K), initially they have a business level that requires one typist and one PC. This allows the production of five documents per day. If there is a rush order requiring 10 documents and the firm can only hire more typists but cannot acquire an additional PC in the short term, the firm must operate with fixed capital. Thus, production in the short run is constrained by the fixed number of PCs, regardless of how many additional typists are employed.