# Consider a firm with the production function f(L,K) = L^{0.5}K^{0.5}. The wage rate and rental...

## Question:

Consider a firm with the production function {eq}f(L,K) = L^{0.5}K^{0.5}. {/eq} The wage rate and rental rate on capital are {eq}w {/eq} and {eq}r {/eq}, respectively.

a. Use the Lagrangian for cost minimization to do derive the long-run cost function for this firm.

b. Suppose the government provides a subsidy of $10 per unit of capital to the firm. Rewrite the long-run cost function.

## Cost minimization

Cost minimization is one of the objectives of the producer side of the market. The main motive of the business is profit motive, but another is cost minimization. The cost is minimal when the inputs and production techniques used are efficient, low cost. Cost act as a limit for production to be effective in production function analysis.

## Answer and Explanation: 1

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**a.**

Total cost function is:

{eq}TC = wL + rK{/eq}

Lagrangian cost minimization:

{eq}\begin{align*} Y = {L^{0.5}}{K^{0.5}}\\ \ell &= wL + rK +...

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Chapter 11 / Lesson 27Learn about the production function. Read the production function definition in economics, learn the production function formula. Plus, see graphs and examples.

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