A firm produces output using labor, L, and capital, K, measured in man-hours and machine-hours, respectively. The firm’s production function f(L, K) = √ √ LK, with associated marginal product functions MPL = √K and MPK = 2 L√ √L . The price of labor is $10 per man-hour and the price of capital is $40 2 K per machine hour. The firm’s production target is 100 units of output per day. (a) Does this firm’s cost minimization problem have an interior or a corner solution? Justify your answer. (b) How many man-hours and how many machine-hours per day are used in the firm’s cost minimizing input combination? (c) What is the minimum cost for this firm of producing 100 units of output per day? *Please fully explain out math process **Production and Cost Minimization Problem**A firm produces output using labor $L$ and capital $K$, measured in man-hours and machine-hours, respectively. The firm's production function is given by:\[ f(L, K) = \sqrt{LK} \]Associated with this production function are the marginal product functions:\[ MP_L = \frac{\sqrt{K}}{2\sqrt{L}} \]\[ MP_K = \frac{\sqrt{L}}{2\sqrt{K}} \]- The price of labor is $10 per man-hour.- The price of capital is $40 per machine-hour.- The firm's production target is 100 units of output per day.**Questions:**(a) Does this firm's cost minimization problem have an interior or a corner solution? Justify your answer.(b) How many man-hours and how many machine-hours per day are used in the firm's cost minimizing input combination?(c) What is the minimum cost for this firm of producing 100 units of output per day?

The Cobb-Douglas production function is based on Paul H. Douglas and C.W. Cobb's empirical analysis of the American manufacturing industry. It is a degree one linear homogeneous production function that takes into consideration two inputs, labour and capital, for the total output of the manufacturing industry.

Answered: A firm produces output using labor, L,…

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Chapter1: Making Economics Decisions

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A firm produces output using labor, L, and capital, K, measured in man-
hours and machine-hours, respectively. The firm’s production function f(L, K) = √ √ LK, with associated marginal product functions MPL = √K and MPK = 2 L√ √L . The price of labor is $10 per man-hour and the price of capital is $40 2 K per machine hour. The firm’s production target is 100 units of output per day.
(a) Does this firm’s cost minimization problem have an interior or a corner
solution? Justify your answer.
(b) How many man-hours and how many machine-hours per day are used
in the firm’s cost minimizing input combination?
(c) What is the minimum cost for this firm of producing 100 units of output
per day?

*Please fully explain out math process

**Production and Cost Minimization Problem**

A firm produces output using labor $L$ and capital $K$, measured in man-hours and machine-hours, respectively. The firm's production function is given by:

\[ f(L, K) = \sqrt{LK} \]

Associated with this production function are the marginal product functions:

\[ MP_L = \frac{\sqrt{K}}{2\sqrt{L}} \]

\[ MP_K = \frac{\sqrt{L}}{2\sqrt{K}} \]

- The price of labor is $10 per man-hour.
- The price of capital is $40 per machine-hour.
- The firm's production target is 100 units of output per day.

**Questions:**

(a) Does this firm's cost minimization problem have an interior or a corner solution? Justify your answer.

(b) How many man-hours and how many machine-hours per day are used in the firm's cost minimizing input combination?

(c) What is the minimum cost for this firm of producing 100 units of output per day?

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