Given the production function: LK/3Show full answers and steps to part g) h) & i) ### Consider a Firm with Production Function**Production Function:** $\frac{LK}{3}$#### (a) Marginal Product of Labor ($MPL$)- **Task:** Find the marginal product of labor, $MPL$.- **Next:** Indicate whether the marginal product of labor is decreasing, constant, or increasing in labor. - **Action:** Explain or show your answer.#### (b) Marginal Product of Capital ($MPK$)- **Task:** Find the marginal product of capital, $MPK$.- **Next:** Indicate whether the marginal product of capital is decreasing, constant, or increasing in capital.- **Action:** Explain or show your answer.#### (c) Marginal Rate of Technical Substitution ($MRTS_{L,K}$)- **Task:** Find and simplify the expression of the marginal rate of technical substitution of labor for capital, $MRTS_{L,K}$.- **Next:** Indicate whether $MRTS_{L,K}$ is decreasing, constant, or increasing in labor.- **Explanation:** What does this $MRTS_{L,K}$ suggest about the shape of the isoquants and the firm’s ability to substitute between inputs?#### (d) Returns to Scale- **Task:** Show whether the firm’s production function exhibits increasing, decreasing, or constant returns to scale. (g) Now, assume the firm operates in the long run. **Formally** (with notation) state the firm’s cost minimization problem (CMP).(h) **Solve** the firm’s CMP in the long run if they want to produce the output target of $ q = 256 $ for given input prices $ w = \$4 $ and $ r = \$3 $. Next, **calculate** the minimized total cost of using their optimal input combination in the long run.(i) **Briefly compare** how the firm’s optimal input decision varies from the short run to long run.

A firm has a choice of either maximization of output given the cost constraint or the minimization of cost given the output constraint. Since in the long run a firm has both the inputs labour and capital as variables so the representation of the cost minimization problem is as follows:wL+rK=C is the equation of the cost constraint.wL:is the total labour cost; rK: is the total capital cost. Q=LK/3 is the production function V= wL+rK+§(Q-LK/3)............(i)Is the representation of the cost minimization problem V=4L+3K+§(256-LK/3)dV/dL= 4+§(-K/3)=12/K=§........(ii)dV/dK= 3+§(-L/3)= 9/L=§..........(iii)dV/d§= 256-LK/3=0.............(iv)Equating equation (ii) and (iii)12/K=9/L12/9=K/L4/3=K/LK=1.333LSubstituting K=1.333L into equation (iv) we get:256-L(1.333)/3= 0256-0.444L =0L=256/0.444L*=576.57K*=1.333(576.57)= 768.57C=4×576.57+3×768.57=2306.28+2305.71C*=4611.99=4612 is the minimized total cost in the long run.The firm's input decision changes from the short run to the long run because in the short run capital is usually the fixed factor of production and only labour is the variable factor of production whereas in the long run both capital and labour are variable factors of production. Therefore from the short run to the long run the firm's decision on optimal factor choice changes.

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Consider a firm with production function: LK 3 (a) Find the marginal product of labor, MPL. Next, indicate whether marginal product of labor is decreasing, constant, or increasing in labor. Explain or show your answer. (b) Find the marginal product of capital, MPK. Next, indicate whether the marginal

Consider a firm with production function: LK 3 (a) Find the marginal product of labor, MPL. Next, indicate whether marginal product of labor is decreasing, constant, or increasing in labor. Explain or show your answer. (b) Find the marginal product of capital, MPK. Next, indicate whether the marginal

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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