Show full answers and steps to part d) e) & f) Consider a firm with production function:LK3(a)Find the marginal product of labor,MPL. Next, indicate whether marginal product oflabor is decreasing, constant, or increasing inlabor. Explain or show your answer.(b)Find the marginal product ofcapital, MPK. Next, indicate whether the marginalproduct of capital is decreasing, constant, orincreasing in capital. Explain or show your answer.(c)Find and simplify the expression of themarginal rate of technical substitution of labor forcapital, MRTSL.K. Next, indicate whetherMRT SL,K is decreasing, constant orincreasing in labor, and explain what does that thisMRTSL,K suggest about the shape of theisoquants and the firm's ability to substitutebetween inputs.(d)] Show whether the firm's productionfunction exhibits increasing, decreasing, orconstant returns to scale.< (e) ₁Suppose the firm operates in the shortrun: Capital is fixed at 24. Solve the firm's costminimization problem (CMP) to find the inputdemand of labor for any output level q in the shortrun.(f) [} Assuming inputs prices arew =: $4 (price of labor) and r = $3 (price ofcapital), use your answer in part (e) to find theoptimal amount of labor the firm uses to produce256 in the short run. Next, find the simplifiedexpression for the firm's short-run total cost9=curve/function and calculate the minimized totalcost of producing the output target.

Production function: Here q denotes the level of output produced using labor L and capital K. In the short run, capital K is fixed at K =24. wage rate w = $4 and r =$3Return to scale refers to the change in output resulting from a proportional change in all inputs. Return to scale examines how simultaneously increasing in all inputs affects the output. Part d)Scale up the production by λ times. This means put L = λL and K= λK, Here λ>1The given production function is homogeneous of degree 2. This implies if all inputs are increased by the same proportion then output will increase by a larger proportion. So, this production function exhibits in increasing return to scale.Part e)In the short run, K =24Short-run production function will be To find the short-run input demand function for labor, solve the above equation for LThe short-run input demand function for labor is represented by equation 1. This is also the cost-minimizing level of input in the short run. Part f)Input demand function for labor in the short run: To produce q =256, the optimal amount of labor is calculated asDetermine the expression for the short-run cost function:Total cost is the cost of labor and capital used in production. Putting L = q/8 and K =24The minimum cost of producing 256 units in the short run is calculated asPart d) Production function exhibits in increasing return to scale.Part e) Short-run input demand for labor is Part f) Cost minimizing level of labor for q = 256 is 32 and short run cost function is C(q) =0.5q+72The minimum cost of producing 256 units is $200

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Economics

(e) ₁ Suppose the firm operates in the short run: Capital is fixed at 24. Solve the firm's cost minimization problem (CMP) to find the input demand of labor for any output level q in the short run. (f) [ 1 Assuming inputs prices are W= $4 (price of labor) and r = $3 (price of capital), use your answer in part (e) to find the optimal amount of labor the firm uses to produce 9 256 in the short run. Next, find the simplified expression for the firm's short-run total cost curve/function and calculate the minimized total cost of producing the output target. =

(e) ₁ Suppose the firm operates in the short run: Capital is fixed at 24. Solve the firm's cost minimization problem (CMP) to find the input demand of labor for any output level q in the short run. (f) [ 1 Assuming inputs prices are W= $4 (price of labor) and r = $3 (price of capital), use your answer in part (e) to find the optimal amount of labor the firm uses to produce 9 256 in the short run. Next, find the simplified expression for the firm's short-run total cost curve/function and calculate the minimized total cost of producing the output target. =

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

Show full answers and steps to part d) e) & f)

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
product of capital is decreasing, constant, or
increasing in capital. Explain or show your answer.
(c)
Find and simplify the expression of the
marginal rate of technical substitution of labor for
capital, MRTSL.K. Next, indicate whether
MRT SL,K is decreasing, constant or
increasing in labor, and explain what does that this
MRTSL,K suggest about the shape of the
isoquants and the firm's ability to substitute
between inputs.
(d)
] Show whether the firm's production
function exhibits increasing, decreasing, or
constant returns to scale.
<

(e) ₁
Suppose the firm operates in the short
run: Capital is fixed at 24. Solve the firm's cost
minimization problem (CMP) to find the input
demand of labor for any output level q in the short
run.
(f) [
} Assuming inputs prices are
w =
: $4 (price of labor) and r = $3 (price of
capital), use your answer in part (e) to find the
optimal amount of labor the firm uses to produce
256 in the short run. Next, find the simplified
expression for the firm's short-run total cost
9
=
curve/function and calculate the minimized total
cost of producing the output target.

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