A cost minimizing firm has the following short run production function: Q=f (L,K

Question

A cost minimizing firm has the following short run production function:
Q=f (L,K)=72L+5L2-0.2L3

a) Briefly explain why this is a short run production function.

b) At what level of employment would diminishing returns set in for the variable input?

c) Fin the levels of employment that define stage two in the production process.

d)If the market determined real wage rate is $20 determine the amount of labor a cost minimizing firm would hire in order to minimize the total cost of production.

2) a) Briefly explain the concept of "Return to scale". What form of "Return toScale" does the follwing production function exhibit? Explain

b) Q=f(L,K)=L0.3 K0.7

Suppose a cost minimizing firm wihses to change its scale of production and needs to know the combination of labor and capita to employ. If the firs's total cost outlay is $14. Fin the this combination using the information below and given that the prices of labor and capita are $1 and $3 pert unit repectively.

Q= 0 1 2 3 4 5 6 7 8

TPI=0 11 20 28 35 41 45 46 46.5

TPK=0 24 45 63 78 87 93 96 97

where TPI AND TPK represent total product of labor and total product of capital respectively.

3) Given the deman and supply schedules ofr semiconductors in a perfectly competivite market:

Qd=60-10P+0.5I, Qs=-10+4P-2C

Where I=average income of the demanders of this production

C=unit cost of the inputs used in the production of this commodity.

a) Find the expression for equillibriud price and equillibrium quantity transceted in this market.

b)Find the initial equillibrium and equillibrioum quantity if I=1000 and C=20

c) Determine the impact of an increase in income from 1000 to 1200 on the initial equillibrium values.

d) Assume that an improvement in the teechnoly of producing this commodity results in an increase in supply by a factor of 5, how will this development affect the initial equillibrium values?0

Explanation / Answer

You're not allowed to post multiple questions. I'll answer the first one. Post the second separately and I can take care of it also. A) This is a short-run production function because K is not included. In the long run, the firm can make tradeoffs between L and K. B) Take the derivative of Q with respect to L, set equal to zero. Q = 72L + 5L^2 - 0.2L^3 dQ/dL = 72 + 10L - 0.6L^2 = 0 use the quadratic formula L1 = (-10 + (10^2 - 4*(-0.6)*72)^0.5)/(2*(-0.6)) L1 = -5.43054855 L2 = (-10 - (10^2 - 4*(-0.6)*72)^0.5)/(2*(-0.6)) L2 = 22.0972152 L1 is irrelevant because we can't hire negative labor. So, diminishing returns sets in after we hire 22 workers. C. We are not given any information about what "stage 2" is. This is not a standard economic term. D. We could solve this easily with a demand curve or information on MPK and r. Cost minimization implies: MPL/W = MPK/r MR = W/MPL But we don't have those. So, we can't find the solution with the information given. We can prove this using calculus. Min 72L + 5L^2 - 0.2L^3 s.t. 10L = C the Lagrange function is: V = 72L + 5L^2 - 0.2L^3 + v(10L - C) where v is a Lagrange multiplier. Set dV/dL = 0 dV/dL = 72 + 10L - 0.6L^2 + v10 = 0 use the quadratic formula L1 = (-10 + (10^2 - 4*(-0.6)*(72+v10))^0.5)/(2*(-0.6)) L2 = (-10 - (10^2 - 4*(-0.6)*(72+v10))^0.5)/(2*(-0.6)) If v = 0, this implies an interior solution and L = 22.0972152 or about L = 22. If v = 1, this implies a corner solution. L = C/10, but we have no information on C.

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