2. A perfectly competitive firm hires capital and labor at competitive price r a

Question

2. A perfectly competitive firm hires capital and labor at competitive price r and w. In the long run, the firm operates at a minimum average cost. Total cost is the sum of the resource expenditures i.e. (rK+wL) and output (Q) is produced according to the production function: Q=(K^)(L^). Average cost (AC) can therfore be expressed as AC=(rk+wL)/((K^)(L^))

a. Determine mathematically and explain the amount of capital and labor that will minimize average cost.

b. Show second order conditions for the minimum.

Explanation / Answer

AC=(rk+wL)/((K^)(L^)) = r*((K^1-)(L^)) + w*((K^)(L^1-))

dAC/dK = (1-)*r*((K^-)(L^)) + *w*((K^-1)(L^1-))

dAC/dL = *r*((K^1-)(L^-1)) + (1-)*w*((K^)(L^-))

Putting dAC/dK and dAC/dL = 0

  -(1-)*r*((K^-)(L^)) =  *w*((K^-1)(L^1-))

  -(1-)*r/ *w = ((K^-1)(L^1-))/(1-)*r*((K^-)(L^))

  (1-)*r/ *w = -((K^2-1)(L^1-2))

now , dAC/dL =0

*r*((K^1-)(L^-1)) = -(1-)*w*((K^)(L^-))

*r/(1-)*w = ((K^)(L^-))/((K^1-)(L^-1))

*r/(1-)*w = -((K^2-1)(L^1-2))

So,

(1-)*r/ *w =  *r/(1-)*w = -((K^2-1)(L^1-2))

(1-)*r/ *w =  *r/(1-)*w

(1-)/ = /(1-)

(1-)(1-) =

1 - - + =

= 1 -

So, Amount of caplital = K^

and Amount of labour = L^1-

d2AC/dK2 = -*(1-)*r*((K^--1)(L^)) + (-1)**w*((K^-2)(L^1-))

d2AC/dL2 = (-1)**r*((K^1-)(L^-2))   - (1-)*w*((K^)(L^--1))

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