Question

In: Economics

Consider the production function ? = ? 1 3? 2 3. Let r and w denote...

Consider the production function ? = ? 1 3? 2 3. Let r and w denote the prices of K (capital) and L (labor). Suppose ?̅ = 27 (fixed) and L is variable.

a. Write down the expression for the short-run production function.

b. Compute the marginal product of labor (??? ) and the average product of labor (??? ). Is the ??? increasing or decreasing? Is ??? >, <, or = ????

c. Find expressions for fixed cost [F], variable cost [VC(q)], average cost [AC(q)] and marginal cost [MC(q)].

d. Find the production efficient level of output (q * ) where unit costs (AC) are cheapest (minimized), your answer will depend on w and r.

e. If a $15 specific tax per unit is imposed, what are the new cost functions (FC, VC, AC and MC)?

Expert Solution

q = L^1/3K^2/3

(a)

In short run, K = 27. Substituting in production function,

q = L^1/3(27)^2/3

q = 9 x L^1/3 [Short run production function]

(b)

MPL = dq/dL = [9 x (1/3)] / L^2/3 = 3 / L^2/3

APL = q/L = 9 / L^2/3

With increase in L, (L^2/3) increases, so MPL is decreasing.

Since 3 < 9,

(3 / L^2/3) < (9 / L^2/3).

In other words,

MPL < APL.

(c)

Short run production function: q = 9L^1/3

L^1/3 = q/9

Cubing:

L = q³/729

Fixed cost (FC) = Total capital Cost = rK = 27r

VC(q) = Total labor Cost = wL = w x (q³/729)

TC(q) = FC + VC(q) = 27r + w x (q³/729)

AC(q) = TC(q)/q = (27r/q) + (wq²/729)

MC(q) = dTC(q)/dq = (wq²/81)

(d)

Average cost is minimized when dAC(q)/dq = 0

dAC(q)/dq = -(27r/q²) + (2wq/729) = 0

27r/q² = 2wq/729

q³ = (729 x 27r)/2w = 9,841.5 x (r/w)

q = 21.43 x (r/w)^1/3

(e)

A specific tax is a form of fixed cost, imposition of which will have no change in VC(q) and MC(q)..

New FC = 27r + 15

New TC(q) = 27r + 15 + (wq³/729)

New AC(q) = TC(q)/q = (27r/q) + (15/q) + (wq²/729)

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