Question

3. Consider Prunella in (1). Let denote w as the wage rate and r as the rental rate.
(a) Derive her total cost function.
(b) Based on (a), what is the marginal cost?

PS:1. Prunella raises peaches. L is the number of units of labour she uses and T is the number of
units of land she uses, her output (bushels of peaches), denoted as Q, is given by
Q =L*T^(1/2)
(a) Consider the short-run decision, in which she has a xed amount of land (T = 9). Does
this production function exhibit the diminishing marginal return on the labour input?
Explain your answer.
(b) If she uses 4 units of labour, then what is the marginal product of labour? What does
it mean?
(c) If she can sell peaches at the price of $5, and she has to pay the wage rate at $4, then
how many units of labour does she employ?

Answer 1

Q = LT^1/2

(3)

Cost is minimized when MPL/MPT = w/r

MPL = ∂Q/∂L = (T/L)^1/2

MPT = ∂Q/∂T = (1/2) x L / (T^1/2)

MPL/MPT = [(T/L)^1/2] / [(1/2) x L / (T^1/2)] = 2 x T / (L^3/2) = w/r

T = L^3/2 x (w/2r)

Plugging in production function,

Q = L^1/2[L^3/2 x (w/2r)]^1/2

Q = L^1/2 x L^3/2 x (w/2r)^1/2

Q = L² x (w/2r)^1/2

L = Q^1/2 x (2r/w)^1/4

T = [Q^1/2 x (2r/w)^1/4] x (w/2r) = Q^1/2 x (w/2r)^3/4

Substituting in total cost function: TC = wL + rT

TC = [w x Q^1/2 x (2r/w)^1/4] + [r x Q^1/2 x (w/2r)^3/4]

TC = Q^1/2 x [w x (2r/w)^1/4] + [r x (w/2r)^3/4]

(b)

Marginal cost = dTC/dQ = [(1/2) x {w x (2r/w)^1/4] + [r x (w/2r)^3/4}] /(Q^1/2)