Question

# 3. Consider Prunella in (1). Let denote w as the wage rate and r as the...

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?
(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?

Q = LT1/2

(3)

Cost is minimized when MPL/MPT = w/r

MPL = ∂Q/∂L = (T/L)1/2

MPT = ∂Q/∂T = (1/2) x L / (T1/2)

MPL/MPT = [(T/L)1/2] / [(1/2) x L / (T1/2)] = 2 x T / (L3/2) = w/r

T = L3/2 x (w/2r)

Plugging in production function,

Q = L1/2[L3/2 x (w/2r)]1/2

Q = L1/2 x L3/2 x (w/2r)1/2

Q = L2 x (w/2r)1/2

L = Q1/2 x (2r/w)1/4

T = [Q1/2 x (2r/w)1/4] x (w/2r) = Q1/2 x (w/2r)3/4

Substituting in total cost function: TC = wL + rT

TC = [w x Q1/2 x (2r/w)1/4] + [r x Q1/2 x (w/2r)3/4]

TC = Q1/2 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}] /(Q1/2)

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