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 = L^{2} 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})**

2. Consider Prunella in (1). In the long-run, she can adjust
both inputs.
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scale? Explain your answer.
(b) Suppose that the wage rate is the same as in (1) and the rental
rate for land is $5. If
she is going to produce 120 peaches, how many units of labour and
land is she going to
choose?
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