Question

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 = √ LT . (a) Consider the short-run decision, in which she has a fixed 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?

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?

I have already answered Question #1, posting it for reference needed to answer #3.

Answer #1

Q = L^{1/2}T^{1/2}

(3)

Cost is minimized when MPL/MPT = w/r

MPL = ∂Q/∂L = (1/2) x (T/L)^{1/2}

MPT = ∂Q/∂T = (1/2) x (L/T)^{1/2}

MPL/MPT = T/L = w/r

T = L x (w/r) = (Lw/r)

Substituting in production function,

Q = L^{1/2}(Lw/r)^{1/2}

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

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

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

T = [Q x (r/w)^{1/2}] x (w/r) = Q x
(w/r)^{1/2}

Substituting in total cost function: TC = wL + rT

TC = [w x Q x (r/w)^{1/2}] + [r x Q x
(w/r)^{1/2}]

TC = Q x [(wr)^{1/2} + (wr)^{1/2}]

TC = Q x 2 x (wr)^{1/2}

TC = 2Q x (wr)^{1/2}

(b)

Marginal cost = dTC/dQ = 2 x (wr)^{1/2}

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