Question

”Adam’s Apples” produces apple cider using labor (L) and apples (A) according to the Cobb-Douglas production function Q = F (L, A) = 2L^1/4 A^3/4 . The price of apples is pA = 2 and the price of labor is W = 10. Adam’s Apples also has a fixed cost for farm buildings, F C = 100.

1. If Adam’s Apples wants to produce 100 gallons of apple cider, Q=100, what is its lowest achievable input cost? (Roadmap: (1) Determine the optimal input choice (L ∗ , A∗ ) by tangency condition, MRTS_LA (L ∗ , A∗ ) = W/pA and production requirement 100 = F (L ∗ , A∗ ). (2) Variable costs are then found by W L∗ + p_aA∗ . (3) Total cost is sum of variable and fixed costs.)

2. Determine Adam’s Apples’ cost function, C (Q). Verify that C (100) gives you same answer as in question 1.

3. Determine Adam’s Apples’ Average Cost function, AC(Q), and marginal cost function, MC (Q)

Answer 1

1) Following tangecnycondition to find the optimal input.

MRTS_LA (L ∗ , A∗ ) = W/pA..........................1

............................................2

from eq 1 and 2

........................................3

A2 =30L    or A =15L

From Production Function and eq 3 which is optimal choice bundle

100 = F (L ∗ , A∗ ) = 2L^1/4 A^3/4

L* = 7.62 and A =15L = 7.62*15 =98.4

Cost = W L∗ + p_aA∗+100

Cost would if we plug these optimal values

Cost = 10*6.55 + 2*98.4+100 =362.3

b)

C(Q) =W L∗ + p_aA∗+100

C(Q) = 10L+2A+100...................................4

Put = Q = 2L^1/4 A^3/4 here A =15L L= Q/(2*15^3/4)

And A =Q15^1/4/2 PLug these values in eq 4

C(Q ) =10*15^-3/4Q/2+2Q15^1/4/2 +100

C(Q) = 5*15^-3/4Q+Q15^1/4 +100   if C(100) =362.3 Same as before

3

AC(Q) =C(Q)/Q = 5*15^-3/4+15^1/4 +100/100

Marginal Cost

MC =dC(Q)/dQ = 5*15^-3/4+15^1/4