Question

1. A firm has the production function () = 1/3, 1/3 Assume that w = $1 a. Derive the firm's short-run total cost function. Using this equation, compuu nction Q=K1313. Assume that w = $10, r = $25 and K-5. of producing 4 units of output. b. Compute fixed cost (FC), variable cost (VC), average fixed cost (APC), average van cost (AVC), and marginal cost (MC) when Q = 4. pute the profit-maximizing level of output when price P=$384. Illustrate graphically.

Answer 1

Do note that part c can also be calculated by making (dπ/dQ) = 0.

Instead what we have done is :

dπ/dQ = (dπ/dL)*(dL/dQ) = 0

d(dπ/dQ)/dQ = Second order derivative of π is been replaced by d(dπ/dL)/dL due to similar calculations.

We know that dL/dQ is only equal to zero when L=0 which is not desirable.

Hence we have used the condition dπ/dL = 0 to make the solving easier.

You could still do the problem using dπ/dQ = 0 though, it will give the same answer.

I've just written the functions in part b in terms of Q as they are being used in the graph and the visualisation becomes easier.
DATE 1) Givent - W = $10 r = $25 K = 5 Price of Labour app Price of Capital Pr a) The shost run TC = total cost function can be derived x.1) + xk) = 10L + 25x5 TC = 10L +125 Also given that 8 = 4 = 8 = k'31"} I 4 -5 [" Cubing both sides we get 64 = 5L 2. LE 64.128 substituting it in equation . π = 10x12.8 + 125 = 128 +125 T = $253 , we get

Total Cost (TC) Variable Cost (Vc) + Fixed Cost (FC). TOL+ 125 = vc +FC - VC=101) TECE 25 Mcdefie) - L125) Now 8 = k'' ! - KES = 533 * 23 3 MO = AC10L +126) all 10* sertatis MC = 30 15 Now, @ 8 = 4, we know that L-645 2 VC = 10L = 128 FC = 125 LAVC = 10L 128= 32 ; AFC =125 - 31.25 Me 115. 30x62", 30 x6 ge

DATE question can e) The profit function of the given I be written as T = Total Revenue - Total Cost Total Revenue Po o l :.T=PQ - TC A = PO - 10L +125) T = P5B 1 - 10L-125 Criven P = $384. T : 384*5'3 XL 210L-125 Por maximisation da o . da 384x5 B 1 Lt - 10 di T ib 3 - 128x $'s too 213 dL OM = 128x51 - [²² - 10 6448x5 2X5 - 12

other arou =L=512 - 11 -102.4 = Q = 5 h L' = gys 8 h 10 = 8 The second order condition is ISME a co dL 126251 (3) 193 Zo w L0 AL- I Hence we have proven that the pt. of profit maximisation = 102.4 is where 8 = 8 Now, =SL

.. VC = (DL 20x8 - 28 Ve = 23 ___AVL = _x = 2g- DANE - 281 IFC = 125 Ctrl MC 3073-30 Meeg IMC Neme Ave 384 P-MR X71 FC AFC