Question

3. Optimal Production under Uncertainty. Assume a firm owner is an expected utility maxmizer and has NM utility function u(x) = ln(x) where x is final wealth. She has initial wealth of w, a constant unit cost of production, c = 5, and fixed costs of co= 2. She faces a maximum production capacity of y = 20. (a) The output price is uncertain: with probability p= .5 the price is 3 and otherwise the price is 9. What is the optimal production level y*? (b) Suppose there is an increase in the fixed cost from 2 to 5, how will the optimal output change? Relate the change to the firm owner's risk attitude. (c) Suppose fixed costs are back to 2, but the owner has a new be- lief about the price. She now believes that the output price will be 4 with prob. .5 and 8 otherwise. What is the new optimal production? Explain your finding.

Answer 1

Given that Solution VNM utility function UCM) = ln (4) where x is final wealth. initial wealth is w unit cost it Production = 5 fixed cost co:2 mai run Voduct Capacity 3 - 26 - a Price Pr = 9 x = w+ P, -P- co} Probability - 5 Price pe 3 : V- EGY) = u(M) = ln (2) P utinos V = E(y) = 0.5 ln {W + 3y -54-2] + 0.5 ln {wt 94-54-2] → *[n[w439 - Jy -2] ++ [1129794-54-2}} 7 *[n[w-214-] + [[wtyy-2)]] v + [in (Way-2) +(wtyy-2)] Apply Desivation with respective y on both sides. t lave met betre) -

-49+ ما - وی (2- 1-ما ) 4 - (1- F9 دما )ة - - او - ماها - و% + ماه دی 0= 9 + - +- % 9+- مابا - ماه = 0 -22 +9 + = 0= - بان) 1 - ماه - =0 a( - 89 -1) = لما-ه = (ما) - مها : - (-) 9% = و و 240 و لم و 10 :: و 20 = 160 / 8 = 9 ) 166> 162 ک دما و با

by Increase in fixed cost C = 5 Probability : 5 Poice P:3, Price P2 = 9 V = E() - 0.560 [w+39-54-53+0:51n[ 13+99-54-] 12{dn{W-24-5] + In { wtqy - 5] v £{dn (W-ry-5) thn (w tyy=5] Apply Desovation on both sides with respect to y duet (ways wanted 0 7 W-2y 5 - 2 (Wtyy 5 2W + 8y-10 Wtyy - 5 & 4 (W-24-5 = 40-by-20

->2w+84-10 = 46-8y-20 → 96-20-84-8y-204/0 - 0 2W-164-10=0 2 (W-84-5)=0 wr8y-5:0 - 8y = W-5 .. . Dittenentiakon clearly , Lyt U(x) = In (A) the owner is Risk averse. RW - - -Y Diferentiatan with resped = - 48 x=3 = P(X) + Again, Differentiation with respect to s' with Risk factor P(x) pe'n con Hence, increased fined cost will induce the conto dachon 12 output.

Co=2 Probability : 0.5 Price P, 4 Price Pr = 8 optimal Production level V: Ely) - 0.5{ ln(w +49-59 -2) +10(1078y - 5y-2)] tl An (w-y-2) 40 (6439-13 Apply derivation on both sides with respect to y be tolyn twig) -- - + 30 wy-2 W134-2

t w-y-2 - 3 W+34-2 - → → - - Coty -2 & 3 (W-Y-2) W-3 +34 +3y –2 +6 -0 -2W + 6y +4 =0 -2(W-3y - 2) =0 W-3y2 = 0 - 3y = W-2 W-2 duction W-2 New optimal Production y* W-2 As compared to a, with a change in Probabity distridution to Prices a: (P, , P) = (3,9), c: (P., B)= (4,8) will increase the optimal aufput. This is due to logarithmic utility function with exhibits a risk avesse individual. So, that even expected price it is same, the optimal Wilih function will be same, due to concauity A In function