Question

3. Optimal Production under Uncertainty. Assume a firm owner is an expected utility maxmizer and has VNM utility function u(x) = In () 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

2019 12 Wednesday ulu = lnn w= final wealth wa initio wealth- c = 5 = constant unit Co = Q = lined costi cost of productions Lunch ỹ = 20 [mox ponduetiero capacity 1 a a) optimal peraderevankeudo V-Ely) - 5 In[w+ 3y - 5y -2] +.5 en [wtqy - sy-2 / = I en I w-ay-2) + ln (w + 4y - 2 ]]

JUIL 13 Thursday 2019 14 Friday ] SAM - W+Y4-2= 2 [w-24-2 10 w+44-2 = 2w -4y - 4 a vlw= enn The owner is ourR eu - - Um) volu] averse. he - Allo = Co -1 ya Lunch h ence increased Jeud cost indire contractor in outputs but y =20 w-2 sibo W S162 2 PM (6 incuare in frud cost from a to 5 1 V-Ely) = new fou ln [ w + Yy_sy -2] + entw + 8y - sy -2]] W+ 4y - 5 = 2[w-ay-sl PH = -5 Uys W-5 #3 -4- W w 2 + 34 - charly y, t < ya W+ 3y - 2 = 30 - 3y - 6 Important Notes y = Qw-4 =W-2 rtant Notes JUNE 2019

Explanation for part c)

As compared to part a) a change in probability distribution of prices, increases the optimal output, this is due to logarithmic utility function, which exhibits a risk averse individual.

such that even though expected price remains same, the optimal changes, due to concavity of the ln function.