Question

UCLA WINTER 2020 Problem Set 6 Due in the beginning of class on Thursday, March 12 1. Consider a small town that has only two Sushi restaurants, A and B. Suppose that it costs $3 to make each sushi (for each store) and that the relation between sales in each month (g. measured in thousands) and price (p, measured in dollars) for each firm is 9i = 12 - P:+Pj. (a) What is each store's monthly profit as a function of PA and po? (b) What are the Nash equilibrium prices, quantities, and profits for A and B? (c) Suppose that the two firms work together and collusively choose a joint best price pc. What single price do they choose to maximize their joint profits? Suppose the two stores are in a repeated relationship, trying to sustain the joint profit-maximizing price calculated in part (c) above. They print new menus each month and thereby commit themselves to prices for the whole month. In any one month, one of them can defect from the agreement, and the other firm will be unable to react until the following month. (d) If one of them holds the price at the agreed level, what is the best defecting price for the other? What are its resulting profits? (e) For what (monthly) discount factors will their collusion be sustainable by using "Grim Trigger" strategies? (f) For what (monthly) discount factors will their collusion be sustainable by using "perfect Tit-for- Tat" strategies? 2. Suppose there is 30% chance that a risk averse person with current wealth of $150,000 will suffer property damage of $30,000 due to a fire. This person has a utility function of U = In(W). (a) What is the actuarially fair price for insurance in this case? (b) Assuming this person does not buy insurance, what is the expected level of wealth? What is the expected level of utility? (c) What is the maximum premium this consumer is willing to pay for full coverage of $30,000? Verify your answer by calculating E(U) when fully insured. (d) Suppose this person can purchase full coverage at the actuarially fair rate. Is the consumer better off by doing so? (e) Would this person be better off, compared to the part (d) outcome, if he purchased insurance to cover half the loss at a fair price? (f) Suppose there is a second person, B, in the same situation as person A. Assume that the probability of fire for A is independent of the probability of fire for B. Suppose that insurance is no longer available, but A and B agree to equally divide any fire losses between themselves. (For example if A has a fire loss of $30,000, then B will give A $15,000.) Find the probability of each of the four possible outcomes: i. A has a fire, B does not have a fire ii. A does not have a fire, B does iii. both A and B have a fire iv. Neither has a fire.

Answer 1

1) Profit =(p-3)*q

Profit A=(12-pa+0.5pb)*(pa-3)=12pa-pa^2+0.5pa*pb-36+3pa-1.5pb=36-pa^2+15pa+0.5pb*pb

And by symmetry,

Profit firmB=36-pb^2+15pb+0.5pa*pb

2)∆ProfitA/∆Pa=-2pa+15+0.5pb=0

Pa=7.5+0.25pb

∆profit B/∆pb=-2pb+15+0.5pa=0

Pb=7.5+0.25pa

Putting pb into pa

Pa=7.5+0.25(7.5+0.25pa)=7.5+1.875+0.0625

Pa=9.375/0.9375=10

Pb=7.5+0.25*10=10

Qa=qb=12-10+0.5*10=2+5=7

Profit A=profit B=(10-3)*7=49

C) because both have same MC so they will charge same price as cartel

joint profits=2(36-pa^2+15pa+0.5pb*pb)=2(36-p^2+15p+0.5p^2=72-p^2+30p

∆profit/∆p=-2p+30=0

P=30/2=15

In collusion they will charge 15$.

D) let say firm A is charging same price and Firm B is defecting

The best defecting price

Pb=7.5+0.25*15=7.25+3.75=11