3,A.) Consider the problem of a big corporation (the principal) buying an intermediate good from a small supplier (the agent), but being unable to specify a level of quality in the contract.

Assume that the agent's utility level in each period depends on the price she receives and the quality she provides. Specifically, per-period utility is determined by equation 10.1 in the textbook (page 523).

Assume that

u--=2u_=2. The termination function that determines the likelihood that the corporation will renew the contract to its supplier is also the same used in the textbook:t=1?qt=1?q.

All other assumptions regarding the principal and the agent are the same as in the textbook (and lecture notes).

(a) What is the lifetime value of the relationship for the agent?

\f(b) Derive the agent's best response function: q=(c) Profit maximization implies that the principal will offer a price satisfying the following condition: 1 p2 =(d) What is the price that the principal will offer to the agent? Answer: (e) What is the quality level that will be provided by the agent? Answer: Application: When Rent-Seeking Does Not Equilibrate a Market: A Housing Bubble 523 Modeling a price bubble and a crash: Stable and unstable equilibria The key to the bubble and the crash is that a change in prices can lead people to expect a future price change. To see how this works, in Figure 9.23 (a) we show the case where starting from an out-of-equilibrium price, REMINDER Getting to an the buyers and sellers converge on an equilibrium price. The green curve equilibrium In Chapter 5 we labeled p(p) tells us the price we will expect later if the price now is as given asked how the two fishermen on the horizontal axis. This is the expected price function. So if the price might get to a Nash today is the low price p, then the expected price later will be somewhat equilibrium if they were at some point not at the higher indicated by P. equilibrium. We showed how If our expectations were correct (which we assume), then that price this could happen and also becomes the new actual price now, indicated by the horizontal arrow to the asked you to think of a case in p = p. But if the price now is at that level, then the expected price (repeating which it would not happen. the above process) will be still higher. This process will continue until the The "not happen" case is what price is p. in which case the price now is also the expected price later. So we are studying here with the unless something changes from outside the model, there are no further price bubble. changes in the price. You can see that had the price been higher than pe initially, then the | process could have brought the price down to pe. This process describes . REMINDER A value in some how market equilibration works, when it works. The price p, is a stable process-for example, a particular price of equilibrium price. housing-such that the In Figure 9.23 (b) we show a different price expectation function. There process is qualitatively are two stable equilibria in panel (b), like point e in panel (a), at points a and different for values higher or c. But there is also point b. If the price is a little above the price at point b, lower than this value-prices then you can see from the green expected price function that people would rising above it or falling below expect the price to be even higher next period. And if the price were higher, it, for example-is called a then prices would continue to rise up to point c. tipping point. If the initial price is a bit below the price at point b then a similar process will lead prices to fall to point a. Point b is called a tipping point. At prices > EXAMPLE Tipping point above the tipping point the economy 'tips' up to point c. At prices below the The temperature 0 degrees tipping point, it tips down to point a. Celsius (32 degrees To see how a bubble can happen imagine that the economy is at point a Fahrenheit) is a tipping point Now think about some change that would result in most people expecting for water, above that point prices to rise next period: for example the government just announced a water is a fluid, below it it program to subsidize housing costs for families with children, or banks just becomes a solid (that is, ice). announced lower interest rates on mortgages, making it easier for people Here, the value of 0 degrees to borrow to buy houses. Celsius is the "critical value" and the "qualitative This would shift up the price expectation curve to the dashed line in the difference" is how water figure. For every price today, the expected price later would be higher. The transitions from being a liquid result would be the stable price equilibrium at point a would no longer exist. (its initial quality) to a solid (a And the same is true of the 'tipping point' b. What would then happen? different quality), therefore The logic of Figure 9.23 (a) tells us that prices would continue rising until demonstrating a qualitative the economy reached point c. This is a price bubble. change