Homework 7 Solutions1a) One self-fulfilling expectations equilibrium occurs at z = 0. Having z = 0 meansf(z) = 0, so reservation prices for all consumers x will be r(x)f (z) = 0. Therefore, ifthe belief is that no one will buy the product (z = 0) then indeed no one will buy theproduct.Two more self-fulfilling expectations equilibria are the solutions to r(z)f(z) =12. Solv-ing this equation gives the equilibria z0≈ 0.15 and z00≈ 0.85.1b) From the graph of r(z)f (z), we can see that r(z)f(z) is greater than the price p = 1/2 ifz0< z < z00, corresponding to upward pressure on demand, and r(z)f(z) < p elsewhere,corresponding to downward pressure on demand.The equilibrium 0 is stable, because if z is slightly higher than 0, then downwardpressure on demand drives z down to 0. The equilibrium z00is also stable, because if zis slightly off from z00then upward pressure on the left of z00and downward pressure onthe right of z00will drive z to z00. Lastly, z0is not stable, and is what we call a tippingpoint; if z is slightly off from z0, then downward pressure on the left of z0and upwardpressure on the right of z0will drive z away from z0(and towards either 0 or z00).1c) If a fraction 0.25 of the population is using the product, then because r(0.25)f(0.25) > pthere will be some people who would like to use the product and aren’t already. Thisdrives demand upward, and so popularity is trending upward. In fact, the popularityshould increase until a fraction z00of the population is using the product.1d) If only a fraction 0.10 of the population were to use the product, then as r(0.10)f(0.10) <p some people would have tried the product and found that it wasn’t worthwhile. Thiswould drive the popularity downward.COMMON MISTAKE: Many students cited upward or downward pressureto answer parts b-d), but didn’t show how they knew there were suchpressures. In order to have a complete explanation you need to considerr(z)f(z) as above or else solve r(x)f(z) = p for x and argue about the size ofx relative to z.1e) Now the price is too high for there to be any equilibrium besides 0; the maximum valueof r(z)f(z) is 1, so there can be no z ∈ [0, 1] with r(z)f(z) = 1.1. The equilibrium 0is still stable in this case, because r(z)f(z) is always less than the price p = 1.1 and sothere will always be downward pressure on demand which drives z to 0. In particular, ifz = 0.25 or z = 0.10, then r(z)f(z) < p and so popularity will be trending downwardstowards z = 0.COMMON MISTAKE: Some students decided that there was no equilib-rium in this case. This is clearly false as long as f(0) = 0, since that’s theonly fact used in showing 0 is an equilibrium in part a).2a) Based on the fact that company A’s market share quickly reverts to 60% as long as itsmarket share stays between 50% and 70%, it is likely that one stable equilibrium is for1company A to have 60% of the market share. We can also conclude that there can beno tipping point in the range of 50% to 70%.Based on the fact that the market share of company A started decreasing after it wasset at 40%, it seems that there is now downward pressure on demand for company A’sservice. This means there must have been a tipping point somewhere in the range of40% to 50%, and company A’s market share is now headed to some stable equilibriumat less than 40% market share.COMMON MISTAKE: Almost everyone addressed company A’s currentsituation, explaining why their market share was decreasing. But manystudents didn’t explain why the market share was reverting to 60% pre-viously. Since company A was expecting this to happen based on earlierobservations, a complete answer would include both why it was happeningbefore and why it isn’t happening now; also, the fact that 60% is a stableequilibrium is essential to justifying why your suggestions in part b) willwork.COMMON MISTAKE: Also note that there is not enough information inthe problem to conclude exactly where the tipping point is. Some peopleclaimed that the tipping point was exactly at 40% or 50%.2b) Company A could temporarily discount the price of its service, which would move thetipping point leftwards. If the price is lowered sufficiently, then company A’s currentmarket share might be above the new tipping point, which would cause there to beupward pressure on demand for company A’s service. After company A’s market sharehas recovered to at least 50%, their original price could be restored and their marketshare would move back to 60% (in this simple model).Another action company A could take is to advertise in order to encourage more peopleto use their service. If this manages to increase their market share above the tippingpoint, then their market share will again revert to 60%.COMMON MISTAKE: Many students correctly suggested lowering theprice, but didn’t say that the discount would be temporary. We can’tassume that company A can afford to lower its price indefinitely.3a) If an invitee is trying to decide whether to go to the party or not, he might look athow other invitees have responded so far. He has his own information about whetherthe party will be good or not, but so do all the other invitees responding. If he seesthat a sufficient majority have chosen not to attend, then he might think they knowsomething that he doesn’t and that the party will be bad for some reason. In thatcase, he would also decide not to attend regardless of his personal information aboutthe party. On the other hand, if a sufficient majority of people are attending, thenhe might think they know the party will be good, and he will therefore also wantto attend, again disregarding his personal information. An information cascade thatcomes from many people making sequential decisions this way would lead to the boomor bust effect.2COMMON MISTAKE: Some people had trouble distinguishing betweenan information cascade and network effects. For example, some said that ifyou can see that your friends are going, then you will want to just go alongwith their decisions. But just considering what your friends have done isn’tenough to explain the boom or bust effect in terms of information cascades,especially since we expect our friends to have information similar to oursand also can just ask our friends about their decisions. In fact, in theexample above it sounds like the main point is that you are more likely togo to a party that your friends attend, which is a direct benefit.Some answers were simply too vague.3b)