# 2015-3 Final Solutions ```2015-3 Final Solutions
1a) Once a consumer owns one unit, having an additional unit is worth at most 50-40=10 for type
R and 35-30=5 for type P. That’s below the firms’ cost of 12.
1b) i. Second-degree PD entails offering a menu of options (e.g. different quantities or qualities)
to consumers. The idea is that different types of consumers will voluntarily pick different
menu items. This can allow a firm to extract more surplus than offering the same
price/product to all, even if it cannot initially tell the different types apart. However, offering
the first-best bundle to all is typically not incentive compatible: there’s usually a type of
consumer that would get positive surplus by choosing another type’s first-best bundle.
ii. No: here, firms can vary neither the quantity nor the observable quality of the good. Thus,
each firm can only offer one menu item: “one unit.”
1c) From A: 𝑤0 + 0.8𝑣𝐻 + 0.2𝑣𝐿 − 𝑃𝐴
From B: 𝑤0 + 0.6𝑣𝐻 + 0.4𝑣𝐿 − 𝑃𝐵
Willingness to pay is the value of P that equates these with 𝑤0 . Thus:
For type R: 48 from A, 46 from B
For type P: 34 from A, 33 from B
1d) Adverse selection arises when asymmetric information causes the part of the informed side
of the market that is worst for the uninformed side to be most likely to show up, at a given
price. There is no adverse selection here: information about the good is perfectly
symmetric, and at a given price, firms don’t care about whether consumers are rich or poor.
1e) Note that type R is indifferent between A and B when 𝑃𝐴 = 𝑃𝐵 + 2, and type P is indifferent
when 𝑃𝐴 = 𝑃𝐵 + 1.
- If 𝑃𝐵 &lt; 10, firm A cannot make positive profit, so it sets 𝑷𝑨 &gt; 𝑷𝑩 + 𝟐 in order to avoid
selling units.
- If 𝑃𝐵 = 10, again firm A cannot make positive profit, so it sets 𝑷𝑨 ≥ 𝟏𝟐 in order to make
zero profit.
- If 𝑃𝐵 ∈ (10,11], firm A can make positive profit from (and only from) type R. It wants to
set 𝑃𝐴 as high as possible without hitting 𝑃𝐵 + 2. Technically, the BR does not exist.
- If 𝑃𝐵 ∈ (11,33], firm A will either try making positive profit just type R, or from both
types. It wants to set 𝑃𝐴 as high as possible without hitting either 𝑃𝐵 + 1 or 𝑃𝐵 + 2.
Again, technically, the BR does not exist.
- If 𝑃𝐵 &gt; 33, firm A can serve all consumers with a price of 34. This yields a profit of 22
per consumer. If it decided to instead only serve type R, it would make an average profit
of at most 0.6(48-12)=21.6. Thus the BR is 34 here.
2a) First-best contracts must involve fixed wage: if Tracy bears any financial risk, the firm could
raise expected profit by instead paying Tracy the certainty equivalent, which must be lower
in expectation because Tracy is risk-averse.
Type E: w=44100. Expected profit = 0.75*90000+0.25*70000-44100 = 85000-44100 =
40900. This is better than the alternative of letting Tracy go, which would yield zero profit.
Type B: For e=1, the required wage is (200+20)2 = 48400. Expected profit = 85000-48400 =
36600. This is better than the alternative of letting Tracy go, which would yield zero profit.
This is also better than the e=0 alternative, which involves w=40000 and would yield
expected profit 0.25*90000+0.75*70000-40000 = 75000-40000 = 35000.
2b) Again, firm can make 35000 by inducing e=0 or 0 by letting Tracy go.
To induce e=1 instead, firm solves:
𝑚𝑎𝑥𝑥,𝑦 85000 − 0.75𝑥 − 0.25𝑦, where 𝑥 is the wage in case of success and 𝑦 is the wage if
project is OK, subject to:
(IR) 0.75√𝑥 + 0.25√𝑦 − 20 ≥ 200
(IC) 0.75√𝑥 + 0.25√𝑦 − 20 ≥ 0.25√𝑥 + 0.75√𝑦
IR must bind: otherwise, can raise profit by reducing 𝑦.
IC must bind: otherwise, can raise profit by reducing 𝑥 and increasing 𝑦 in such a way that
the expected utility (left-hand side of IR and IC) is unchanged, which insures that IR would
still hold. This must decrease expected wage (and thus raise expected profit) because it
reduces the risk borne by Tracy without increasing expected utility, and Tracy is risk-averse.
We now have a system of two equations and two variables. Solving gives √𝑥 = 230 and
√𝑦 = 190, so expected profit is 85000 − 0.75𝑥 − 0.25𝑦 = 36300. This is better than the
other options. Thus the firm offers 52900 if the project is successful, 36100 if it is OK.
2c) Note that the contract from part b gives expected utility 220 to type E, which is greater than
the reservation utility of 210. In this case, the firm makes 36300.
The firm can instead induce e=0 by type B. Then it would pay a fixed wage for the same
reason as in part a. This fixed wage could be:
- 0, in which case both types quit, and the firm makes 0;
- 40000, in which case type E quits, and the firm makes 35000(1-p);
- 44100, in which case both types work, and the firm makes 40900p + (75000-44100)(1-p)
= 30900 + 10000p.
Thus, if p≤0.54, the contract from part b is second-best; if p≥0.54, a fixed wage of 44100
is second-best.
2d) Both types get their reservation value in the observable case. If p≤0.54, type E does better
(220&gt;210); if p≥0.54, type B does better (210&gt;200). The intuition is that the firm can take
the entire surplus from only one of the two types, and will therefore let a type keep some
surplus when it’s sufficiently rare.
2e) In a separating equilibrium, the firm would offer workers showing signs w=44100. Type B
will therefore be willing to fake signs if 𝑐 &lt; 44100 − 40000 = 4100. Thus, if 𝒄 ≥ 𝟒𝟏𝟎𝟎, a
separating equilibrium exists.
2f) Type E actually wants to mimic type B: it gets expected utility 220 by doing so and getting
the contract from part b. Since type E can either show signs or not for free, it is impossible
to maintain separation.
3a) Market Q = 6000.
To find Qeff, note that the MPB is 1000-0.1Q, while the MPC is 100+0.05Q. From a global
perspective, the externality is 60, so MSC is 160+0.05Q. Setting MSC=MSB=MPB gives
Qeff=5600.
To find Q*, do the same as for Qeff but with an externality of 30. This gives Q*=5800.
3b) Yes: non-tradable permits are likely to be Pareto inefficient, since trades are Pareto
improvements. Trades increase surplus by shifting production from producers that are unable
to use the right to pollute to generate much surplus (i.e. for whom it’s not very costly to curb
emissions) to those that are able to do so (i.e. for whom curbing emissions is very costly).
Note: Whether permits are given or auctioned is irrelevant, although auctioned permits would
not need to be traded if all producers participate in the auction and nobody’s willingness to
pay for permits changes after the auction.
3c) 30: the demand for permits is MPB-MPC because that’s the amount of profit that a marginal
permit allows the producer to make (MPB is the price).
3d) Under the treaty, each country’s net surplus is the area of the trapezoid between demand and
supply from 0 to 5600, minus the externality of 30*5600*2 = 336000.
Area of trapezoid = 5600*(900+60)/2 = 2688000, so net surplus = 2352000.
Deviation means producing 5800; then, Area = 5800*(900+30)/2 = 2697000.
In the period of deviation, externality = 30*(5600+5800) = 342000; net surplus = 2355000.
Then, if grim trigger (note that this is a repeated prisoner’s dilemma), externality =
30*5800*2 = 348000; net surplus = 2349000.
We know that grim trigger provides the maximum incentives for cooperation: no harsher
deterrent of deviations is available. Thus to maintain cooperation, we need:
2352000
𝛿
≥ 2355000 + 2349000
1−𝛿
1−𝛿
This simplifies to 𝜹 ≥ 𝟎. 𝟓.
4. Let “C” denote the country adopting carbon pricing.
Production-based: C might shift to cleaner industries, but other countries without carbon
pricing might fill the gap in dirtier industries. (This could even increase emissions if other
countries’ new production is done in a dirtier way than in C.) Or C might shift to cleaner
inputs, but due to the price drop of dirtier inputs, other countries might shift towards them.
Consumption-based: C would consume less polluting goods (or goods produced in less
polluting ways). This might cause the world price of polluting inputs to drop relative to that
of cleaner ones. As a result, other countries may decide to consume more polluting goods (or
goods produced in more polluting ways).
5. The Expected Utility Theorem identifies necessary and sufficient conditions under which
individuals’ choice over lotteries correspond to the maximization of an expected utility
function (i.e. preferences over lotteries are represented by expected utility). It is important
because it provides a foundation for using expected utility, which is both intuitive (it’s
probably among the first ways that come to mind if you’re trying to assign utilities to
lotteries) and practical (once we have utilities for outcomes, we automatically get utilities for
all lotteries over those outcomes, which is a much larger set of objects).
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