Economics Homework Exercise 2-7

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Symmetric Information and Competitive Equilibrium

Neil Wallace

January 3, 2017

1 Introduction We are all familiar with the general idea of uncertainty. We are uncertain about tomorrow’s weather, about whether we will wake up with a headache tomorrow morning, and about whether someone’s estimate of the labor re- quired to repair our car is correct. Considerable effort is directed toward coping with uncertainty. Some farmers have costly irrigation systems in or- der to make output less dependent on variations in rainfall. And many of us buy insurance of various sorts to limit our exposure to some kinds of un- certainty. Moreover, there are government programs like disaster aid and unemployment insurance that are intended to offset some of the effects of uncertainty. Here is an example of the kind of setting we will study. There are N

people labelled 1, 2, ...,N. Rainfall is uncertain and it can either be high or low, just two possibilities. We denote the level of rainfall by s ∈ {H,L}, where we use the letter s as a shorthand for state or state-of-the-world and where H stands for high and L for low. We suppose that each person has some land that will without effort bear a crop of some amount of rice. The size of the crop will depend on whether rainfall is high or low. For person n, we denote the size of the rice crop by (wnH,wnL), where wns is the crop on n’s land if the state is s. We assume that wns > 0, but, otherwise, make no other special assumptions about it. In particular, we want to assume that some land does better with high rainfall and other land does better with low rainfall. If s = H, the total crop is

∑N n=1 wnH, denoted WH; if s = L, the

total crop is ∑N

n=1 wnL, denoted WL. Here is the sequence of actions in this

world. First people make deals or trade. Then the state s is realized and everyone sees it. Then, people honor the deals they made. Then everyone eats the rice they end up with and that is the end. Our goal is to describe the deals or trades that people make in this world.

2 The state-preference or contingent-commodity approach

This approach to modelling trade in the above and related settings was in- vented in the 1950’s. As applied to the above example, it takes the standard two-goodstaticmodel (withthegoods, say, beingflourandsugar)andapplies it to the above setting. Rather than treat the two goods as flour and sugar, it treats the two goods as rice-if-rainfall-is-high and rice-if-rainfall-is-low. These are called state-contingent commodities or, more simply, contingent commodities. Missing from the above example is a description of what people like, their

preferences. We start out by simply pursuing what we would do for flour and sugar and apply it to rice-if-rainfall-is-high and rice-if-rainfall-is-low. But, to save space, just as we would assign numbers to flour and sugar– for example, calling flour good 1 and calling sugar good 2, we now do the same for our two contingent goods: we call rice-if-rainfall-is-high good 1 and call rice-if-rainfall-is-low good 2. Using this notation, person n’s crop is denoted (wn1,wn2) and WH = W1 and WL = W2. Next, we introduce symbols for what each person ends up eating, which

we call consumption. In doing this, we carry over the idea of contingent commodies. Thus, we denote by (cn1,cn2) what person n ends up consuming, where cn1 is the amount of rice that person n consumes if rainfall is high and cn2 is the amount of rice that person n consumes if rainfall is low. This may seem strange because only one level of rainfall will occur. However, remember the sequence of actions. First, people make deals or trade; then, the level of rainfall is realized; then deals are honored (payments are made or received); and, finally, consumption occurs. When deals are made, a person has to think about consumption in both contingencies– just as you would do if you were thinking about buying renter’s insurance. We assume that people care only about their own consumption and that

we can represent their preferences either using symbols by a utility function that has as arguments the quantities of the two goods consumed, denoted

(cn1,cn2) for person n, or graphically by an indifference curve map with good 1 measured on the horizontal axis and good 2 on the vertical axis. That is, we assume that person n has a utility function Un(cn1,cn2) and that it satisfies all the usual assumptions that we would make if the goods were flour and sugar. And, that function gives rise to an indifference-curve map that satisfies all the usual assumptions about the shape of indifference curves.

Exercise 1 Sketch a few indifference curves for person n that satisfy the usual assumptions made for two goods in intermediate microeconomics.

Exercise 2 Suppose that Vn(cn1,cn2) = aUn(cn1,cn2) + b, where a and b are constants and a > 0. Show that Vn and Un give rise to the same indifference- curve map.

Theabove isacompletedescriptionofa two-goodpure-exchangeeconomy with private ownership of the total resources, provided that we assume, as we do, that W1 and W2 represent not only the sum of what people start out owning, but also the total resources in the economy. However, before we analyze that economy and present a model of trade, we should call attention to one feature of the above specification. We said that rainfall could either be high or low. However, we said nothing about the likelihood of it being high or low. Does it seem strange that that likelihood appears nowhere in the above description?

Exercise 3 Suppose person n thinks that the probability that rainfall is high is zero. What in the above description of person n would be affected by that belief? What if person n thinks that the probability that rainfall is high is close to zero?

3 Allocations, feasible allocations, and Pareto effi cient allocations

In this economy, an allocation is a list consumption pairs, one pair for each person. (It is an answer to the question: Who gets what?) We denote it by a list (c11,c12), (c21,c22), ..., (cN1,cN2) or (cn1,cn2) for n = 1, 2, ...,N. An allocation is feasible if it does not more than exhaust the available re- sources. That is, (cn1,cn2) for n = 1, 2, ...,N is feasible if

∑N n=1 cn1 ≤ W1

and ∑N

n=1 cn2 ≤ W2. In order to define what it means for an allocation to be Pareto effi cient, we start by defining Pareto superiority. Consider two allocations, (cn1,cn2) for n = 1, 2, ...,N and (c̆n1, c̆n2) for

n = 1, 2, ...,N. We say that (cn1,cn2) for n = 1, 2, ...,N is Pareto superior to (c̆n1, c̆n2) for n = 1, 2, ...,N, if someone, say person j, strictly prefers (cj1,cj2) to (c̆j1, c̆j2) and there is no person i who strictly prefers (c̆i1, c̆i2) to (ci1,ci2). In other words, (cn1,cn2) for n = 1, 2, ...,N is Pareto superior to (c̆n1, c̆n2) for n = 1, 2, ...,N if (cn1,cn2) for n = 1, 2, ...,N would win unanimous consent if the alternative is (c̆n1, c̆n2) for n = 1, 2, ...,N. Wesaythatallocation (cn1,cn2) for n = 1, 2, ...,N isParetoeffi cient (some

say Pareto optimal) if two conditions hold: (i) (cn1,cn2) for n = 1, 2, ...,N is feasible; (ii) there is no feasible allocation that is Pareto superior to it. Given our interpretation of the goods as contingent commodities, notice

that each person is judging what they get before the state occurs, before they see whether rainfall is high or low.

4 Competitive equilibrium (CE) with contin- gent commodities

We call the pair (wn1,wn2) person n’s endowment. Also, as noted above, we assume that everything is owned by someone. That is,

∑N n=1 wn1 = W1 and∑N

n=1 wn2 = W2, where Wi is the total amount of rice theeconomyhas if state i occurs. In order to define a CE, we need one other concept: affordability at given prices. We let pi denote the price of good i (measured in abstract units of account

per unit of good i). We always assume that pi > 0. We say that person n can afford to buy the pair (cn1,cn2) at the price (p1,p2) if

p1cn1 + p2cn2 = p1wn1 + p2wn2. (1)

As is well-known, CE is a theory of allocations and relative prices. That is, there is no hope of determining both p1 and p2. This follows because (p1,p2) and (p′1,p

′ 2) with p

′ 1 = αp1 and p

′ 2 = αp2 where α > 0 determine the same

affordable set.

Exercise 4 Let (p′1,p ′ 2) satisfy p

′ 1 = αp1 and p

′ 2 = αp2 for any α > 0. Show

that if (ĉn1, ĉn2) is affordable at (p1,p2), then (ĉn1, ĉn2) is affordable at (p′1,p ′ 2).

In order to avoid the implied indeterminacy, we impose the condition that p1 + p2 = 1. This is one standard way to normalize prices, a way that will turn out to be convenient. Before we move on, let’s offer several interpretations of (1) and of trade given our interpretation of the goods as contingent commodities. As is standard, the righthand side of (1) is the value of what person n

starts out owning and the lefthand side is the value of what person n ends up with. According to (1), the two have to be equal. That is, trading at a price does not change the value of what a person has. We can rewrite (1) as

p1(cn1 −wn1) = p2(wn2 − cn2).

This says that if person n ends up with more of good 1 than they started with, then they necessarily end up with less of good 2 than they started with. Put differently, in such a case, person n bet on outcome 1 and, necessarily in this two-outcome setting, bet against outcome 2.

Exercise 5 Suppose that the only market is a betting market and that the available odds on outcome 1 are 5 to 2. Let’s take this to mean that if person n bets x amount of outcome-2 good on outcome 1, then person n collects (5/2)x amount of outcome-1 good if outcome 1 occurs and pays out x amount of outcome-2 good if outcome 2 occurs. Find the numerical values for (p1,p2) that give rise to the same trading opportunities as this betting market. (Recall that we are imposing p1 + p2 = 1.)

Exercise 6 With good 1 on the horizontal axis and good 2 on the vertical axis, sketch with labels all the pairs (cn1,cn2) that satisfy (1). (Remember that a line is determined in one of two ways: by a point and a slope, or by two points. In either case, those should be expressed in terms of (wn1,wn2) and (p1,p2).)

ThedefinitionofCEis standard. Anallocation (ĉn1, ĉn2) forn = 1, 2, ...,N and a price p̂ = (p̂1, p̂2) is a competitive equilibrium (CE) if two conditions hold: (i) (ĉn1, ĉn2) for n = 1, 2, ...,N is a feasible allocation; (ii) for each person n, (ĉn1, ĉn2) is liked by n as well as anything else that n can afford at the price p̂.

Exercise 7 Consider the special case in which everyone has the same indif- ference curve map and the same endowment– that is, (wn1,wn2) = (W1/N,W2/N) for all n. Describe a CE.

In general, not much can be said descriptively about a CE. Soon, we will specialize the description of indifference curve maps and will, then, be able to say more. There is, though, one important result that we can state (and prove): a version of Adam Smith’s invisible-hand proposition.

Proposition 1 If (ĉn1, ĉn2) for n = 1, 2, ...,N is a CE allocation, then (ĉn1, ĉn2) for n = 1, 2, ...,N is Pareto effi cient.

Here is an outline of a proof by contradiction. Because (ĉn1, ĉn2) for n = 1, 2, ...,N is a CE allocation, there is a price p̂ = (p̂1, p̂2) such that (ĉn1, ĉn2) for n = 1, 2, ...,N and p̂ is a CE. If (ĉn1, ĉn2) for n = 1, 2, ...,N is not Pareto effi cient, then there exists another allocation, call it (c̆n1, c̆n2) for n = 1, 2, ...,N that is Pareto superior to (ĉn1, ĉn2) for n = 1, 2, ...,N and is feasible. It follows from Pareto superiority that there is some person, call them

person i, who strictly prefers (c̆i1, c̆i2) to (ĉi1, ĉi2). Therefore,

p̂1c̆i1 + p̂2c̆i2) > p̂1wi1 + p̂2wi2. (2)

Exercise 8 Interpret (2) and explain in a few sentences why it holds.

It also follows from Pareto superiority that

p̂1c̆j1 + p̂2c̆j2) ≥ p̂1wi1 + p̂2wi2 for all j 6= i. (3)

Exercise 9 Interpret (3) and explain in a few sentences why it holds.

Next, we sum (2) and (3) over all j 6= i and obtain

p̂1C̆1 + p̂2C̆2 > p̂1W1 + p̂2W2, (4)

where C̆i = ∑N

n=1 c̆ni for i = 1 and i = 2.

Exercise 10 Explain why (4) is inconsistent with feasibility of (c̆n1, c̆n2) for n = 1, 2, ...,N.

That completes the proof.

5 von Neumann and Morgenstern expected utility

Ideally, everything done so far is review. What follows may be new. It is fairly standard to specify preferences under uncertainty in a more special way than done so far. One such way was invented by von Neumann and Morgenstern in the book in which they invented game theory. We will use such preferences for a lot of what we do. Then, in the third part of the course, you will read some work that is critical of the von Neumann and Morgenstern specification and, moreover, critical of generalizations of it– criticisms that fall under the heading behavioral economics. The formulation we use assumes that there is an objective probability

distribution over the set of states. In the example set out above, this is a probability distribution over the set {H,L} or, using our numerical labels for the two states, over the set {1, 2}. By objective, we mean that it is shared by everyone. We denote these probabilities π = (π1,π2). (If there were S states, then π = (π1,π2, ...,πS).) We assume that πi > 0 and, of course, that all the π’s sum to unity: π1 + π2 = 1 (or with S states,

∑S s=1 πs = 1).

We saw above in an exercise that a person’s view about the likelihood of the different states would affect the shape of the person’s indifference curve map or, equivalently, would affect their utility function, Un(cn1,cn2). We now make that explicit by writing the utility function as Un(cn1,cn2; π), which is nothing but notation that indicates that the objective distribution π matters for how person n orders various combinations of cn1 and cn2. The expected utility hypothesis is that Un(cn1,cn2; π) has a special form; namely, that

Un(cn1,cn2; π) = π1un(cn1) + π2un(cn2), (5)

where the functionu representshowpersonn valuescertainorsureconsumption– of rice in the above example. Unless we say otherwise, from now on we assume (5) and a un function

(defined on positive real numbers) that satisfies the following properties: it is strictly increasing; it has a tangent at each point which we denote u′n (the derivative of the function un or marginal utility), and that the function u′n is decreasing (diminishing marginal utility). We will make extensive use of the following property of the indifference

curve map implied by (5) and those assumptions: with outcome-1 good mea- sured on the horizontal axis and outcome-2 good measured on the vertical

axis, the slope of the indifference through the point (cn1,cn2) is given by the formula

− π1u

′ n(cn1)

π2u′n(cn2) . (6)

Exercise 11 What is the slope of the indifference through the point (cn1,cn2) if cn1 = cn2? What inequality does the slope of the indifference through the point (cn1,cn2) satisfy if cn1 > cn2? What inequality does the slope of the indifference through the point (cn1,cn2) satisfy if cn1 < cn2?

Exercise 12 Is formula (6) consistent with your discussion in exercise 3? Explain.

Exercise 13 Suppose un(x) = x(1/2), the positive square-root function. For this function, the derivative function, u′n(x), is the function (1/2)x

−(1/2). In the same graph, sketch x(1/2) and (1/2)x−(1/2). For this case, express the slope of the indifference through the point (cn1,cn2) in terms of two ratios, (π1/π2) and (cn2/cn1). Interpret the way the slope of the indifference through the point (cn1,cn2) depends on those two ratios.

Exercise 14 Suppose un(x) = ln x, the natural logarithm of x. For this function, the derivative function, u′n(x), is the function 1/x. In the same graph, sketch ln x and 1/x. For this case, express the slope of the indifference through the point (cn1,cn2) in terms of two ratios, (π1/π2) and (cn2/cn1). Interpret the way the slope of the indifference through the point (cn1,cn2) depends on those two ratios.

Exercise 15 Assume the expected utility hypothesis in (5) and that fn(x) = aun(x)+b. Show that Un(cn1,cn2; π) = π1un(cn1)+π2un(cn2) and Vn(cn1,cn2; π) = π1fn(cn1) + π2fn(cn2) have the same indifference curve map.

Exercise 16 Suppose that u(c) = c(1/2). Consider the equation π1u(c1) + π2u(c2) = b for given π1, π2, and b. There are many pairs (c1,c2) that satisfy this equation. Sketch such pairs in a diagram.

Exercise 17 Let u(c) = c1/2 and let c2 > c1 > 0 be given. Let c̄ = π1c1 + π2c2, where π2 = 1 − π1. (i) Sketch the function u and indicate in your sketch the points (c1,u(c1)), (c̄,u(c̄)), and (c2,u(c2)). (ii) Draw the line that connects the two points (c1,u(c1)) and (c2,u(c2)) and represent that line algebraically. [Hint: Find the slope and intercept in terms of the two points, (c1,u(c1)) and (c2,u(c2)).] (iii) Use that algebraic result to show that the point (c̄,π1u(c1) + π2u(c2)) is on that line.

If so, it follows from your diagram that

π1u(c1) + π2u(c2) < u(c̄). (7)

This inequality holds for any u function that is strictly increasing at a di- minishing rate. Notice that π1c1 + π2c2 = c̄ is the expected value of the random vari-

able that takes the value c1 with probability π1 and takes the value c2 with probability π2. Inequality (7) says that the expected utility of that random variable is less than the utility of getting the expected value of that random variable with probability one. That is called risk aversion. If the u func- tion is such that π1u(c1) + π2u(c2) = u(c̄) for all (c1,c2), then that is called risk-neutrality.

Exercise 18 Show that π1u(c1) + π2u(c2) = u(c̄) for all (c1,c2) if and only if the u function has the form u(x) = ax + b.

Exercise 19 Sketch a u function with the property that π1u(c1) + π2u(c2) > u(c̄) for all (c1,c2) with c1 < c2. Sketch the indifference-curve map implied by such a u function.

In what follows, unless we explicitly say otherwise, we assume that u is strictly increasing at a diminishing rate; that is, we assume risk aversion.

6 Some descriptive aspects of CE under the expected utility hypothesis

Because expected utility is a special case of the model with contingent com- modities, the definition of a CE is unchanged. Also, proposition 1 continues to hold. The expected utility assumption does allow us to derive some new descriptive aspects of a CE.

Proposition 2 Assume the expected utility hypothesis. (i) If W1 = W2, then (a) there is a CE with p = π and (b) there is no other CE. (ii) If W1 > W2, then any CE has p1/p2 < π1/π2. (iii) If W1 < W2, then any CE has p1/p2 > π1/π2.

Exercise 20 Prove proposition 2.

The last exercise and the next two use the results of earlier exercises in which you derived conclusions about the slope of indifference curves implied by the expected utility hypothesis.

Exercise 21 Assume the expected utility hypothesis and un(x) = x(1/2) for all n. Describe the CE consumption of person n in terms of the following objects: (wn1,wn2), (W1,W2), and π.

Exercise 22 Assume the expected utility hypothesis and un(x) = ln x for all n. Describe the CE consumption of person n in terms of the following objects: (wn1,wn2), (W1,W2), and π.

7 More than two states A variant of the high-low rainfall model would have more levels of possible rainfall– say, S levels. Then, we would have S states, one state for each level of rainfall. Conceptually, nothing changes when there are more than two states, but we cannot easily use diagrams. With three states, we would need three-dimensional diagrams; with more than three, diagrams are impossible. Here is how we generalize the concepts introduced above so that they apply to a general finite number of states, which we denote S. An allocation is a list of consumption bundles: (cn1,cn2, ...,cnS) for n =

1, 2, ...,N. An allocation is feasible if ∑N

n=1 cns ≤ Ws for s = 1, 2, ...,S, where Ws is the total amountof thegoodavailable in state s. Let π = (π1,π2, ...,πS) be the probability distribution over the S states. Person n has a utility function Un that satisfies

Un(cn1,cn2, ...,cnS; π) =

S∑ s=1

πsun(cns), (8)

where un satisfies the general assumptions made above for the case S = 2. The definition of Pareto effi ciency is unchanged and we will not repeat it. We do want to state the definition of a CE. First, we assume an initial

ownership or endowment pattern. We let (wn1,wn2, ...,wnS) denote what person owns initially and assume that

∑N n=1 wns = Ws for s = 1, 2, ...,S.

Next we say that person n can afford (cn1,cn2, ...,cnS) at p = (p1,p2, ...,pS) with ps > 0 and

∑S s=1 ps = 1 if

S∑ s=1

pscns = S∑ s=1

pswns.

Then, an allocation (ĉn1, ĉn2, ..., ĉnS) for n = 1, 2, ...,N and a p̂ is a CE if the allocation is feasible and if each person n likes (ĉn1, ĉn2, ..., ĉnS) as well any (cn1,cn2, ...,cnS) affordable at p = p̂. Finally, the proposition that a CE allocation is Pareto effi cient holds for

the generalization to S states. A proof follows exactly the steps outlined above for S = 2.

7.1 Independent risks and insurance

Here is a different variant of our model of rice crops, one with independent risks. Westudythisbecause it resembles thekindof situation inwhichpeople buy insurance. We again suppose that there are N people. But now, each person either gets a high crop wH with probability π or a low crop wL with probability 1 − π. Moreover, these outcomes are determined independently for each person. Also, each person maximizes expected utility and has the same u function that satisfies the general assumptions made above. The sequence of actions is as above. In particular, first everyone is to-

gether and they make deals or trade. Next, all N outcomes are realized and everyone observes all N outcomes. Finally, payoffs are made in accord with the deals or trades arranged earlier and people eat what they end up with. Let’s start by describing the set of states for this world. We call a list of

crop realizations arranged by person number a profile. Thus, if N = 3, one such profile is (H,L,L) which means that person 1 realized the high crop, and persons 2 and 3 realized the low crop. A state is a realized profile.

Exercise 23 List all the possible profiles when N = 2. List all the possible profiles when N = 3. How many states are there when N = 2? How many are there when N = 3? Express the number of states in terms of N.

Exercise 24 Suppose N = 2. Without being formal, make a guess about the kind of deal that would have the following properties? Both people are willing to make the deal; the deal gives the same expected utility to both people; and there is no other deal that would have those properties and would give both higher expected utility.

Exercise 25 Suppose you are an analyst for FEMA (Federal Emergency Management Agency). Would you use the above model with person n re- placed by region n and L replaced by disaster and H replaced by no disaster? Explain.

If we treat each profile as a state, then there are many states. With N people, there are 2N different states. Purchasing claims contingent on the entire profile of outcomes does not look much like the purchase of insurance. Moreover, although CE is defined for any number of traders, we tend to think of it as a reasonable description only when there are many traders. But, consider the profile (H,L,L,...,L). What we expect to see is that person 1 will sell claims on output in this state and that everyone else will buy a bit. However, that makes person 1 look like a monopolist because they are the only seller of claims on output in that state. Instead of starting out with complete contingent claims, let’s make a

guess about a good allocation in the class of feasible allocations that treats similarly situated people in the same way. Our answer will generalize to N people what you may have found above for the case of N = 2. We define an aggregate state by the number of people who get the H re-

alization. That number could be any integer between 0 and N; that is, given this definition, there are N + 1 aggregate states, which is (much) smaller than the number of profiles. If the aggregate state is k, then the economy realizes total output, kwH + (N −k)wL, which we denote by W(k). Our guess about a good allocation is that each person gets to consume W(k)/N in aggregate state k. Before we argue that this a good allocation, let’s interpret it as the kind of allocation that would be achieved by an insurance arrangement that is organized as what is called a mutual. In doing that, we make one innocuous change to the way most insurance companies operate. We assume that no payments are made before the uncertainty is realized. Here are the payments made by the mutual. The payment to person

n– a negative payment should be interpreted as something like an ex post premium payment– depends on only two things: the crop realized by person n and the aggregate state k:

payment to person n in aggregate state k

 

W(k)/N −wH if n realizes wH

W(k)/N −wL if n relaizes wL . (9)

Exercise 26 Verify that (9) implies that total payments are zero.

Exercise 27 Most observers think that insurance companies offer very unattrac- tive annuities. Why might that be? One form of annuity is called a “tontine.” Read a bit about tontines and explain the sense in which a tontine is a mutual form of organization.

Here is why the above allocation is a good allocation among allocations that treat similarly situated people in the same way. If the aggregate state is k, then feasibility implies that total consumption cannot exceed W(k). The allocation described above has each person consuming W(k)/N. That is, it makes a person’s consumption independent of both the person’s ID number and the person’s individual realization, H or L, and dependent only on the aggregate state. Given that nothing about the person, preferences or endowments, depends on ID number, it seems sensible that allocations not depend on it. But, why make it independent of whether the person realized H or L? Well, each person is risk-averse, so that also may seem reasonable. Indeed, we can easily argue that it is desirable. To do that, let’s introduce some alternatives and argue that they are

not desirable. Suppose the aggregate state is k. One alternative to having everyoneconsume W(k)/N is toallowconsumption toalsodependonwhether a person realized H or L. Let cH denote the consumption of those who realize H and let cL denote the consumption of those who realize L. There are k who realize H and N −k who realize L. Feasibility requires

kcH + (N −k)cL ≤ W(k) (10)

or, equivalently,

(k/N)cH + [(N −k)/N]cL ≤ W(k)/N. (11)

Given aggregate state k, the probability of a person having the high outcome is k/N and that of having the low outcome is (N − k)/N. Therefore, the conditional expected utility of a pair (cH,cL) (conditional on the aggregate state being k) is

(k/N)u(cH) + [(N −k)/N]u(cL) (12) It follows from the general properties of the function u that cH = cL = W(k)/N maximizes (12) subject to (11).

Exercise 28 With cH measured on the horizontal axis and cL measured on the vertical axis, sketch the pairs (cH,cL) that satisfy (11). In the same diagram, sketch some of the indifference curves (level curves) of (12).

It is possible to show that the allocation that gives each person W(k)/N in aggregate state k is a CE when the set of states is the set of profiles. The details, which go beyond what is expected of most readers of this exposition, appear in Cass, Chichinisky, and Wu, Econometrica, vol 64 (March 1966), pages 333-341.

Exercise 29 Describe a world in which there are individual risks of the sort just described and also risks coming from something like high and low rainfall that affects everyone. What sort of insurance would you expect to see in such a world?

7.2 Arbitrage (asset) pricing theory

Consider again the two-state rainfall model and assume that in addition to making trades contingent on the outcome of the rainfall, each person can buy or sell pieces of land before rainfall is realized. There are N pieces of land and the owner of piece n gets (wn1,wn2); that is, wn1 amount of rice if rainfall is high and wn2 if rainfall is low. If wn1 6= wn2, then piece of land n is what would ordinarily be called a risky asset. We can use arbitrage to compute the price of a given piece land in terms of the prices of rice contingent on the rainfall outcome, the prices we defined above. The general version of doing that is called arbitrage-pricing theory (or APT as our finance friends like to call it). Suppose p = (p1,p2) is the price of outcome contingent rice. The APT

says that the price-of-land n is p1wn1 + p2wn2. In order to argue that that is true, assume that any person can either buy land-n or go-short land-n. (To go-short land-n means promising to pay out wn1 amount of rice if rainfall is high and wn2 if rainfall is low.) Let v(wn1,wn2) be the pre-outcome price of land-n (in abstract units of account).

Exercise 30 Suppose contingent rice can be bought or sold at p = (p1,p2). What purchases and sales would be profitable (without bearing risk) if v(wn1,wn2) < p1wn1+p2wn2? What purchases and sales would be profitable (without bear- ing risk) if v(wn1,wn2) > p1wn1 + p2wn2?

Now let’s leave rice, land, and two states behind, and exposit APT more generally. We assume that there are S states and that p = (p1,p2, ...,pS) denotes the price vector of outcome-contingent goods. We define assets by their payoffs. In particular, an asset is a vector of outcome-specific payoffs– say, (a1,a2, ...,aS) with the interpretation that the owner of this asset receives as units of the good if outcome s occurs. (If as < 0, then the owner must pay out as units of the good if outcome s occurs.) The price of the asset is the pre-outcome price at which the asset can be bought or sold. We denote that price v(a1,a2, ...,aS).

Proposition 3 The only price of the asset consistent with no arbitrage prof- its is

v(a1,a2, ...,aS) = S∑ s=1

psas. (13)

Exercise 31 Argue that proposition 3 is true. (Hint: Show that each person would profit by buying the asset if v(a1,a2, ...,aS) <

∑S s=1 psas and that no-

one wants to buy it; then show that each person would profit by selling it short if v(a1,a2, ...,aS) >

∑S s=1 psas.)

Consider two assets: (a1,a2, ...,aS) and (b1,b2, ...,bS), where bs = zas for s = 1, 2, ...,S for some number z > 0. Show that in a CE, v(b1,b2, ...,bS) = zv(a1,a2, ...,aS).

7.2.1 The Modigliani-Miller theorem

We can also use proposition 3 to price the debt and equity of limited li- ability corporations. We define the corporation as an asset in the sense that ownership of the entire corporation entitles the owner to a vector of outcome-specific payoffs– say, (x1,x2, ...,xS). As part of limited liability, we now assume that xs ≥ 0 for each s. We next define debt of the corporation. Debt is a liability with a promised

pay-off that is not contingent. The amount of debt is measured by the magnitude of that promised pay-off. Let P ≥ 0 be the promised pay-off. The magnitude P is to be distinguished from what the owners of the debt actually get. (After all, junk bonds are called junk because it is understood that holders of junk bonds will not always receive the promised pay-off.) Let min{a,b} denote the smaller of a and b. The owners of the debt of

the corporation (x1,x2, ...,xS) with promised pay-off P receive the vector (min{x1,P}, min{x2,P}, ..., min{xS,P}). We let D denote the pre-outcome value of this debt. Using proposition 3, we have

D =

S∑ s=1

ps min{xs,P}. (14)

Now we define the equity of the corporation. Consider a corporation (x1,x2, ...,xS) with debt having the promised pay-off P. The owners of the equity of this corporation receive the vector (max{x1 −P, 0}, max{x2 − P, 0}, ..., max{xS −P, 0}), where max{a,b} means the larger of a and b. We

let E denote the pre-outcome value of this equity. Using proposition 3, we have

E = S∑ s=1

ps max{xs −P, 0}. (15)

The Modigliani-Miller Theorem concludes that the total value of a cor- poration, D + E, does not depend on how much debt it has. This conclusion follows from (14) and (15).

Exercise 32 Show that min{a,b} + max{a − b, 0} = a. (Hint: Consider each of the following exhaustive cases: a > b, a = b, and a < b.)

Exercise 33 (i) Use the result of the last exercise and (14) and (15) to compute D + E. (You have proved the Modigliani-Miller Theorem.)

Exercise 34 We are describing a corporation by a vector of payoffs, (x1,x2, ...,xS). Suppose S = 2. (i) How would you describe the "riskiness" of a corporation? (ii) Are less risky corporations worth more than more risky corporations? Explain.

Because D is the pre-outcome value of the debt and P is the promised pay-off, the ratio P

D is the (gross) promised yield or (gross) rate-of-return

on the debt. As we next show, this yield is in general increasing in P, our measure of the amount of the debt. We draw this conclusion by studying the ratio D

P . If b > 0, then min{a,b}

b = min{a

b , 1}. Therefore, from (14), if P > 0,

then D

P =

S∑ s=1

ps min{ xs P , 1}. (16)

Exercise 35 According to (16), higher debt as measured by P either does not affect the promised yield or increases the promised yield. Describe the circumstances in which each conclusion holds.

Exercise 36 Competition is sometimes described as a situation in which people face prices at which they can buy and sell any amount. Our model is competitive in the sense that we are assuming that people can trade any amounts at the price vector of outcome-contingent goods, which we denote p. Despite that, the last exercise shows that a limited liability corporation faces a promised yield on debt that can be increasing in the amount of debt it sells. How do you reconcile that result with competition?

Exercise 37 In the US, the base on which the federal corporate income tax is levied excludes interest payments. In the context of the above model, the base for the tax in state s is xs − (P − D). Show that a positive tax rate makes E + D an increasing function of P.

Exercise 38 At times, limited liability corporations are given loan guaran- tees. In a full analysis, we would have to describe how the guarantee is financed. Here we will ignore that. If the guarantee is genuine, then the owners of the debt of the corporation (x1,x2, ...,xS) with promised pay-off P will actually receive P in every outcome. Suppose that is the case. (i) Give a formula for the pre-outcome value of such debt. (ii) Give a formula for D + E for such a corporation. (iii) Does D + E depend on P and in what way? (iv) Suppose the guarantor of the debt imposes some maximum amount of debt. Given that P is set at the maximum, does the value of the equity of the corporation depend on the riskiness of the corporation?

Exercise 39 Read a bit about what is called the Volcker Rule. Draw an anlogy between the setting of the last exercise and the setting that motivates the Volcker Rule.

These results are the starting point of the study of corporate finance. That study involves various departures from the above model. After all, the above model does not even tell us anything about the circumstances in which the institution of limited liability is desirable. In the setting of the above model, eliminating that institution would have no consequences.

8 Concluding remarks That completes our study of risk-sharing when there is symmetric informa- tion. We will next turn to the study of models of asymmetric information. We will, however, maintain the expected utility hypothesis and the assump- tion that people agree about the relevent probability distributions.