Symmetric Information and Competitive Equilibrium

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Symmetric Information and Competitive Equilibrium

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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=1wnH , denoted WH ; if s = L, the

total crop is ∑N

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

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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-good static model (with the goods, say, being flour and sugar) and applies 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

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(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.

The above is a complete description of a two-good pure-exchange economy 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

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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 . We say that allocation (cn1, cn2) for n = 1, 2, …, N is Pareto effi 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=1wn1 = W1 and∑N

n=1wn2 = W2, whereWi is the total amount of rice the economy has 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).

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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).)

The definition of CE is standard. An allocation (ĉn1, ĉn2) for n = 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.

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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.

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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 function u represents how person n values certain or sure consumption– 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

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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) = lnx, the natural logarithm of x. For this function, the derivative function, u′n(x), is the function 1/x. In the same graph, sketch lnx 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). Consi

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