Microeconomics of Banking: Lecture 2

Transcription

1 Microeconomics of Banking: Lecture 2 Prof. Ronaldo CARPIO September 25, 2015

2 A Brief Look at General Equilibrium Asset Pricing Last week, we saw a general equilibrium model in which banks were irrelevant. For the next few lectures, we will briefly examine the general equilibrium asset pricing model. This is the foundation for nearly all models of asset pricing and finance. We will see one of the main results in asset pricing, the mutuality principle. This principle will be revisited later when we look at asymmetric information models of borrowing and lending.

3 The reading for this part of the course can be found in Microfoundations of Financial Economics by Lengwiler, Chapters 2.1, , and We will start with a review of consumer theory, then briefly discuss general equilibrium, decisions under uncertainty, and risk aversion.

4 Review of Consumer Theory Suppose there are two goods to choose from. Let X, Y be the amounts of each good in the market basket. As before, assume that the consumer s preferences are described by a utility function U(X, Y ), with the following properties: U(X, Y ) is continuous and differentiable. U(X, Y ) is increasing in both X and Y. For any utility level u, the set of bundles X, Y that give utility U(X, Y ) u forms a convex set. Prices for the two goods are P x, P y respectively. Consumer s income is I.

5 Constrained Optimization We can formulate the consumer choice problem as a constrained optimization problem: max U(X, Y ) subject to P xx + P y Y = I X,Y U(X, Y ) is the objective function, and P x X + P y Y = I is the constraint To find the solution to this problem, we can use the method of Lagrange multipliers, which goes as follows: Rewrite the budget constraint as: P x X + P y Y I = 0 Form the Lagrangian equation, which is simply the objective function U(X, Y ) minus the constraint multiplied by a variable λ. L(X, Y, λ) = U(X, Y ) λ(p x X + P y Y I ) λ is called a Lagrange multiplier.

6 Constrained Optimization L(X, Y, λ) = U(X, Y ) λ(p x X + P y Y I ) We want to maximize this equation with respect to X, Y, and λ. Write down the partial derivatives with respect to each variable and set it to 0: L X = U X λp x = 0 L Y = U Y λp y = 0 L λ = P xx + P y Y I = 0 Note that the third equation is simply the budget constraint. U(X,Y ) x is the marginal utility with respect to X.

7 Constrained Optimization L X = U X λp x = 0 L Y = U Y λp y = 0 L λ = P xx + P y Y I = 0 Combining the first two equations, we get the MRS = price ratio condition for optimality: U X U Y = P x P y We can rearrange this to get the equal marginal principle: λ = U X P x = U Y P y

8 The optimal choice, x = (X, Y ), is where the indifference curve is tangent to the budget line. The slope of the indifference curve (= MRS) is equal to the slope of the budget line (= the price ratio). Another way to state this is U = λ(p x, P y ) T : the gradient of the utility function is a scalar multiple of the price vector Suppose the consumer has an initial endowment of goods ω = (X ω, Y ω ). Then this is equivalent to having an income of I = P x X ω + P y Y ω.

9 General Equilibrium The consumer choice problem takes prices as given, and assumes the consumer is a price-taker. But what determines prices? In previous courses, you have seen supply and demand curves; the equilibrium price is determined by the intersection of supply and demand. A supply and demand graph for a single market is called a partial equilibrium model, since it does not take into account the possible effects of other markets. A general equilibrium model is one that takes all prices and quantities on all markets into account simultaneously. We ll look at the simplest possible case of a general equilibrium model, with 2 agents and 2 goods.

10 General Equilibrium There are two goods. Two agents with smooth, quasiconcave utility functions: u 1 (x 1 1, x 1 2 ) and u 2 (x 2 1, x 2 2 ) Each agent has an initial endowment of each good: e 1 = (e 1 1, e1 2 ) and e 2 = (e 2 1, e2 2 ) The total endowment of each good in this economy is (e e2 1, e1 2 + e2 2 ) We can plot all possible allocations of goods among the two agents in an Edgeworth box.

11 The width of the box is the total endowment of good 1. The height of the box is the total endowment of good 2.

12 The bottom left corner is the origin (0 of both goods) for agent 1. The upper right corner is the origin for agent 2. Every point on the box corresponds to a possible allocation of goods between agent 1 and agent 2.

13 Agent 1 s indifference curves increase away from the lower left corner. Agent 2 s indifference curves increase away from the upper right corner.

14 General Equilibrium A Walrasian equilibrium is a price vector p = (p 1, p 2 ) and an allocation (x1 1, x 2 1), (x 1 2, x 2 2 ) such that: Each agent s allocation (x i 1, x2 i ) maximizes his utility, given prices p and initial wealth p e i Markets clear: i x i m = i e i m for each good m = 1, 2. At equilibrium, the first order conditions for each agent require that the ratio of marginal utilities equals the price ratio: u 1 x 1 u 1 x 2 = u 2 x 1 u 2 x 2 = p 1 p 2

15 An allocation on the Edgeworth box is a Walrasian equilibrium if neither agent has any incentive to trade. This occurs when the indifference curves of both agents are tangent to each other.

16 The price vector is then the slope of the indifference curves at the equilibrium allocation point. You can think of this as two consumer choice problems being solved simultaneously, using the same prices.

17 The market clearing condition ensures that the net demand of agent 1 is exactly balanced by the net demand of agent 2.

18 Contingent Claims We can extend consumer choice framework to handle time and risk by expanding our definition of what a good is. We can add the time of availability to the definition of a good. For example: instead of two goods apple and orange, we can define the good 1 apple today and 1 orange, one week from now. We can also define a good to be contingent on a random event. Examples: umbrella when it is raining vs. umbrella when it is not raining 1 unit of grain when the harvest is good vs. 1 unit of grain when the harvest is bad income when there is a recession vs. income when there is an expansion These are called contingent claims.

19 Contingent Claim Economy We call the collection of all possible random outcomes the states of the world. Let S denote the possible states of the world, and s S is an individual state of the world. Time is finite: t = 0, 1,..., T For each t, there is a partition ɛ t of S (i.e. decompose S into a collection of non-intersecting subsets) This partition represents the information about the state s available at time t. At t, all agents know that an event e ɛ t has occurred, but not necessarily which individual state s e has occurred.

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21 Two-Period Model At t = 0, there is complete uncertainty: the only information agents have is that all states of the world are possible. At t = 1, uncertainty is resolved: all agents know exactly which state of the world has occurred.

22 Decisions Under Uncertainty When talking about random outcomes (states of the world), we also need to know the probability of each state. Suppose the set of states of the world is {s 1, s 2,...s S }, where S is finite. The probability of state s i occuring is denoted as π i. Each π i 0, and π π S = 1. Suppose the goods in the consumer s market basket are: income in state of the world i, for i = 1...S. Denote the amount of each good as x 1,..., x S. How should the consumer take probabilities into account in his utility function?

23 von Neumann-Morgenstern Utilities We will assume that consumers utility functions have a specific form, called expected utility or a von Neumann-Morgenstern utility. U(x 1,..., x S ) = S i=1 π i v(x i ) A market basket over income in uncertain states of the world is sometimes called a lottery. The function v(x) is the utility function for nonrandom outcomes. We assume that the consumer has the same v(x) for every state. Therefore, U(x 1,..., x S ) is the expected value of v(x).

24 Risk Aversion We will frequently make a further assumption, that consumers are risk averse. Suppose there are S states of the world, and a consumer is offered two possible lotteries: 1. x 1,..., x S 2. x,..., x, where x is the expected value of x: E(x) = S i=1 π i x i A risk-averse consumer prefers the second lottery to the first one. U(x 1,..., x S ) < U( x,..., x) A consumer is risk-averse iff his utility function over nonrandom outcomes, v(x), is concave. If u(x) is (strictly) concave, then u (x) is (strictly) decreasing.

25 Risk Aversion A risk-neutral consumer is indifferent between any lottery that has the same expected value. U(x 1,..., x S ) = U( x,..., x) A consumer is risk neutral iff his utility function over nonrandom outcomes, v(x), is linear.

26 Suppose there are two states of the world: low and high. The consumer s wealth can be: x = x low or x = x high respectively.

27 Let E(x) = π low x low + π high x high, the expected value of x. A risk-averse consumer gets higher expected utility from the lottery (E(x), E(x)) than from (x low, x high ).

28 CE(x) is called the certainty equivalent of x; it is the certain (i.e. does not vary between states) amount of wealth that gives the same amount of expected utility as (x low, x high ).

29 Example: Buying Insurance Suppose a consumer has initial wealth w 0, and is considering whether to buy insurance for his car. There are two states of the world: crash and ok. If the state of the world is crash, the consumer suffers a loss in wealth of L. Suppose the probability of crash is α. Then the probability of ok is 1 α. The consumer can buy insurance from an insurance company. For every unit of insurance bought, the insurance company will pay 1 unit of wealth in the crash state.

30 Example: Buying Insurance Suppose the price of insurance is p. We will assume that the price of insurance is actuarially fair, that is, the insurance company makes zero expected profit (e.g. under perfect competition). The profits of the insurance company per unit of insurance sold in each state are: crash: (p 1) ok: p Expected profit is α(p 1) + (1 α)p. Setting this to 0, we get p = α.

31 Example: Buying Insurance How much insurance should the consumer purchase at a price of α? Let v(w) be the consumer s utility function over nonrandom outcomes, assumed to be strictly concave. Let x be the amount of insurance purchased. Consumer s wealth in each state is: crash: w0 αx L + x ok: w0 αx Expected utility is: αu(w 0 αx L + x) + (1 α)v(w 0 αx)

32 Example: Buying Insurance We maximize with respect to x. Differentiate with respect to x and set to 0: (1 α)αv (w 0 αx L + x) α(1 α)v (w 0 αx) = 0 v (w 0 αx L + x) = v (w 0 αx) Since v ( ) is strictly decreasing, then this equality can only hold when the arguments to v ( ) are equal: w 0 αx L + x = v (w 0 αx) x = L Therefore, the consumer chooses perfect insurance: wealth is the same in all states of the world. Wealth is equal to w 0 αl in both crash and ok states.

33 Example: Buying Insurance Risk-averse agents will try to eliminate risk (i.e. differences in wealth across states). We will see this principle many times in the course.

34 Finance Economy A finance economy combines the GE model with risk averse utility functions. Suppose there is only one consumption good, call it money, income or wealth. A financial asset is a security that entitles its holder to a specified payout for each possible state of the world. Suppose that asset j is specified by: r j = (r j 1,..., r j S )T Whoever holds 1 unit of asset j will receive r j s at t = 1, if the state of the world happens to be s. A storage asset (e.g. cash) would be (1,..., 1) T. A riskless bond with nominal yield 1 + r would be (1 + r,..., 1 + r) T. An asset is called risk-free if it gives the same payoff in every state.

35 Finance Economy In the real world, stocks or equities are a claim of ownership over a fraction of a firm. This entitles the stockholder to a fraction of the profits of the firm. A bond is a loan, which may or may not be repaid. In the model, stocks and non-government bonds are modeled as risky assets, that is, their payouts will be different in different states of the world.

36 Finance Economy For example, if the possible states of the world are recession and expansion, then a stock could give a high payout in the expansion state and a low payout in the recession state. For bonds, the states of the world might be success or bankrupt. The bond gives the promised payout in the success state, and gives a zero payout in the bankrupt state. A riskless bond is a special kind of bond, that gives the same payout in all states. Obviously nothing is completely riskless in real life, but for our purposes we can treat the sovereign debt of rich nations with no history of default as riskless.

37 Arrow securities The simplest asset is one that pays 1 unit in exactly one state of the world s, and zero in all other states. e s = (0, 0,..., 1,...0) T This is called the Arrow security for state s. Any financial asset can be represented as a linear combination of Arrow securities.

38 Two-Period Economy By convention, we will say that t = 0 is state s = 0, and the states at t = 1 are s = 1, 2,..., S. Let y s denote the amount of consumption in state s. Assume agents have a utility function v(y 0 ) + δe [v(y)] = v(y 0 ) + δ π s v(y s ) pi s is the probability that state s occurs. v( ) is a vnm utility function. δ (0, 1) is the discount factor. This type of utility function is time-separable, i.e. additive in the utility for t = 0 and t = 1. S s=1

39 Efficient Risk-Sharing Suppose there are two agents, S = {1, 2}, and agents are risk-averse: v i ( ) is strictly concave. Agents are endowed with some amount of securities that pay off at t = 1. Assume there is no aggregate risk: the sum of endowments for each state s is constant. There may be idiosyncratic risk: the endowment for an individual agent may differ across states. The mutuality principle states that an efficient allocation in this situation will diversify away idiosyncratic risk. Agents will consume the same amount in both states; they will only bear aggregate risk.

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41 Assume each agent s utility is: u i (y 0, y 1, y 2 ) = v(y 0 (i)) + δ π s v(y s (i)) y s (i) is the amount consumed in state s by agent i. S s=1 Assume the same aggregate income in both states: y 1 (1) + y 1 (2) = y 2 (1) + y 2 (2) = W At equilibrium, both agents MRS are equal to each other and the price ratio. u 1 y 1 (1) u 1 y 2 (1) = u 2 y 1 (2) u 2 y 2 (2) π 1 v 1 (y 1 (1)) π 2 v 1 (y 2 (1)) = π 1v 2 (W y 1 (1)) π 2 v 2 (W y 2 (1)) v 1 (y 1 (1)) v 2 (W y 1 (1)) = v 1 (y 2 (1)) v 2 (W y 2 (1))

42 u 1 y 1 (1) u 1 y 2 (1) = u 2 y 1 (2) u 2 y 2 (2) π 1 v 1 (y 1 (1)) π 2 v 1 (y 2 (1)) = π 1v 2 (W y 1 (1)) π 2 v 2 (W y 2 (1)) v 1 (y 1 (1)) v 2 (W y 1 (1)) = v 1 (y 2 (1)) v 2 (W y 2 (1)) By assumption, v 1, v 2 are strictly concave, therefore v 1, v 2 are strictly decreasing. v 1 (x) The function f (x) = is strictly decreasing, so if two values v 2 (W x) x, x give f (x) = f (x ), then x = x. Therefore, y 1 (1) = y 2 (1) and both agents consume the same amount in each state.

43 Mutuality Principle Lengwiler Box 5.1 (Mutuality Principle): An efficient allocation of risk requires that only aggregate risk be borne by agents. All idiosyncratic risk is diversified away by mutual insurance among agents. The marginal aggregate risk borne by an agent equals the ratio of his absolute risk tolerance to the average risk tolerance of the population. The mutuality principle can fail if: Beliefs are heterogeneous (different agents have different subjective probabilities of states) if market frictions (e.g. trading costs, short sale constraints) impede Pareto efficiency if markets are incomplete (we ll get to this in a few minutes)

44 Mutuality Principle This principle has many applications in different fields of economics. In international macro, many papers try to test efficient risk sharing among different countries, and explain if/why it does not occur In labor, test efficient risk sharing among workers, retirees, health insurance consumers, etc Many of the models we have seen in this course want to explain banks as a way to implement some sort of risk-sharing. However, risk-sharing is not the only motivation for financial transactions.

45 Next Week Homework 1 will be posted on the website later today. It will be due in 2 weeks.