microeconomics of banking: lecture 10

Microeconomics of Banking: Lecture 10

Prof. Ronaldo CARPIO

Dec. 4, 2015

Prof. Ronaldo CARPIO Microeconomics of Banking: Lecture 10

Administrative Stuff

▸ HW 4 is due next week.

▸ Final is on Dec. 18.


Review of Last Week

▸ In asymmetric information models, one party (e.g. seller, employee)has relevant information that another party (e.g. buyer, employer)does not know.

▸ In a model with adverse selection, the ”low-quality” type is the onlyone that is traded on the market, since the buyer does not know thetrue type of the object.

▸ One way to mitigate this problem is with a costly signal.

▸ We saw a signaling model of education with ”low” and ”high” typesof workers.

▸ Education could be used as a signal to differentiate betweenworkers, even though it was assumed to have no effect on ability.

▸ A separating equilibrium is an equilibrium where different typeschoose different actions, and therefore can be distinguished fromone another.

▸ A pooling equilibrium is one where different types choose the sameaction, and cannot be distinguished from one another.


Review of Last Week

▸ We saw a model of risk-averse entrepreneur/borrowers with privateinformation about the quality of their projects.

▸ If the project quality θ were not private, then higher-quality projectswould be sold to investors at a higher price.

▸ If the project quality is private, we can have a situation with adverseselection, where only low-quality projects are sold.

▸ Entrepreneurs with high-quality projects can use self-financing as acostly signal to differentiate themselves.


Example of Principal-Agent Problem

▸ There is a single insurance company and a single consumer who isconsidering whether to buy car insurance.

▸ The consumer might get into an accident, causing a random loss of`, where ` ∈ {0,1, ...,L}.

▸ The consumer can choose two levels of ”effort” to drive safely:e ∈ {0,1}. 1 gives a higher disutility to the consumer.

▸ The probability of getting into an accident of loss ` depends on e:π`(e), assumed to be strictly positive for all `, e.

▸ Assume π`(0)π`(1)

is strictly increasing in ` ∈ {0,1, ...,L}.

▸ This means that conditional on accident ` being observed, theprobability that e = 0 was chosen is higher than the probability thate = 1 was chosen.


Example of Principal-Agent Problem

▸ Assume the insurance company is risk-neutral. Therefore, it onlycares about the expected value of its payoff.

▸ Assume the consumer has vN-M utility function u(⋅), which isstrictly increasing and strictly concave.

▸ This implies the consumer is risk-averse, and is always willing toaccept some decrease in average payoff in exchange for a decreasein variance.

▸ Let d(e) denote the disutility of effort. Assume d(1) > d(0).

▸ Consumer’s vN-M utility function is: u(⋅) − d(e)

▸ Assume the consumer’s initial wealth is w , which is large enough topay for insurance.


Insurance Contract

▸ A contract specifies an action in each observable state of the world(i.e. each random outcome that is observable to all parties).

▸ The insurance company can observe the loss `, but not the effort e.Therefore, the contract can only specify actions for values of `.

▸ The contract is a tuple (p,B0,B1, ...,BL), where:

▸ p is the price of the contract, paid by the consumer to theinsurance company in the beginning

▸ B` specifies the payment from the insurance company to theconsumer, if accident ` occurs

▸ Assume that consumers have a reservation utility u that they canget elsewhere (perhaps from a competing insurance company).

▸ We want to find a contract that maximizes the insurance company’sprofits, while still being acceptable to the consumer.


Insurance Contract

▸ We can model this as an extensive-form game:

1. First, the insurance company proposes a contract, which is acollection of L + 1 real numbers (p,B0,B1, ...,BL), to theconsumer.

2. The consumer can accept or reject.3. If the consumer rejects, the insurance company gets a payoff of

0, and the consumer gets his reservation utility u.4. If the consumer accepts, the consumer pays p to the company.5. The consumer then chooses an effort level e = 0 or e = 1.6. Nature chooses the loss level ` ∈ {0,1, ...,L}, where the

probability distribution depends on e.7. The insurance company pays B` to the consumer. Final payoff

is p −B`.8. Consumer’s final payoff is his utility, u(w − p − ` +B`) − d(e).


Symmetric Information Case (e is observable)

▸ Suppose e is observable. Then e can be specified in the contract.

▸ The insurance company wants to solve the following problem:

maxe,p,B0,...,BL

p −L

∑`=0

π`(e)B` subject to

L

∑`=0

π`(e)u(w − p − ` +B`) − d(e) ≥ u

▸ This problem maximizes over the variables (e,p,B0, ...,BL). We candecompose it into two steps:

▸ Take e as given, and maximize over (p,B0, ...,BL), to get afunction of e.

▸ Maximize this function over e.


Symmetric Information Case (e is observable)

▸ Take e as given.

maxp,B0,...,BL

p −L

∑`=0

π`(e)B` subject to

L

∑`=0

π`(e)u(w − p − ` +B`) − d(e) ≥ u

▸ Lagrangian is:

L(p,B0, ...,BL) = p−L

∑`=0

π`(e)B`−λ [u −L

∑`=0

π`(e)u(w − p − ` +B`) − d(e)]


First Order Conditions

▸ First order conditions:

∂L

∂p= 1 − λ [

L

∑`=0

π`(e)u′(w − p − ` +B`)] = 0

∂L

∂B`= −π`(e) + λπ`(e)u

′(w − p − ` +B`) = 0 for ` ∈ {0, ...,L}

∂L

∂λ= u −

L

∑`=0

π`(e)u(w − p − ` +B`) − d(e) = 0

▸ The insurance company will lower the payouts as far as possible, sowe can assume the constraint is binding and λ > 0.

▸ The middle equalities implies u′(w − p − ` +B`) = 1/λ for all `,therefore B` − ` is constant for all `.

▸ This is a classic risk-sharing result:▸ the risk-averse agent will fully insure (receive the same utility

in all states/outcomes)▸ the risk-neutral party will take on risk (receive different utilities

in different states/outcomes)Prof. Ronaldo CARPIO Microeconomics of Banking: Lecture 10

Optimal Contract for Symmetric Information

▸ There are many possible optimal contracts, since p can be adjustedto compensate for B`.

▸ One possible optimal contract is: B` = ` for ` = 0, ...,L

▸ For each e, the optimal p is determined by the equationu(w − p(e)) = d(e) + u

▸ Then, the optimal e maximizes the expected profit of the insurancecompany:

p(e) −L

∑`=0

π`(e)`


Asymmetric Information Case

▸ Suppose e is not observable. The insurance company must designthe contract so that the consumer will voluntarily choose thecompany’s desired e.

▸ This introduces an additional incentive constraint, that the choiceof e is utility-maximizing for the consumer:

maxe,p,B0,...,BL

p −L

∑`=0

π`(e)B` subject to

L

∑`=0

π`(e)u(w − p − ` +B`) − d(e) ≥ u

L

∑`=0

π`(e)u(w −p − `+B`)−d(e) ≥L

∑`=0

π`(e′)u(w −p − `+B`)−d(e

′)

▸ where e ≠ e′.


Optimal Policy for e = 0

▸ In the symmetric information case, the optimal policy given e = 0must satisfy the equality

u(w − p) = d(0) + u for ` = 0, ...,L

▸ Adding the incentive constraint to this problem does not increasethe feasible set, so if a maximizer of the symmetric problem alsosatisfies the incentive constraint, then it is also a maximizer of theasymmetric case.

▸ The incentive constraint for e = 0 is:

L

∑`=0

π`(e)u(w −p − `+B`)−d(e) ≥L

∑`=0

π`(e′)u(w −p − `+B`)−d(e

′)

d(0) ≥ d(1)

▸ which is true by assumption.

▸ The optimal policy for e = 0 is the same as in the symmetric infocase.


Optimal Policy for e = 1

▸ Form the Lagrangian with two constraints (the incentive constraint,and the constraint ensuring e = 1).

▸ It can be shown that both constraints are binding: consumer isindifferent between e = 0 and e = 1

▸ The optimal policy must have the property: ` −B` is strictlyincreasing in `

▸ This no longer provides full insurance. It specifies a deductiblepayment that increases with size of loss

▸ This is necessary to incentivize consumer to choose high effort.


Delegated Monitoring (2.4 in Freixas & Rochet)

▸ When there is asymmetric information, then efficiency can beincreased with monitoring, which can mean:

▸ screening of projects before deciding to lend to them▸ preventing opportunistic behavior of a borrower (e.g. replacing

management who are not performing well)▸ punishing borrowers who fail to meet their contractual

obligations

▸ Specialized firms are likely to be more efficient at monitoring thanindividual lenders. This provides a clear justification for the role offinancial intermediaries.


Delegated Monitoring

▸ This is based on the paper: Diamond (1980), ”FinancialIntermediation and Delegated Monitoring”

▸ There are n identical, risk-neutral firms that are seeking to finance aproject.

▸ Assume the riskless interest rate is R.

▸ Each firm requires an investment of 1, and the cash flow of eachfirm is an i.i.d. random variable y .

▸ All agents agree on the distribution of y , but it is only observed bythe entrepreneur of the firm, but not the lender.

▸ Assume y > 0, and E(y) > R +K , where K > 0 will be defined later.

▸ The entrepreneur can lie and misreport the cash flow of the firm.

▸ How can the lenders and borrowers come to an agreement?


Approach 1: Optimal Financial Contract

▸ Let’s try to come up with a contract that is incentive-compatible(both sides will make a positive expected payoff by agreeing to it).

▸ A complete contract is a rule that specifies what each party shoulddo, for each observable state of the world.

▸ Here, only the entrepreneur’s chosen repayment z is observable, so acontract specifies the actions of the borrower and lender for everypossible z .

▸ In the absence of outside enforcement, a contract is only useful ifboth sides voluntarily agree to follow its terms.

▸ Therefore, any contract that is actually agreed to must be a Nashequilibrium: the parties have no incentive to deviate from the termsof the contract.



▸ Suppose y is the actual cash flow of the firm.

▸ z(y) is a function that specifies what the entrepreneur will repay,given y .

▸ φ(z) is the penalty that is imposed by the lender when theentrepreneur repays z .

▸ An optimal penalty function must satisfy these conditions:

▸ Penalty must be high enough to deter underpayment▸ Penalty must be small to avoid welfare costs



▸ Consider this penalty function: φ∗(z) = max(h − z(y),0), where h isthe contractually agreed level of repayment

▸ The sum of repayment and penalties is always h; therefore, theentrepreneur is indifferent between lying and telling the truth

▸ We’ll assume that the entrepreneur will not lie in this situation.



▸ What should h be? Consider this value of h: the lowest value thatprovides lenders with an expected return of R.

▸ h is the smallest value that satisfies

P(y < h)E [y ∣y < h] + P(y ≥ h)h = R

▸ Consider z(y) = min(h, y).

▸ The entrepreneur passes on all the cash flow if y < h; otherwise, herepays h and keeps the rest for himself

▸ This is like a standard credit contract: if the borrower cannot repay,the lenders seize the firm and liquidate the assets

▸ This contract is incentive-compatible for both the lender and theborrower:

▸ the lender’s expected return is equal to R, by the aboveequation

▸ the borrower is indifferent between lying and not lying, asshown previously, so will report the true y



▸ The final specification of the contract is:

▸ The borrower agrees to repay h, which satisfies

P(y < h)E [y ∣y < h] + P(y ≥ h)h = R

▸ If he repays some amount z < h, then the lender will impose apenalty of h − z , otherwise the penalty is zero

▸ This is not socially efficient, since the entrepreneur has to pay apenalty even when he is telling the truth.

▸ Entrepreneurs could be made better off without making lendersworse off, if y were observable.

▸ An alternative solution: Monitoring by lenders


Approach 2: Contract with Monitoring

▸ Suppose there are m lenders per firm, and that each lender canspend K > 0 to observe y .

▸ Welfare is increased if mK < E [φ∗(z(y))], i.e. if the total cost ofmonitoring is less than the expected amount of penalty.

▸ Monitoring is preferable if m is small, and if the expected penalty ishigh (e.g. if there is a high probability of low returns)


Approach 3: Delegated Monitoring

▸ If monitoring is delegated to a financial intermediary (FI),monitoring costs can be reduced to K

▸ However, this brings up an additional problem: how can the lenderstrust the FI? There must be an additional level of monitoring

▸ We can combine the two previous approaches: the lenders have anoptimal contract with penalties between themselves and the FI

▸ And, the FI monitors the borrowers.

▸ If the FI deals with a large number of firms with independentreturns, then the probability of low returns goes to 0, by the Law ofLarge Numbers

▸ Then, the expected penalties go to zero as well.


Coalitions of Depositors (Ch. 2.2 in Freixas & Rochet)

▸ This is from Diamond & Dybvig (1983), ”Bank runs, depositinsurance, and liquidity”

▸ One way to think about depository institutions like banks: pools ofliquid assets (e.g. cash).

▸ Individual households need cash due to idiosyncratic shocks (e.g. tofinance consumption).

▸ These households can combine their resources in a coalition, todiversify against risk.

▸ As long as these households’ shocks are not correlated across time,then the total cash reserve needed to satisfy these shocks increasesless slowly than the total number of households.


The Model

▸ There are three time periods: t = 0,1,2.

▸ There is 1 good.

▸ A continuum of consumers who are identical at t = 0 are endowedwith 1 unit of the good.

▸ At t = 1, the consumers learn what type they are.

▸ Type-1 consumers are called ”early” consumers, and consumeonly in t = 1, with utility u(c1)

▸ Type-2 consumers are called ”late” consumers, and consumeonly in t = 2, with utility u(c2)

▸ Let π1, π2 be the probability of being Type-1 or 2. The expectedutility at t = 0 is

U = π1u(c1) + π2u(c2)


The Model

▸ At t = 0, a consumer can choose to consume his endowment, orsave it in two ways:

▸ Store it for 1 period. This is equivalent to a riskless asset witha net return of 1.

▸ Invest amount I in a long-run technology, which gives a netreturn of R > 1 at t = 2, but it can be liquidated early, to givea return of L < 1 in t = 1.

▸ For example, a long-dated certificate of deposit, or a long-termconstruction project.


Optimal Allocation

▸ First, let’s look at the optimal allocation, which maximizes expectedutility at t = 0.

maxc1,c2,I

π1u(c1) + π2u(c2) subject to

π1c1 = 1 − I , π2c2 = RI ⇒ π1c1 + π2c2R

= 1

▸ The first order condition is

u′(c∗1 ) = Ru′(c∗2 )

▸ The marginal rate of substitution between consumption at t = 1,2equals the return on the long-run technology.


Autarky: No Trade Between Agents

▸ Suppose that trade between agents is not allowed.

▸ At t = 0, each agent chooses the amount to store, and the amountto invest I .

▸ Then at t = 1, their type is revealed.

▸ Type-1 agents’ income will be the amount they stored, plus LI(the liquidation value of what was invested in t = 0)

c1 = 1 − I + LI = 1 − I (1 − L)

▸ Type-2 agents’ will continue their storage until t = 2. Then,their income will be the amount they stored, plus RI (thelong-term return on what was invested in t = 0)

c2 = 1 − I + RI = 1 + I (R − 1)



▸ Type-1 agents’ income will be the amount they stored, plus LI (theliquidation value of what was invested in t = 0)

c1 = 1 − I + LI = 1 − I (1 − L)

▸ Type-2 agents’ will continue their storage until t = 2. Then, theirincome will be the amount they stored, plus RI (the long-termreturn on what was invested in t = 0)

c2 = 1 − I + RI = 1 + I (R − 1)

▸ At t = 0, all investors will maximize expected utility subject to thesetwo constraints.

▸ Note that c1 ≤ 1, with equality if I = 0.

▸ c2 ≤ R, with equality if I = 1.



▸ Note that c1 ≤ 1, with equality if I = 0.

▸ c2 ≤ R, with equality if I = 1. Since both cannot hold, then

π1c1 + π2c2R

< 1

▸ This is inefficient, since some resources are lost due to earlyliquidation.


Introduce a Financial Market for Bonds

▸ Suppose that agents are allowed to trade a riskless bond with otheragents.

▸ We model this by introducing a financial market at t = 1: agentscan trade 1 unit of the good for 1 riskless bond that has price p(endogenously determined), and returns 1 at t = 2.

▸ We will assume agents are price-takers.

▸ Since this is after agents’ types have been revealed, Type 1 agentswill trade with Type 2 agents.

▸ Both Type 1 and Type 2 agents chose the same amount of I , sincetheir type was not yet revealed.



▸ Both Type 1 and Type 2 agents chose the same amount of I , sincetheir type was not yet revealed.

▸ Type 1 agents will sell bonds at t = 1. They will repay these bondsat t = 2 with the returns of the long-term technology:

c1 = 1 − I + pRI

▸ Type 2 agents will buy bonds at t = 1 with the amount they stored,and get extra consumption at t = 2:

c2 =1 − I

p+ RI =

1

p(1 − I + pRI )



▸ Type 1 agents will sell bonds at t = 1. They will repay these bondsat t = 2 with the returns of the long-term technology:

c1 = 1 − I + pRI

▸ Type 2 agents will buy bonds at t = 1 with the amount they stored,and get extra consumption at t = 2:

c2 =1 − I

p+ RI =

1

p(1 − I + pRI )



c1 = 1 − I + pRI

c2 =1 − I

p+ RI =

1

p(1 − I + pRI )

▸ Note that c1 and c2 are linear functions of I , and c2 =c1p

.

▸ The only interior equilibrium occurs when p = 1R

.

▸ If p > 1R

, then I = 1 and there is an excess supply of bonds: Type-1agents will sell bonds, but no one will buy them

▸ If p < 1R

, then I = 0 and there is excess demand of bonds: Type-2agents want to buy bonds, but no one will sell them



▸ At equilibrium, c1 = 1, c2 = R, I = π2.

▸ This Pareto-dominates the autarky solution since there is noliquidiation, therefore no wasted resources.

▸ However, the first-order condition u′(c∗1 ) = Ru′(c∗2 ) is not satisfied.

▸ Efficiency is acheived when there is perfect insurance: the marginalutility is equalized across all possible states of the world (that is, allpossible random outcomes).

▸ In a complete market (i.e. there is a security corresponding to everyoutcome), risk-averse agents will perfectly insure.

▸ This market, however, is incomplete, since there is no securitycorresponding to the outcome of the liquidity shock (Type 1 vs.Type 2).


Financial Intermediation

▸ Suppose we know what the optimal allocation (c∗1 , c∗

2 ) is.

▸ Assume there is a financial intermediary that offers this depositcontract:

▸ In exchange for 1 unit at t = 0, the FI will pay either c∗1 att = 1, or c∗2 at t = 2.

▸ Out of the 1 deposited, the FI will store π1c∗

1 , and invest 1−π1c∗

1 inthe long-term technology.

▸ The FI achieves the Pareto-optimal allocation.

▸ Note that this requires that only Type-1 depositors withdraw att = 1; this is true in Nash equilibrium, since Type-2 depositors wouldlower their payoff if they withdrew early.

▸ There is another Nash equilibrium where all Type-2 depositorswithdraw; this is a bank run and will be studied later in the course.

▸ One problem with this model is that the FI cannot coexist with afinancial market.


What Can Banks Do That Investors Can’t Do ByThemselves?

▸ In the first week, we saw that in a general equilibrium with completemarkets (i.e. a security exists for every possible state of the world),there is no role for banks.

▸ If consumers can directly lend to borrowers, or buy and tradesecurities, then there is no need for a financial intermediary.

▸ If we want to model financial intermediaries correctly, we need tospecify something that individual consumers cannot do bythemselves.

▸ We have seen three models that provide this rationale:▸ A single depositor cannot diversify against liquidity shocks, but

a coalition of many depositors can;▸ If there is asymmetric information between borrowers and

lenders, a coalition of lenders can decrease the cost of capital;▸ If lenders have to monitor borrowers, then a financial

intermediary that deals with many lenders can decrease theaverage cost of monitoring.


Summary of Ch. 2

▸ We’ve seen 3 models where financial intermediaries can providesomething that individual investors cannot do for themselves.

▸ In each of the 3 models, the individual borrower-lender relationshipis inefficient, due to some sort of asymmetric information.

▸ If we assume that coalitions of borrowers or lenders can diversifyaway this inefficiency, then we have a rationale for financialintermediaries.


Administrative Stuff

▸ HW 4 is due next week.

▸ Final is on Dec. 18.


microeconomics of banking: lecture 10

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