EC3101 Consolidated InformationTable of ContentsWeek 1..................................................................................................................8Present and Future Value..................................................................................8The Intertemporal Choice Problem....................................................................8The Intertemporal Budget Constraint............................................................8Intertemporal Utility.......................................................................................9Optimal Consumption.....................................................................................9Optimisation with Inflation................................................................................9Comparative Statistics.....................................................................................10Changes in Interest Rate and Inflation Rate.................................................10Income and Substitution Effects......................................................................10Important assumption...................................................................................10Week 2................................................................................................................11Expected Utility...............................................................................................11Expected Utility Function.............................................................................11Log Transformation of Cobb-Douglas Function............................................11Affine Transformations.................................................................................11Expected Money/Value....................................................................................12Risk Aversion and Risk Loving.........................................................................12Measures of Risk Aversion...............................................................................12Certainty Equivalence...................................................................................12Risk Premium................................................................................................12Arrow-Pratt Measure of Risk Aversion.........................................................12Week 3................................................................................................................13Insurance.........................................................................................................13States of Nature............................................................................................13Contingencies...............................................................................................13Consumption and Premium...........................................................................13Budget Constraint.........................................................................................14Utility............................................................................................................14Optimal Consumption...................................................................................14Optimising Utility and Insurance.....................................................................15Competitive Insurance..................................................................................15Optimal Insurance........................................................................................15Fair Insurance...............................................................................................15

Unfair Insurance...........................................................................................16Monopoly.............................................................................................................17Differences between Perfect Competition and Monopoly................................17Monopoly Market.............................................................................................17Introduction..................................................................................................17Profit Maximisation.......................................................................................17Why is P>MR for monopolist?.......................................................................17Elasticity of Demand........................................................................................18Monopolistic Pricing........................................................................................18Mark-up Pricing............................................................................................18Week 4................................................................................................................19Inefficiency of Monopoly..................................................................................19Price Discrimination........................................................................................19Perfect Price Discrimination.........................................................................19Third-degree Price Discrimination................................................................20Barriers to Entry..............................................................................................21Natural Monopolies.........................................................................................21Tax Levied on Monopolist................................................................................23Quantity Tax..................................................................................................23Lump-sum Tax...............................................................................................23Week 5 and 6.......................................................................................................24Differences between Perfect Competition and Monopoly................................24Types of Oligopoly Markets.............................................................................24Cournot Competition........................................................................................24Stackelberg Competition.................................................................................25Collusion..........................................................................................................25Case 1: Both firms have constant marginal cost..........................................25Case 2: Only one firm has a constant marginal cost.....................................25Case 3: Both firms have increasing marginal costs......................................25Punishment – Grim Trigger Strategy............................................................26Bertrand Competition......................................................................................26Case 1: A firm sets price ¿ pm.......................................................................26Case 2: A firm sets price ≤ pm.......................................................................26Week 8................................................................................................................27Pareto Optimality.............................................................................................27Pareto Improvement.....................................................................................27Games..............................................................................................................27Definition and Assumptions..........................................................................27

Nash Equilibirum..........................................................................................27Strategies.........................................................................................................28Strictly Dominant and Dominated Strategies...............................................28Weakly Dominant and Dominated Strategies...............................................28Link between Nash Equilibrium, Stratgies, and Best Response......................28Solving Simultaneous Games...........................................................................28Best Reponse Method...................................................................................28Iterative Deletion of Dominated Strategies..................................................28Representation of Simultaneous Games..........................................................29Week 9................................................................................................................29Repeated Games..............................................................................................29Infinite Games...............................................................................................29Finite Games.................................................................................................29Representation of Sequential Games...............................................................30Solving Simultaneous Games...........................................................................30Backward Induction......................................................................................30Entry Deterrence.............................................................................................30Non-credible Threats....................................................................................31Mixed Strategy Nash Equilibria (in Sequential Games)..................................31Pure Strategy Nash Equilbiria......................................................................31Mixed Strategy Nash Equilbiria....................................................................31Example........................................................................................................32Week 10..............................................................................................................33Rubenstein Bargaining....................................................................................33Discount Factor............................................................................................33Assumptions..................................................................................................33How to Solve.................................................................................................33Example........................................................................................................34Week 11..............................................................................................................35Definition of Externality...................................................................................35Pareto Optimality..........................................................................................35Production Externality.....................................................................................35Profit Maximisation.......................................................................................36Ways to Achieve Economic Efficiency..............................................................36Internalising the Externality.........................................................................36Coase Theorem.............................................................................................37Tragedy of the Commons.................................................................................37Reasons for Tragedy.....................................................................................38

Modern-day Tragedies of the Commons.......................................................38Solving the Problem......................................................................................38Week 12..............................................................................................................39Goods...............................................................................................................39Excludability and Rivalry..............................................................................39Types of Goods..............................................................................................39Provision of Public Goods................................................................................39Pareto Optimal Allocation of Public Good.....................................................40Asymmetric Information..................................................................................42Information in Competitive Market..............................................................42Imperfect Information...................................................................................42Adverse Selection............................................................................................42Buying Lemons and Peaches........................................................................43Adverse Selection with Quality Choice.........................................................44Ways to Reduce Adverse Selection...............................................................44Week 13..............................................................................................................45Signalling.........................................................................................................45Education......................................................................................................45Separating Equilbirum..................................................................................46Moral Hazard...................................................................................................46Incentive Contracting......................................................................................47Examples of Contracts that Satisfy the Incentive-Compatiblity Constraint.....48Rental Contracts...........................................................................................48Wage Contracts............................................................................................48Take it or Leave it Contracts........................................................................48Examples of Contracts that Do Not Satisfy the Incentive-Compatiblity Constraint........................................................................................................49Fixed Fee......................................................................................................49Revenue-sharing Contracts...........................................................................49Incentive Contracts with Uncertainty..............................................................49Rental Contracts...........................................................................................49Revenue-sharing Contracts...........................................................................49

Week 1Present and Future ValueThe present value of a good is the value of a future good in the present. The future value of a good is the value of a present good in the future.Let r be the interest rate between periods. Let the future value of a good in period n≥2 be FV n, where the current period is period 1. Then,

PV=F V n

(1+r )n−1

The Intertemporal Choice ProblemThe Intertemporal Budget Constraint Let m1 and m2 be the endowments the agent receives in periods 1 and 2 (in

terms of goods) Let c1 and c2 be the consumption of a good c, in periods 1 and 2 respectively Let p1 and p2 be the prices of consumption in periods 1 and 2.The budget constraint must satisfy either

PVLR=PVLC⇒ p1m1+p2m2

1+r=p1 c1+

p2 c21+r

FVLR=FVLC⇒ (1+r ) p1m1+ p2m2=(1+r ) p1c1+ p2 c2

Intertemporal Utility To find the indifference curve farthest from the origin that still satisfies the budget constraint, we look for the indifference curve tange to the budget constraint.

Optimal Consumption Slopeof Budget Constraint=Slope of Indifference Curve(1+r )

p1p2

=MRS=MU c1

MU c2

Optimisation with InflationDefine the inflation rate by π where p2=p1 (1+π )

Then, the budget constraint becomes(1+r ) p1m1+(1+π ) p1m2=(1+r ) p1c1+(1+π ) p1 c2

c2=−(1+r )

(1+π )c1+

(1+r )(1+π ) (m1+ (1+π )

(1+r )m2)

where the slope now (from Fisher’s Equation) is−(1+r )(1+π )

=−(1+ρ )⇒ ρ=r−π1+π

where ρ is the real interest rate. At low inflation rates (π ≈0 ) , ρ≈ r−π .Comparative StatisticsSince the budget constraint is −(1+r )

(1+π ), the budget line flattens if r falls or π

increases (since it becomes less negative).Changes in Interest Rate and Inflation Rate

Interest Rate Inflation Rate Borrowing Saving Budget LineIncrease Decrease

Costlier More rewarding SteeperA borrower may become a saver as he is worse off A saver remains a saverDecrease Increase

Less costly Less rewarding FlatterA borrower remains a borrower A saver may become a borrower as he is worse offThe budget line pivots about the endowment point (c1 , c2)=(m1 ,m2 ) because changes in interest rate or inflation rate do not affect endowment (points of no borrowing or saving).

Income and Substitution Effects Income Effect

An increase in r increases income, increasing consumption in both periods. Substitution Effect

An incresae in r makes consumption more attractive in period 2 (c2 increases)

Important assumption Since endowment is in units of the consumption good, where there is inflation, the consumer’s salary in period 2, m2 p2, increases to ensure he can still afford m2.

Week 2Expected UtilityExpected Utility Function Expected utility of a lottery is a weighted sum of the utility from consumptions in each possible state where the weights are the probabilities of each state occuring.Let u ( x ) be the utility function for any consumption x. Then, the expected utility (or von-Neumann-Morgenstern utility) function is any function of the form

EU=π1u (c1 )+π 2u (c2 )+…+πnu (cn ) ,where∑i=1

n

π i=1

Log Transformation of Cobb-Douglas Function Although c1π 1 c21−π1 is not an expected utility function, its log transformation

ln EU=π1 ln c1+ (1−π1 ) ln c2

is an expected utility function. The log transformation changes the scale of utility relative to the Cobb-Douglas function, while preserving the ranking. That is if A≻B with Cobb-Douglas, A≻B with log transformation.Thus, while individual utility changes, since only the ranking of preferences matter, the log transformation captures utility in the same way as the Cobb-Douglas utility.Affine Transformations

EU=π1u (c1 )+π 2u (c2 )+…+πnu (cn )

The expected utility function maintains the expected utility properties under positive affine transformations. That is v=au+b, then the following is also an expected utility function,

EU=π1 v (c1 )+π2 v (c2 )+…+πn v (cn )

Expected Money/ValueExpected money of a lottery is the probability weighted average of the outcomes of a lottery

EM=π1m1+π2m2+…+πnmnU (EM )=u (π 1m1+π 2m2+…+πnmn )

Risk Aversion and Risk LovingRisk Type EU vs U(EM) Preference Utility Function Function Class Marginal Utility in

cAverse U (EM )>EU Average payout over lottery Concave Injective(one-to-one map)DecreasingLoving U (EM )<EU Lottery over average payout Convex IncreasingNeutral U (EM )=EU Indifferent Linear ConstantMeasures of Risk Aversion

Certainty Equivalence The certainty equivalence value is defined as the amount a consumer must receive with certainty to attain the same level of utility as a risky asset. Thus, given a utility function u ( x ),

U (CE )=EU

The larger the certainty equivalence, the more the consumer values the lottery.Risk Premium Risk premium is a measure of risk-aversion, as it is the willingness to pay to avoid risk (discount willing to be taken to exchange risk for certainty).

RP=EM−CE

The larger the risk premium, the more the agent hates uncertainty.

Arrow-Pratt Measure of Risk Aversion The Arrow-Pratt measure shows us the degree of risk aversion using the curvature of utility.

−u' ' (w )u ' (w )

The larger the curvature, the higher the risk aversion.Week 3

InsuranceStates of Nature Different outcomes of some random event Let damage be denoted d with probability πd

Let no damage be denoted nd with probability πnd=1−πd

Contingencies A contingent consumption plan is a specification of what will be consumed in each different state of nature (and only one is consumed at any time). Let consumption with damage be denoted cd

Let consumption with no damage be denoted cnd

Consumption and Premium Let the consumer’s wealth be $m, and any loss from damage be $ L.The premium of an insurance is its price. If $1 of damage insurance costs $γ<$1 and if the agent buys $ K of damage insurance, he pays $γK and receives $ K in the event of damage

First derivative normalises the scale of utility functions to ensure measure is the same even for affine transformations

Budget Constraint Without Insurance cd=m−L

cnd=m

With Insurance cd=m−L−γK+K=m−L+(1−γ )K

cnd=m−γKThus, the budget constraint is given by

cnd=−γ1−γ

cd+m−γK1−γ

Utility EU=πdu (cd )+πnd u (cnd )

dEU=π d MU (cd )∂cd+πnd MU (cnd )∂cnd=0

Thus, ∂cd

∂cnd=

πd MU (cd )πnd MU (cnd )

Optimal Consumption γ1−γ

=πd MU (cd )π ndMU (cnd )

We find the optimal consumption point to solve for the optimal value of K , that is, the optimal amount of insurance to be bought.Optimising Utility and InsuranceCompetitive Insurance The insurance firm’s expected profit function is

Profit=Revenue−InsurancePayout if damage¿γ K−πd K¿ (γ−π d ) K

If the market for insurance is competitive, the firms can enter the market freely, and they enter until there are 0 expected profits. That is, with free entry,

( γ−π d ) K=0

γ=πd

If the price of $1 insurance, $γ , equals the probablity of damage, πd, then insurance is fair.Optimal Insurance

γ1−γ

=πd MU (cd )π ndMU (cnd )

πd

1−πd=

π d MU (cd )πnd MU (cnd )

πd

π nd=

πd MU (cd )πnd MU (cnd )

MU (cd )=MU (cnd )

Since marginal utility is the same, and utility functions are injective, cd=cnd.Fair Insurance For fair insurance, γ=πd and thus, cd=cnd. That is, when one fully insures against uncertainty, all risk is eliminated. The consumer consumes the same level in either state, d or nd.A consumer fully insures if they are risk averse, maximises expected utility, and is offererd fair insurance.Unfair Insurance If insurers make positive expected profit, that is,

( γ−π d ) K>0

γ>πd

The insures chager more than the cost of insuring. Thus, 1−γ<1−π d=πnd

γ1−γ

>π d

πnd

Since rational choice requires γ1−γ

=πd MU (cd )π ndMU (cnd )

πd MU (cd )π ndMU (cnd )

>π d

πnd⇒

MU (cd )MU (cnd )

>1

MU (cd )>MU (cnd )

For all risk averse utility functions, since marginal utility is decreasing in c, we have

cd<cnd

In this case, the consumer only partially insures, but still consumes more in the no damage state.

MonopolyDifferences between Perfect Competition and Monopoly

Perfect Competition MonopolyIndustry Structure Fragmented, no seller has significant market share Concentrated, one firm dominates entire marketPricing Price takes Price makersBarriers to Entry None SignificantExamples Raw commodities Patented drugsA perfectly competitive market is able to achieve an efficient outcome if firms care only about maximising profits, as this maximises total surplus (no deadweight loss).Monopoly MarketIntroduction A monopolised market has a single seller who can decide for itself what price to charge. However, a monopolist faces the constraint of a downward sloping market demand curve, and hence, can only increase output by reducing price.

Profit Maximisation π (q )=r (q )−c (q )=p (q )q−c (q )

π ' (q )=Marginal Revenuep (q )+ p' (q )q

M¿

The profit maximising output occurs whenπ ' (q )=0Marginal Revenue=r ' (q )=c ' (q )=MarginalCost

Why is P>MR for monopolist?We have MR (q )=p' (q )q+ p (q ). Since p' (q )q<0,

MR (q )< p (q )

Elasticity of DemandLet price elasticity of demand be ϵ . Then,

ϵ= dqdp (q )

p (q )q

Since MR (q )=p (q )+ p ' (q )q, we then haveMR (q )= p (q )+ dp (q )

dqq=p (q )[1+ dp ( q )

dqq

p (q ) ]=p (q )[1+ 1ϵ ]Since ϵ<0,

MR (q )=p (q )[1− 1|ϵ|]

And for a profit maximising firm,

¿ Increasing output also lowers the price of all units, so revenue decreases by this change in price multiplied by all units¿ Increasing output by one unit raises the revenue by the price of

p (q ) [1− 1|ϵ|]=MC (q ) *For perfectly competitive markets, |ϵ|=∞ (perfectly elastic).

Hence, MC (q )=p (q )

Monopolistic PricingIf |ϵ|<1 (inelastic demand), then

MR (q )=p (q )[1− 1|ϵ|]<0

Since MC (q )>0, MR (q )≠MC (q ) for all q if |ϵ|<1. Thus, a profit maximising firm will not operate on the inelastic portion of the demand curve.Intuitively, on the inelastic portion, since the magnitude of the percentage fall in quantity demanded is less than the percentage rise in price, when price is increased, revenue rises. Hence, the firm can always do better by decreasing quantity until it reaches elastic demand.Mark-up Pricing

p (q ) [1− 1|ϵ|]=MC (q )⇒ p (q )= 1

1− 1|ϵ|

MC (q )

where 1

1− 1|ϵ|

is called the price mark-up over marginal cost.

Week 4Inefficiency of MonopolyIn a perfectly competitive market, the efficient output level qe satisfies p (q )=MC (q ) and total gains to trade is maximised.However, in a monopoly, there is deadweight loss as there are gains lost due to lower output. A monopolist only cares about his own profit, and not total gains-to trade. Since increasing output reduces producer surplus, the monopolist does not transact with the consumer.

Price DiscriminationPerfect Price Discrimination

Perfect price discrimination can fix this inefficiency. If the monopolist can charge a different price to each consumer, then they can charge on the demand curve until p (q )=MC (q ).

While this is efficient as it maximises total surplus, the monopoly absorbs all the surplus as producer surplus, and consumer surplus is 0.

Third-degree Price Discrimination A firm can only practice perfect price discrimination if it has information about every consumer, which is impossible to get. Instead, they can practice third-degree price discrimination, splitting the consumers into different groups that pay different prices. To do so, firms must1. Be able to identify the demand curve for specific group of consumers2. Ensure there is no arbitrage across groups. That is, consumers in one group

cannot resell the goods to consumers in another group. (Also a property of perfect price discrimination. This assumption is relaxed for second-degree price discrimination.)

No price discriminationMonopoly uses an aggregate demand function for the whole market to maximise profit.MR may not ¿MC for all markets, and hence, the monopoly should move quantity around.

Third-Degree Price DiscriminationMonopoly treats each group as a separate market, and maximises profit when

M R1=M R2=…=M Rk=MC

charging each market i a different price.

Method to find equilibrium price and quantity for uniform pricing:Let the firm with higher demand be Firm 1, and the lower demand be Firm 2.1. Plot both markets’ demand functions. Find the Q=Q at which the Firm 1

curve’s price coincides with the Firm 2 curve’s intercept.2. Split market demand into two ranges, one for 0<Q≤Q and Q>Q. Now,

Q={ P1 (Q ) 0<Q≤QP1 (Q )+P2 (Q ) Q>Q

3. Invert the above function (to get P), and then, π. Solve for profit maximising condition, and find the quantity produced. Sub this back to P, for the correct domain, and find price.

4. Using the price, sub it back into the individual demand functions for q1 and q2.

Barriers to EntryMonopolies exist because firms have barriers to entry – factors that allow an existing firm to earn positive profit while making it unprofitable for newcomers to enter the industry. If this did not exist, firms would enter until there are 0 profits.Reasons for barriers to entry:i. Legal: Patents and licenses issued by governments

This ensures the quality of products, while allowing firms to recoup costs and incentivising innovation. However, this denies access to the markets.

ii. Strategic: Pricing Limit Pricing (Entry deterrent pricing)

Monopoly charges low price to make it unprofitable for new players to enter

Predatory PricingLarge firms lower price to drive smaller rivals out of the market.

(Competition and Consumer Commission of Singapore monitors such behaviour and pursues legal action against firms who create such barriers to entry.)

iii. Structural Positive Network Externalities

The value of a product increases with the size of the market captured by a particular firm.

Natural MonopolyNatural MonopoliesNatural monopolies arise when the total cost a single firm would incur is less than the combined total cost that two or more firms would incur if the output were divided among them equally. Firms exhibit economies of scale as there is high fixed cost but low marginal

cost, with decreasing average total cost. Demand is small relative to cost structure.For a natural monopoly, since ATC is falling, MC must be below ATC. Like any profit-maximising monopolist, the natural monopolist causes a deadweight loss producing when MR (q )=MC (q ).

If the government wishes to maximise efficiency, it can command the natural monopoly to produce at the socially efficient level of output, when

p (q )=MC (q )

However, at this point qe, ATC (qe)> p (qe )

This causes the firm to make an economic loss, and it leaves the market. Thus, instead, the government can make the monopoly produce at

p (q )=ATC (q )

At this point qa, deadweight loss is minimised.

However, there are problems with such a policy: Enforcing such a policy is costly, and the regulator must go through the

financial records of the company to ascertain its ATC. Companies have an incentive to “pad” their financials to make it seem as if

their ATC is high, so that the government will allow them to charge higher prices.

Companies have no incentive to be cost efficient since they make zero profits no matter what.

Tax Levied on MonopolistQuantity Tax A quantity tax is distortionary as it reduces the quantity produced by the monopolist. A tax of $ t /¿ouput raises the marginal cost of production by $ t. Hence, the profit function becomes

π (q )=p (q )q−(c (q )+tq )

Thus now, marginal cost has increased from MC (q ) to MC (q )+t .The profit maximising condition has changed from MR (q )=MC (q ) to

MR (q )=MC (q )+t

The quantity tax causes a drop in output and a rise in price, causing a fall in demand for inputs.In all cases, the tax passed on to consumers ¿ p (qt )−p (qm )

However, if elasticity is constant across the entire demand curve,p (q t )−p (qm )= 1

1+ 1ϵ

(MC (q )+t )− 1

1+ 1ϵ

MC (q )= t

1+ 1ϵ

=t ϵ1+ϵ

Since |ϵ|>1, ϵ1+ϵ

>1, and hence, the monopolist passes on more than the tax to the consumer.Lump-sum Tax A lump-sum tax is non-distortionary and does not affect the quantity decision of the firm, as long as the lump-sum tax does not exceed the firm’s profit. However, it reduces the profits of the firm as compared to a scenario without tax. Hence, the profit function becomes

π (q )=p (q )q−(c (q )+T )

Week 5 and 6Differences between Perfect Competition and Monopoly

Perfect Competition Oligopoly MonopolyIndustry Structure Fragmented, no seller has significant market shareA small number of firms dominate the market

Concentrated, one firm dominates entire marketPricing Price takes Depends Price makersBarriers to Entry None Some SignificantExamples Raw commodities Cola, Airlines, Cell Phone Service Patented drugsTypes of Oligopoly Markets

Cournot Competition Firms produce identical products Firms compete by idependently and simultaneously choosing output levelsMethod to find Cournout Equilibrium:1. Obtain each firm’s (residual) demand.2. Calculate profit maximising condition using MR=MC or π ' (q )=0.3. Find each firm’s best response function by expressing the optimal output of

each firm as a function of the other firm’s quantity.

Do they cooperate?

Do they compete on price or quantity?

Do they act simultaneously or sequentially?Cournot

Stackelberg

Do they act simultaneously or sequentially?Bertrand

Sequential Price GameCollusion

QuantityPric

e

Simultaneous

No

Yes

SequentialSimultaneo

usSequenti

al

4. The point of intersection of the best response functions is the Cournot Equilibrium.

At this point, neither firm has an incentive to deviate.Stackelberg CompetitionStage 1: Stackleberg leader sets profit maximising quantity based on the Stackleberg follower’s best response function.Stage 2: Follower sets its quantity based on its best response function, and the leader’s quantity.Method to find Stackleberg Equilibrium:1. Find the Stackleberg follower’s best response function (in terms of leader’s

output).2. Plug this into the demand function, and solve for the leader’s profit

maximising quantity.3. Use the Stackleberg follower’s best response function to find its profit

maximising quantity.CollusionIn a collusion, 2 firms in a duopoly act as a single monopolist firm, or a cartel.Let demand be P (Q ), and costs of Firm 1 and Firm 2 be C1 (q1 ) and C2 (q2 ).Case 1: Both firms have constant marginal cost Firm with the lower marginal cost always produces. They split profits according to an agreement.

Case 2: Only one firm has a constant marginal cost WLOG, assume Firm 1 has the constant marginal cost of c ≥0. Let q be such that c=M C2 (q ). Then, Firm 2 only produces up till and including q. Firm 1 produces the remaining Q−q, if any.To determine Q, determine the profit maximising quantity under the domain constraints of

π={P (Q )Q−C2 (Q ) 0<Q≤qP (Q )Q−C1 (Q ) Q>q

Case 3: Both firms have increasing marginal costs Both firms will produce. Solve for q1=f 1 (Q ) and q2=f 2 (Q ) using

MC1 (q1 )=MC2 (q2 ) such that q1+q2=Q

Find the Q that maximises the profit function π (Q )=P (Q )Q−C1 ( f 1 (Q ) )−C2 ( f 2 (Q ) ).Note: MC1 (q1 )=MC2 (q2 )=MR (Q )

However, collusion is not a stable equilibrium. Firms will be tempted to cheat. In such a case, they can be punished, assuming that firms meet repeatedly for eternity, by not cooperating again.Punishment – Grim Trigger Strategy Let the firm’s profit per period be πu if it is in the cartel πc if it cheats in the first period (by optimising according to their best reponse function) π p in each period after it cheats (default is Cournot)

Let the firm’s discount factor be δ∈ [0,1 ]. This captures how much the firm cares about future profits relative to the current period. The lower it is, the less the firm values the future.

πcoop=πu+δ πu+δ2πu+…=

πu

1−δ

πcheat=πc+δ π p+δ2π p+…=π c+δ

π p

1−δ

Firm will cheat if and only if πcheat>π coop

Bertrand CompetitionWe make the following assumptions Products are identical for all firms Each firm’s marginal production cost is constant and identical (MC=c )

No fixed costs All firms set their prices simultaneously We assume that any single firm can satisfy the entire market.Let pm be the profit maximising price assuming a firm acts as a monopoly in the market.Case 1: A firm sets price ¿ pm

This firm can be treated as not being a competitor, and the other firm can behave as a monopoly, setting its price ¿ pm.Case 2: A firm sets price ≤ pm

Both firms compete on prices till p=MC

Week 8Pareto OptimalityAn allocation is good if it is Pareto optimal. A Pareto-efficient allocation is an allocation where there is no way to make some individual better off without making someone else worse off.A Pareto optimal outcome need not be fair or equitable.Pareto Improvement An allocation is Pareto improving if it harms no one and helps at least one person. If an allocation is Pareto optimal, then the only way to make someone better off is to make someone worse off. This is not a Pareto improvement.GamesDefinition and Assumptions By definition, a game must have Players Strategies Outcomes with related Payoffsand has the following assumptions RationalityAll agents playing the game are rational and everyone knows that everyone is rational, utility-maximising. Complete InformationEveryone knows the game and knows everyone else knows the game.Nash Equilibirum A Nash equilibirium is when all players are playing their best response to what the other players will do, and is a situation where no agent has an incentive to unilaterally deviate.

A Nash equilibirum need not be Pareto optimal. Rational pursuit of self-interest can lead to an outcome that is sub-optimal for everyone. Individual rationality may not bring about the social optimum.StrategiesStrictly Dominant and Dominated Strategies A strictly dominant strategy is a strategy that yields the highest payoff to a player no matter what stratgies the other players choose.A strictly dominated strategy is a strategy that is inferior to at least one other strategy across all the strategies of the other players.Weakly Dominant and Dominated Strategies A weakly dominant strategy is a strategy that is at least as good, if not better, across all the strategies of the other players.A weakly dominated strategy is at best, only equally as good as another strategy across all the strategies of the other players.Link between Nash Equilibrium, Stratgies, and Best ResponseAn outcome is a Nash equilibrium if and only if all players are playing their best response. Strictly dominated strategies are never part of a Nash equilibrium, but a weakly dominated strategy may be.Solving Simultaneous GamesBest Reponse Method Find each player’s best strategy in response to all strategies played by another player. The Nash equilibrium occurs whenever both players’ best responses overlap.

Iterative Deletion of Dominated Strategies Compare all strategies of any player across all strategies of another player. Delete strategies that are strictly dominated to narrow down the Nash equilibrium. Do not delete weakly dominated strategies as they can be part of a Nash equilibrium.

Representation of Simultaneous Games

The above is known as the payoff matrix or the normal representation form. The strategies are listed along the rows and columns, while the outcomes or payoffs are {a1 , a2,…,d1, d2 }.Then, the Nash equilibria are written as {Player :Strategy , Player :Strategy ,… }. Note that outcomes are not listed in the Nash equilibria.

Week 9Repeated GamesInfinite Games If a game is repeated infinitely, it is possible to develop a punishment to induce cooperation.For instance, in a collusion, we hae seen the “Grim Trigger Strategy”, which is also known as the “Nash Reversion Strategy”. Both parties collude until one cheats, after which the other party also cheats. The outcome reverts to the Nash equiblirum {Cheat, Cheat}.Finite Games In finite games, there is no possiblity of punishment using “Nash Reversion Strategy”. Since there is a last round, after which there is no punishment for cheating, both parties would cheat in the last round. However, since they are already

cheating in the last round, there is no punishment for cheating in the second last round, and would cheat then as well. By induction, collusion is not sustainable as everyone will cheat all the time.An alternative punishment is “Tit-for-tat”. The strategy is to cooperate in the first round, and for subsequent rounds, copy whatever the other play did in the previous round. Thus, punishment is only temporary, rather than forever.Representation of Sequential Games

As opposed to simultaneous games, sequential games have an order of actions. Instead of the payoff matrix, games are represented in extensive form as shown above. Note that here, the Nash equilibria are written as {Player :Outcome ,Player :Outcome ,…}, as opposed to using strategies when dealing with the normal representation form.Solving Simultaneous GamesBackward Induction Determine the last player’s responses first for all strategies possible, and move up towards the first player’s responses. This is known as the subgame perfect Nash equilibirum.

Entry DeterrenceSuppose an incumbent monopolist faces the threat of entry by another firm, and that the entrant decides first if it wants to enter or not. If the entrat doesn’t enter, it gets a payoff of 0, while the monopolist gets 10. If it enters, the monopolist can choose to fight or accommodate, and the payoffs are as shown.

By Backward Induction, the Nash equilibrium is {Monopolist : Accomodate , Entrat :Enter }

Non-credible Threats Before the game starts, the Monopolist can threaten the entrant that it will fight if the entrant chooses to enter. However this is a non-credible threat.The entrant knows that if it chooses to enter, the incumbent monopolist’s best reponse is to accommodate, as fighting would result in a lower payoff for the monopolist. Hence, it can safely enter the market without having to risk fighting the monopolist and receiving a payoff of −1<0.Mixed Strategy Nash Equilibria (in Sequential Games)Pure Strategy Nash Equilbiria Up till now, all the Nash equilibira considered were Pure Strategy Nash Equilbiria (PSNE). A PSNE is a Nash equilibirum where players select a course

of action with probablity 1.Mixed Strategy Nash Equilbiria A Mixed Strategy Nash Equilbiria (MSNE) is a Nash equilibirum where players select a course of action with probablities p such that 0≤ p<1.In a game with MSNE, players randomise over their strategies, and play each strategy with probablities less than one. Their strategy thus is a mix of the various different strategies. It is still a Nash Equilibrium, and is hence characterised by no incentive for unilateral deviation. An MSNE cannot be a PSNE, and vice-versa. If a question asks to find all the Nash equilibria of a game, find both the

PSNE and MSNE. A game can have any combination of no PSNE or MSNE, one PSNE or MSNE, or more than one PSNE or MSNE.

Example

In the game of “Rock, Paper, Scissors”, there is no PSNE. However, there is a MSNE. Suppose that players play mixed strategies with the following probabilities.Player A plays i. P (Rock )=pii. P (Paper )=qiii. P (Scissors )=1−p−q

Player B playsi. P (Rock )=rii. P (Paper )=siii. P (Scissors )=1−r−s

Hence, if Player B plays Rock, his expected payoff is p (0 )+q (−1 )+ (1−p−q ) (1 )=1−p−2q

if Player B plays Paper, his expected payoff is p (1 )+q (0 )+(1−p−q ) (−1 )=−1+2 p+q

if Player B plays Scissors, his expected payoff is p (−1 )+q (1 )+(1−p−q ) (0 )=−p+q

By symmetry, Player A has similar payoffs in terms of r and s. There is no incentive to unilaterally deviate if the payoffs from all outcomes are the same. Hence,From Player B’s payoffs: 1−p−2q=−1+2 p+q=−p+q⇒ p=q=1

3

From Player A’s payoffs: 1−r−2 s=−1+2 r+s=−r+s⇒r=s=13

Thud, the MSNE where no player has an incentive to deviate to play any given strategy exclusively is when

{p=13 , q=13 , r=13 , s=13 }Week 10

Rubenstein BargainingRubenstein Bargaining entails a specific structure where offers are made sequentially over a set of rounds. If no decision is made after the set number of rounds, then a pre-determined allocation is agreed upon.The player with a higher pre-determined allocation has a greater bargainging power, and is hence able to secure a better allocation than the other player through the backward induction process. The allocation in the event of a failure to reach an agreement is only relevant to the person accepting or rejecting the offer in the final period.Discount Factor If a player has a discount factor 0<α<1, then, $1 in the next period of the game is worth $α in the current period.Assumptions If a player is indifferent between acception the offer and not, they will choose the option preferred by the opponent.How to Solve The idea is that the opponent can give an arbitrarily small amount to the other player to make them accept the option preferred by the opponent, which can be rounded down to 0.

This amount is solved for using backward induction.

Example Suppose there is a bargaining game where 2 players, Player A and Player B, choose a value in an interval [0 , x ], and Player A offers first. Let the allocation in the event of a failure to reach an agreement after n rounds be ( A ,B )=(a ,b ). Futher suppose that Player A accepts/rejects in the final round, that is, n is even. The game is modelled as follows.

In the first round, Player A offers x1, and hence Player B gets x−x1. Player B can choose to either accept the offer (Y ) or reject (N ). If he accepts, the allocation is ( A ,B )=(x1 , x−x1 ). If he rejects, Player B then offers x2, and hence Player A gets x−x2. Player A can choose to either accept or reject, and so on. If the game has been played for n rounds, and they do not agree on an allocation, then it defaults to ( A ,B )=(a ,b ).

Week 11Definition of ExternalityWikipedia: An externality is the cost or benefit that affects a part who did not choose to incur that cost or benefit.Perloff: An agent’s outcome is directly affected by the actions of other agents rather than indirectly through changes in prices.Crucially, an externality impacts somebody who is not a participant in the activity that produces the external cost or benefit. An externally-imposed benefit is a positive externality, and an externally imposed cost is a negative externality.Pareto Optimality Externalities may cause markets to fail and result in Pareto inefficiencies. Too much resources are allocated to an activity which causes a negative

externality Too few resources are allocated to an activity which causes a positive

externalitProduction ExternalitySuppose there are two price-taking firms A and B producing different products. Further suppose that Firm A’s production results in pollution, a negative externality that affects Firm B.Denote the price and costs of Firm A ' s and B' s output as pA , pB and c A , cB respectively, and their output as a and b. Futher denote the amount of pollution by Firm A as x. Then,

Π A (a , x )=pA a−c A (a , x )

ΠB (a , x )=pBb−cB (b ,x )

where c A increases in a and decreases in x wile cB increases in b and x. Note that although the pollution from Firm A enters as a cost to Firm B, while Firm A is able to choose its amount of pollution x, Firm B cannot choose x.

Profit Maximisation Thus, we have the following first order conditions to find the profit maximising conditions.For Firm A

Π A (a , x )∂a

=0⇒ p A=∂c A (a , x )

∂aΠ A (a , x )

∂ x=0⇒

∂c A (a , x )∂ x

=0

For Firm BΠB (b , x )

∂b=0⇒ pB=

∂cB (b , x )∂b

However, economic efficiency is not achieved as total profit is not as high as it can be.Ways to Achieve Economic EfficiencyInternalising the Externality Suppose instead that the firm merge and behave as one, with profit function

Π (a ,b , x )=Π A+Π B=pA a−c A (a , x )+ pBb−cB (b , x )

and first order conditions

Π (a ,b , x )∂a

=0⇒ pA=∂cA (a , x )

∂aΠ (a ,b , x )

∂b=0⇒ pB=

∂cB (b , x )∂b

Π (a ,b , x )∂ x

=0⇒∂c A (a , x )

∂ x+∂cB (b , x )

∂x=0

The merger of the firms causes a fall in the marginal benefit of pollution to the firm as it considers the cost of pollutions on Firm B’s output as well. Hence, the merger internalises the externality, and induces economic efficiency, and increases total profit.Similarly, if the firms impose a positive externality, they can internalise it as well.

Coase Theorem The Coase theorem states that bargaining will lead to an efficient outcome regardless of the initial allocation of property if property rights of an externalist are clearly defined and there are no transaction costs.Suppose that the rights to the object being polluted (water, air, etc.) is assigned to either one of the firms, who can then sell these rights at a price of px x+F , for some fixed fee F. Suppose Firm B has these rights. If F=0, Firm B will not sell any pollution rights to Firm A. Hence, depending on the profit functions, Firm A must offer some F>0 so that it can buy rights to pollute.If instead the property rights were assigned to Firm A, total profits would still be the same, but the division of profits will differ. Any firm would prefer to have the propert right assigned to them for additional revenue.Tragedy of the CommonsModel the Tragedy of the Commons using the example of a grazing cows. Consider a grazing area owned “in common” by all members of a village, where each villager has a cow. When c cows are grazed, total milk production is f ( c ) , f '>0 , f ' '<0. That is, there are diminishing returns to cows grazing. The profit function for the village is

Π (c )=f (c )−pc c

where the price of milk is 1 and the cost of grazing a cow is $ pc. Hence, if the village behaves as a collective, rational individual, then it chooses c=c¿ such that f ' (c¿ )=pc. That is MR=MC.

However, suppose that instead, each individual decides they want graze a cow individually. The private cost is pc and private benefit is f (c )

c, assuming each cow is equally

productive. Notice that, as long as c<c, f (c )c

> pc. That is, the private benefit exceeds cost. Hence, entry continues

until f (c )c

=pc at c= c, which results in overgrazing.Reasons for Tragedy The reason why this tragedy occurs is that when the additional villager puts his cow out to graze, he gains income of f (c )

c− pc, but the income of villagers

already grazing cows falls. This is because f (c )c

is falling as c increases beyond c¿. The villager who adds the extra cow takes no account of the cost inflicted upon the other villagers.Modern-day Tragedies of the Commons Some examples are Over-fishing the high seas Over-logging forests on public

lands Over-intensive use of public

parks Urban traffic congestion

Solving the Problem Assigning property rights could solve the problem. If everything that people care about is owned by someone who can control its use and, in particular, exclude others from overusing it, then there are by definition, no externalities. A merger could solve the problem as well. Other solutions are

i. TaxationIf we tax each individual the amount equal to the decrease in returns he imposes on others, that is, the cost of the externality, then he is forced to take into account the social cost of his actions, internalising it.

ii. Quantity LimitIf we know what the optimal quantity is, we can cap production at that level.

iii. Cap-and-tradeWithout regulation, all firms pollute until pollution no longer reduces their costs. Suppose every firm values pollution differently. Then, the government can assign permits to firms to pollute at a certain level, and firms who don’t value the pollution can trade their permits and make revenue. This way, the government caps pollution without needing to gather information on the various costs of pollution and who abates, which is an expensive process.

Week 12GoodsExcludability and Rivalry Excludable

A good is excludable if it is not impossible to exclude someone from consuming it

RivalA good is rivalrous if one person’s consumption of the good diminishes another person’s consumption of the good.

Types of Goods Excludable Non-excludableRivalrous Private Goods

Examples:Common goodsExamples:

Food, Clothing, Books Fish stocks, Public highways when congestedNon-rivalrous

Club GoodsExamples:Golf courses, Cinemas

Public goodsExamples:National defence, Public highways when not congested

Non-rivalrous can mean indivisible or very large/abundant. National defence is indivisible, and a golf course is very large/abundant.Provision of Public GoodsWith private goods, the free market allocation leads to optimal production and allocation of the good through MRS=¿ price ratio, or MB=MC. However, private provision may lead to under-provision of a public good.This is because people have an incentive to free-ride on the production of the public good by others since public goods are non-excludable. To overcome this market failure, intervention is necessary. For instance, the government can coordinate the provision of the good instead.

Pareto Optimal Allocation of Public Good Suppose there are n individuals with endowments ωi. They can consume public good G with cost c (G ) or private good x i with price px. Then, WLOG, we want to find

maxx i ,G

i=1 ,… ,n

u1 ( x1 ,G ) suchthat (ui (x i ,G )=ui for i=2 ,…,n )∧( px∑i=2

n

x i+c (G )=∑i=2

n

ωi)That is, we want to find an allocation where individual 1 is as well-off as possible given all other n−1 individual’s utility. Hence, the Lagrangian is

L (x1 ,…, xn ,G )=u1 (x1 ,G )−λ [∑i=2n

u i (x i ,G )−ui]−μ [px∑i=2

n

xi+c (G )−∑i=2

n

ωi]Finding the first-order conditions, we have

∂L∂x1

=∂u1 (x1 ,G )

∂ x1−μ px=0

∂L∂x i

=−λ∂ui (xi ,G )

∂ xi−μ px=0 , for i≠1

∂L∂G

=∂u1 (x1 ,G )

∂G−λ∑

i=2

n ∂ui ( xi ,G )∂G

−μ ∂c (G )∂G

=0

Hence, we have1μ∂u1 (x1 ,G )

∂G− λ

μ∑i=2n ∂ui (xi ,G )

∂G−∂c (G )∂G

=0

and since ∂u1 (x1 ,G )

∂ x1=μ px⇒

1μ=px( ∂u1 (x1,G )

∂x1 )−1

−λ∂ui (x i ,G )

∂ x i=μ px⇒

λμ=−px ( ∂u i (x i ,G )

∂ xi )−1

We have

px

∂u1 (x1 ,G )∂G

∂u1 (x1 ,G )∂x1

+ px∑i=2

n∂ui (x i ,G )

∂G∂ui (x i ,G )

∂x i

=∂c (G )∂G

px∑i=1

n∂ui (xi ,G )

∂G∂ui (xi ,G )

∂x i

=∂c (G )∂G

That is,∑i=1

n

MR S i=MC (G )

px

where MR Si is the rate of substitution of x i for G, or how much x i a person is willing to give up in exchange for an additional unit of G, keeping utility constant. If G were a private good, the optimal allocation would be MR Si=MC (G ) , i=1 ,…,n

. That is, each individual consumes such that MB=MC. However, since G is both non-excludable and non-rivalrous, each unit of G can be fully consumed by all individuals. Hence, if there is an increase in the public good by one unit, there is an increase in cost by MC (G ), and also increase in the benefit to all individuals by the marginal benefit each. So, the increase in benefit is the sum of the marginal benefits of all individuals.There will be under-provision if any one individual provides the public good by themselves as they would ignore the benefit the public good brings to every other individual.

Asymmetric InformationInformation in Competitive Market In purely competitive markets, all agents are fully informed about traded commodities and other aspects of the market. Both sides of the market (buyers and sellers) know all information there is to know about the product and each other. In reality, however, there are market where agents don’t have all the information about the product or about other agents.Imperfect Information Markets where one side and/or the other is imperfecty informed are markets with imperfect information. Typically, these are markets where the products or agents have qualities that are difficult to assess.Imperfectly informed markets with one side better informed than the other are markets with aysmmetric information.Adverse SelectionAdverse selection occurs when agents with more or better information in the market exploit the agents with less infromation, leading to inefficient market outcomes. In any market, there are two types of goods – “Lemons” and “Peaches” – which are, respectively, of good and bad quality.Suppose that forLemons Sellers accept: $ Ls

Buyers pay at most: $ LB

Peaches Sellers accept: $ Ps

Buyers pay at most: $ PB

Trade for Lemons will occcur between LS and LB, and similarly, PS and PB for Peaches. Suppose that no buyer can tell a Peach from a Lemon, but know that

the fraction of Peaches is q and that of Lemons is 1−q. Sellers, of course, can. Suppose agents are risk-neutral (EU=U (EM ) ). Then,

EV=q PB+(1−q ) LB

Buying Lemons and Peaches If there are a lot of Peaches (high q) such that EV >PS>LS, all sellers can pretend that they sell Peaches, and negotiate a price between LS and EV . All sellers gain from being in the market, but buyers who get Lemons lose.On the other hand, if q is such that EV <Ls, then no Peach seller will sell their product as they cannot negotiate a price of at least LS since no buyer will pay more than their EV . They will exit the market, and hence, all buyers know that the remaining sellers own Lemons only, and would hence pay at most PB.In this scenario, too many Lemons crowd out the Peaches from the market. Only bad products are adversely selected in the market, and gains-to-trade are reduced since no Peaches are traded. That is, the presence of Lemons inflicts an exernal cost on buyers and Peach owners when there is asymmetric information.Hence, for Lemons to not crowd out the Peaches, we need

EV=q PB+(1−q ) LB≥PS

q (PB−LB )+LB≥ PS

q≥PS−LB

PB−LB

That is, the fraction of Lemons should be at most

1−q≤1−PS−LB

PB−LB=

PB−PS

PB−LB

Adverse Selection with Quality Choice Consider a market with high and low-quality laksa, which is a credence good. That is, its quality cannot be ascertained until it has been purchased. Suppose that the marginal cost of production of high and low-quality laksa are $4 and $3.50 respectively, and the buyer’s valuation are $6 and $3 respectively. Further suppose that laksa is sold at the buyer’s valuation.With perfect information, low-quality laksa is never sold since cost > valuation. However, in the case of imperfect information, assume buyers’ valuation is ther EV . Suppose that every seller decides to only sell high quality laksa. Then, seller’s make a profit of $6−$ 4=$ 2. However, anyone seller has an incentive to deviate and switch to producing low-quality laksa and make an extra $0.50, without having an impact on the proportion of good vs bad sellers. Since all sellers have the same incentive, an equilibrium with only high-quality laka is unsustainable.Let the fraction of high-quality laksa be q. In the case of an equilibirum where both types of laksa are sold, for there to be high quality laksa sold, we needEV ≥ 4. Thus,

EV=6q+3 (1−q )=3 (q+1 )

3 (q+1 )≥4⇒ q≥ 13

However, even in this scenario, every sellers resons that they can switch to low-quality laksa to increase profit by $0.50. Hence, there is also no equilibirum where both types of laksa are sold. Futher, in this scenario, since the cost of producing low-quality laksa exceeds buyers’ valuation of it, low-quality laksa is never sold as well. Hence, there is

nothing sold, and adverse selection has destroyed the entire market.Ways to Reduce Adverse Selection Banning low-quality products Critics such as food bloggers to review food Crowd-sourced aggregators allowing buyers to skim reviews

Week 13SignallingA labour market has two types of workers – high-ability, H , and low-ability, L. A high-ability worker’s marginal product is wH and a low-ability worker’s is wL, where wH>wL.Let the fraction of high-ability workers be 0≤q<1, and that of low-ability be 1−q. Each worker is paid his expected marginal product, given that the labour markert is competitive. If firms know each worker’s type, they pay each worker their marginal product, wH or wL. The difference, wH−wL is known as the wage premium. However, if firms cannot tell workers’ types, then every worker is paid the same wage rate, that is the expected marginal product

wP=qwH+(1−q )wL

This is known as a pooling equilibrium, as firms cannot distinguish between worker types, and thus, pool all types of workers together and pay them the same wage. Clearly, wP<wH. Hence, high-ability workers have an incentive to find a credible signal.

Education Suppose that workers can acquire education, e, to signal their ability. Education costs a high-ability worker cH per unit and a low-ability worker c L per unit. Suppose that c L>cH, and that education has no effect on worker productivity (it only acts as a signaller).For education to be an effective signal, high and low-ability workers must obtain different levels of education, and in fact, high-quality workers must obrain more education.This is known as a separating equilibrium as firms can separate high and low-ability workers. Workers with the same education level are assumed to have the same ability level.

Separating Equilbirum Let eH be the education obtained by high-ability workers. In a separating equilbrium, there are 2 conditions that must be fulfiled.i. wH−cH eH>wL

This condition ensures that the utility obtained by high-quality workers after obtaining education is higher than if there do not obtain an education and earn wL.

ii. wH−cLeH<wL

This condition ensures that low-ability workers do not pass off as high-ability workers by ensuring that the utility they obtain after obtaining the same amount of education as high-quality workers is less than if they do not obtain an education and earn wL.

Solving these 2 equations gives the range of eH that high-ability works can obtain to effectively signal their ability. Hence, eL=0 as any eH>eL>0 would incur cost but bring no benefit.Hence, signalling improves information in the market to the benefit of the high-ability workers. However, it reduces total surplus as cost as to be expended to acquire the signal.Moral HazardAdverse Selection arises because agents cannot determine other agent’s type. Moral Hazard, on the other hand, arises because agents cannot determine other agent’s actions.Suppose a worker is hired by a principal to do a task. Only the worker knows the effort he exerts, which affects the principal’s payoff. Once the worker gets

the job, he has incentive to exhibit moral hazard and slack off.The “principal-agent problem” is the problem if the principal can design an incentive contract that induces a worker to exert the amount of effort that maximises the principal’s payoff, eliminating moral hazard.

Incentive ContractingLet x denote the worker’s effort. The principal’s revenue is the ouput of the worker’s effort, which is y=f ( x ) for some function fof x. An incentive contract is a function s ( y ) specifying the worker’s payment (salary) when he produces an output y. Thus, the principal’s payoff is

Π p= y−s ( y )=f (x )−s ( f ( x ) )

Denote ~u the reservation wage of the worker. That is, the opportunity cost of working for the principal. Denote the worker’s cost of effort c ( x ). Then, the worker’s payoff is

Πw=s ( f ( x ))−c ( x )

To get the worker’s participation, the contract, s ( y ), must offer the worker a utility of at least ~u. This is known as the participation constraint.

s (f ( x ) )−c ( x )≥~u

However, to maximise its utility, the principal will design the contract such that Πw=~u. Thus, the principal’s problem is

maxx

Π p=f ( x )−s ( f ( x ) ) such that s (f ( x ) )−c ( x )=~u

Therefore, we have Π p=f ( x )−s (f ( x ) )=f ( x )−c (x )−~u. At maximim, Π p' =0. That is,

f ' ( x )=c ' (x )

The contract that maximises the principal’s payoff equalises the worker’s maginal effort cost to the principal’s marginal revenue from worker effort. Let x¿ be the amount of effort such that f ' ( x )=c ' (x ).

Thus, to ensure the worker exerts the right amount of effort, x=x¿ must be most preferred by the worker. Hence, the contract s ( y ) must satisfy the incentive-compatibality constraint.

s ( f ( x¿ ) )−c ( x¿ )≥ s ( f (x ) )−c ( x ) ,∀ x ≥0

Examples of Contracts that Satisfy the Incentive-Compatiblity ConstraintRental Contracts The principal keeps a lump-sum R for themselves, and the worker gets all the revenue above R.

s (f ( x ) )=f ( x )−R

Hence, Πw=s ( f ( x ))−c ( x )=f ( x )−R−c (x ). At maximum, Πw=0⇒ f ' ( x )=c ' ( x ).This contract is incentive-compatible. The principal chooses R such that Πw ≥~u.

f ( x¿ )−R−c ( x¿)=~u

R=f ( x¿)−c ( x¿)−~u

Wage Contracts In a wage contract, the payment to the worker is, for some wage per unit w, and lump-sum K ,

s (f ( x ) )=wx+K

Hence, , Πw=s ( f ( x ))−c ( x )=wx+K−c ( x ). At maximum, Πw=0⇒w=c ' ( x ).The contract is incentive compatible if w=f ' ( x ). The principal chooses K such that Πw ≥~u.

f ' ( x¿ ) x+K−c ( x¿ )=~u

K=~u− f ' ( x¿ ) x¿+c ( x¿ )

Take it or Leave it Contracts Choose x=x¿ and be paid a lump-sum L, or choose x≠ x¿ and be paid zero. The worker’s payoff, Πw, from choosing x=x¿ is L−c ( x¿) and x≠ x¿ is −c ( x ). Hence, the worker chooses x=x¿.The contract is incentive compatible. The principal chooses L such that Πw ≥~u.

L−c (x¿)=~u

L=~u+c (x¿)

Examples of Contracts that Do Not Satisfy the Incentive-Compatiblity ConstraintFixed Fee The principal pays the worker a fixed fee that is not a function of his effort L. That is,

s (f ( x ) )=L

Hence, Πw=s ( f ( x ))−c ( x )=L−c ( x ). At maximum, Πw=0⇒−c' ( x )=0. The contract is not incentive compatible.Revenue-sharing Contracts The worker receives a share of the revenue

s (f ( x ) )=αf ( x ) ,0<α<1

Hence, Πw=s ( f ( x ))−c ( x )=αf ( x )−c ( x ). At maximum, Πw=0⇒ α f ' ( x )=c ' ( x ). Since α<1

, x≠ x¿, and the contract is not incentive compatible.Incentive Contracts with UncertaintySuppose that now, y=f ( x )+ϵ instad, where ϵ represents uncertain events that affect the production process beyond the control of the worker. Then, effort does not necessarily translate into output.Rental Contracts Since the principal receives a fixed fee, the worker bears all the risk of the noise. If the worker is more risk averse than the owner, the worker will exert effort x<x¿. Then, a rental contract may no longer be incentive compatible.Revenue-sharing Contracts Since the worker doesn’t bear all the risk of the noise, he is more willing to share some of the profits from the production since this offloads some of the

risk to the principal. In such a case, a revenue-sharing contract can be incentive compatible.