Advanced Micro Revision

Lecture 1: General Equilibrium and Exchange Economy

Budget Constraint

Xpx + Ypy < M

Utility Function

UA (XAE YAE) for Anna, UB (XBE YBE) for Ben

Pareto Efficient Allocations

ICs are tangent MRSA = MRSB = - PX / PY

First Welfare Theorem

Competitive market mechanism results in Pareto Efficient allocations in all markets Requires information on individual preferences, individual endowment and relative prices of X and Y

Walrasian Equilibrium Condition

Aggregate excess demand function is continuous and therefore small changes in the prices of goods only causes small changes in the demand for goods Convex Preferences OR Consumers are small relative to the size of the market, therefore they have no market power and are price takers

Lecture 2: Calculus for General Equilibrium

Check if Allocation is Pareto Efficient

1. Find formula for MRSA = MRSB 2. Sub. (XA, YA) and (XB, YB) into the formula 3. If MRSA = MRSB therefore Pareto Efficient

Example

MRS = - Mux / MUy If U(X, Y) is the utility function therefore: MUx = dU / dX & MUy = dU / dY Utility is an ordinal concept so any monotonic transformation preserves the original function

Finding the Contract Curve

1. Find MRSA and MRSB and set them equal to each other 2. Use: XA + XB = X and YA + YB = Y to eliminate XB and YB 3. Solves for YA as a function of XA 4. Sketch in Edgeworth Box

Example 1

Q: U (XA, YA) = XA.YA and U (XB, YB) = XB.YB, there are 10 units of X and 10 units of Y in the economy

1. Along the contract curve: MRSA = - YA/XA = MRSB = - YB / XB

2. YA / XA = 10 YA / 10 XA because (YB = 10 YA etc)

3. Therefore YA = XA

4. Diagonal

Example 2

Q: Anna: U (XA, YA) = XA2 + YA and Ben: U (XB, YB) = XB2 + YB, there are 20 units of X and 20 units of Y in the economy

1. MRSA = -2XA = MRSB = -2XB

2. XA = XB therefore XA = 20 XA therefore 2XA = 20 therefore XA = 10

Lecture 3: Calculus for General Equilibrium

Finding the CGE

1. Write down the budget constraint for each consumer 2. Write down each consumers constrained optimisation problem 3. Use lagrangian optimisation to find each consumers demand curves for goods X and Y 4. Write down the market clearing conditions and solve for relative prices (Demand = Supply) 5. Find the CGE allocation

Q: Anna U (XA, YA) = XA1/3 YA2/3 and Ben U (XB, YB) = XB1/3YB2/3. The economy has 30 units of X and 30Y. The initial endowment (E) is: XA = 30, YA = 0 and XB = 0, YB = 30. Fix PY to 1, which just leaves PX, which will be written as P (PY = 1 is the numeraire rule).

1. XAPx + YAPy < 30Px and therefore XAP + YA < 30P XBPx + YBpy < 30 Py and therefore XBP + YB < 30

2. Max. U (XA, YA) = XA1/3 YA2/3 subject to XAP + YA < 30p Max. U (XB, YB) = XB1/3YB2/3 subject to XBP + YB < 30

3. Max. LA = XA1/3 YA2/3 - (XAP + YA 30P)

1. dLA / dXA = 1/3 XA 2/3 YA2/3 P = 0 2. dLA / dYA = 2/3 XA1/3 YA-1/3 = 0 3. dLA / d = - (XAP + YA 30P) = 0

Eliminate by dividing 1 by 2: YA/2XA = P therefore YA = 2XAP Sub into (3): 3XAP = 30P therefore XA = 10, YA = 20P

Max. LB = XB1/3 YB2/3 - P = 0

1. dLB / dXB = 1/3 XB 2/3 YB2/3 P = 0 2. dLB / dYB = 2/3 XB1/3 YB-1/3 = 0 3. dLB / d = - (XBP + YB 30P) = 0 Therefore YB / 2XB = P and YB = 2XBP

4. XA + XB = 30 therefore 10 + 10/P = 30 therefore 10P + 10 = 30P therefore 20P = 10, therefore Px = 0.5 and Py = 1.

5. Sub P = 0.5 back in: XA = 10, YA = 20P = 10 and XB = 10/P = 20, YB = 20

Lecture 4: Monopoly Power and Externalities

Monopoly In the Edgeworth Box

1. Monopolist has market power and sets relative prices, Anna dictates Px/Py to Ben 2. Anna traces out all of Bens optimal consumption choices at different prices, his Price Offer Curve 3. Anna chooses prices that maximise her Utility subject to Bens P. Offer Curve. Not Pareto efficient because ICs are not tangential. Anna sells less than the CGE and at a higher price, UA > UB.

Perfectly Price Discriminating Monopolist

1. Anna sells each unit of Y at a different price, a price at which Ben is indifferent between buying and not buying. 2. Anna extracts Bens entire consumer surplus. Pareto Efficient because ICs are tangential. 3. For Anna: PPDM > GME > GCE

Externalities

1. Distort economy because these are typically not sold in a market. 2. Level of externality depends on initial endowment and bargaining process. 3. Solution depends on preferences and legal situation. 4. Solution: Coase Theorem and Creating Markets for Externalities.

Importance of Property Rights

1. An efficient point can be reached if property rights are clearly defined and there are established mechanisms for negotiation. 2. But: Often not clearly defined, social norms prevent private bargaining. 3. Imperfect information about source of externality. 4. Coordination failure, bargaining is costly in money and time, free riding.

Coase Theorem

1. If preferences are quasi-linear and property rights are clearly defined, then bargaining leads to the same efficient outcome, irrespective of how property rights are distributed.

Lecture 5: Production Economies

Robinson Crusoe Economy Dual role both a consumer and a producer. Must allocate his time (T) between leisure (X) and labour (Y). Trade off between X and Y. Robinsons choice determined by Technology (production function for Y) and his Preferences.

Technology Diminishing returns to labour Upper limit on Robinsons labour due to time constraint Time spent in leisure = T Y MRS = Marginal Product of Labour (MPL) Pareto Efficiency: MRS = MPL = PE Further along production function, the more shallow it is, MU of labour low and therefore pays to work less (MRS > MPL) At start of production function, steep curve, MU of labour high and therefore pays to work (MPL > MRS).

Production Possibilities Frontier All feasible combinations of X and Y MRS = MRT (Rate at which you can transform one good into another)

Example: Finding the PPF Preferences: U (X, Y) = lnX + 2lnY Technology: If he devotes t hours to gathering coconuts: Y = t1/2 1. Write down Robinsons time constraint ant sub in technology formula X + t = 72 ( t = working hours)

2. Sub in technology formula to eliminate t: Y = t1/2 Y2 = t X + Y2 = 72 Y = (72 X)1/2 (PPF equation) (Therefore min.Y when X = 72, maxY when X = 0)

3. Finding the MRT for Y = (72 X)1/2 MRT = dY/dX = -1/2 (72 X)-1/2

Finding the MRS: U (X, Y) = lnX + 2lnY MRS = - Mux / MuY = (1/x) / (2/Y) = - Y / 2x

Does MRT = MRS? MRS = - Y / 2x = - 1/12 MRT = - 1/12 MRS = MRT = Optimal, Robinson works 36 hours.

Finding the MPL Find MPL Find MRS between labour and coconuts: MRS = - MUt / MUy Equate

Firms = pC wL (Profit = price*coconuts wage*labour) Rearrange for C: C = / p + w/p * L (Coconuts = profit/price + real wage*labour) Isoprofit line gives all combinations of C and L given profit (Gradient = w/p) Isoprofit line = Budget line of individual

Production of Two Goods: Fish and Coconuts MRS = MRT = - PF / PC Equilibrium production point pins down the dimension of Edgeworth box (Optimal when IC is tangent to PPF)

Lecture 6: Comparative Advantage and International Trade

Comparative Advantage Difference in state of technology means different production function Time constraint: tF + tC = 10

Robinson Tech: F = 5tF, C = 10tC, tF + tC = 10 Robinsons Production Set: F/5 + C/10 = 10 (x10): 2F + C = 100 (solve for C): C = 100 2F this is the PPF equation MRT = Opportunity cost = -2

Friday Tech: F = 10tF, C = 5tC Fridays Production Set (Foregin): F/10 + C/5 = 10 (x10): 2C + F = 100 (solve for c): C = 50 -1/2 F MRT = -1/2 (To get 2F, only sacrifice 1 C)

Economy Wide (Joint) PPF Most efficient allocation of resources. Kinked PPF. Adding a third worker makes 2 kinks. Large number of workers gives the standard, concave PPF.

International Trade Rapid growth post WW2, barriers to trade came down (GATT/WTO), transport, information better and cheaper. Protectionism still exists, anti-trade lobby, DOHA stalled Debate: Growth booster or restrictor (Depends on country, institutions)

International Trade Key Questions What channels? Winners and losers? Traditional explanations focus on C.A

Ricardian Model: Technological differences (Labour only) Closed economy: Different autarky (production = consumption) relative prices reflects different opportunity costs and thus C.A leads to gains from trade Open economy: World production = world consumption. Under free trade there is a common world relative price that clears the market. Free trade allows consumption outside PPF, higher social utility. Complete specialisation according to CA means consumers can reach a higher IC.

Hecksher-Ohlin Model: Factor abundances (Labour and capital) (2 * 2 * 2) Two Sectors: i = (X, Y) Two Countries: j = (H, F) Two Factors: L and K Factor Prices: w and r Assumptions: (Factors mobile across sectors, immobile across countries) (Identical standard preferences across countries) (Perfect competition in goods and factor markets) (Focus on incomplete specialisation case, both goods still produced under free trade) (Tech identical across countries, constant returns to scale) Let Home be capital abundant: K / L > K* / L* Good 2 is relatively capital intensive: KY / LY > KX / LX for all w/r A country will have a C.A and a lower relative autarky price for the good which uses relatively intensively its relatively abundant factor C.A where factors are abundant Free trade relative prices lie between autarkic relative prices of Home and Foreign: A(Px / PY)F < FT(Px / PY) < A(Px / PY)H

Implications of HO Model Each factor exports factor abundant C.A good Rybczynski Theorem: For given prices, an increase in factor endowment, increases production of the good which uses that factor intensively and decreases production of other factor Factor Price Equalisation Theorem: Assuming incomplete specialisation, free trade equalises relative, nominal and real factor prices (As if youre trading the labour used in a good when you export a good) (Trade as a substitute for migration)

Stopler Samuelson Theorem An increase in the relative price of a good (e.g. through Trade Liberalisation) raises the real return of the factor used intensively in the production of that good and reduces the real return to the other factor Winners and Losers

Home: P of Y ^ Nominal real wage and relative return to capital ^ Nominal, real and relative return to labour falls r ^ and w falls

Foreign: P of X ^ Nominal, real and relative return to labour ^ Nominal, real and relative return to capital falls r falls and w ^

Mechanism: Let Py = 1, Px rises with trade and therefore Py falls. X expands and absorbs more resources (opp for Y) Excess demand for labour relative to capital drives up w relative to r Each industry substitutes K for L and uses less labour intensive techniques, therefore increasing the MPL) MPL rises and so does the real wage (MPL = real wage = w/p)

Policy Implications of SS Theorem Pareto Criterion: Change makes 1 person better, no one worse off SS doesnt satisfy Pareto Criterion. However, overall welfare higher so a Pareto improvement. Overall welfare higher, winners could compensate losers Responses to increased inequality: Broaden ownership of high-priced factors Responses to real wage declines: Saftey nets, adjustment assistance

Lecture 7: Social Choice and Welfare

FWT, SWT are both related to PE but need to consider equity, important for policy decisions

Social Change Rule

Lecture 8: Auctions

Practical challenges to the design and implementation (Klemperer 02)

1. Mechanism design problem: To choose which auction design will produce the most surplus. Asymmetric info problem makes it impossible to achieve PE and PM. Collusion between bidders, Too few bidders, Inappropriate reserve price, Loopholes, Credibility of the rules, Market Structure.2. Solution: Robust rules, Anonymous bidding, Sealed bids, Anti-trust

Two Objectives

1. Pareto Efficiency An auction that results in the good being received by the bidder with the highest value. Example: 2 bidders with v1 > v2 > s = 02. If person 2 receives the good for (max) price p = p2, then both can gain if person 2 sells the good to person 1 for any price between v2 and v1 (Gains from trade not exhausted)3. Profit Maximisation An auction that yields the highest expected profit to the seller. Price equals the highest valuation.

English Auction Example

1. With pR = s, guarantees PE as the good goes to the person with the highest valuation. The winner pays the value of the second highest bidder + the minimal bid increment. Four cases: (10, 10) (10, 100) (100, 10) (100, 100):

With pR = 100. E () = (0 + 100 + 100 + 100) = 75. Therefore the profit maximising reservation price is 100. Due to incomplete information, in the first scenario the item doesnt sell. Therefore not completely Pareto Efficient, gains from exchange arent exhausted. Trade off between PM and PE.

Dutch Auction Example

1. No guarantee the good will be awarded to the person with the highest valuation, in general a Dutch Auction is not PE. Similarly, in a sealed bid auction, the optimal bid depends on the bidders beliefs about the values of other bidders > Not PE.

Vickrey Auction Example Suppose pR = 0, If everyone bids their true value, the winner will be the person with the highest valuation therefore PE. Winner pays second highest bid and therefore not PM.1. Two bidders: Values v1 and v2, write down bids b1 and b22. Expected payoff of bidder 1 is: Pr(b1 > b2)[v1-b2] B1 > b2: Probability of 1 winning auction V1 b2: Difference between value and the second highest bid Suppose v1 > b2. By setting b1 = v1, Pr (b1 > b2) = 1 Suppose v1 < b2. By setting b1 = v1, Pr (b1 > b2) = 0 Ensures bidders bid their true value, PE but not PM.

Lecture 9: Risk and Expected Utility 1

Risks and Expected Utility We always face risky alternatives, lottery, insurance, whether to adopt new technologies, whether to join a trade union. We need a theory to understand how agents choose between lotteries. > Expected utility theory (EU).

Expected Utility Theory A lottery is a random variable with a probability distribution Many different outcomes, n possibilities l = (c1, c2 cn ; 1, 2 n) c = consumption, = probability

Example 1 Flip a coin; Heads you win 10, Tails you lose 1 l = (10, -1; , )

Example 2 You have 100, and lottery tickets are 1. There is a one in thousand chance of winning, in which case you win 1000. l = (99, 1099; 999/1000, 1/1000)

Example 3 You have 100 and can take one (or none) of these lotteries. Which do you prefer? L1 = (21, -19; ) L2 = (69, -36; , ) L3 = (125, -84; , ) Expected Payoff: E(L1) = (21) + (-19) = 1 E(L2) = (69) + (-36) = 16.5 E(L3) = (125) + (-84) = 20.5 People care about the spread of outcomes, more concerned with downside risk than the upside. Depends on your risk preference and initial level of wealth

Expected Utility Individual will optimise by choosing the lottery that maximises his expected utility Suppose David has an initial wealth of 100 and his utility function is U(M) = M. How would he rank l1, l2, l3? EU(l1) = 121 + 81 = 11/2 + 9/2 = 10 EU(l2) = 169 + 64 = 13/2 + 8/2 = 10.5 (Therefore David chooses 2) EU(l3) = 225 + 16 = 15/2 + 4/2 = 9.5 None: U(100) = 100 = 10

Atttitudes to Risk Risk averse: Concave e.g. U(W) = W Risk loving: Convex e.g. U(W) = W2 Risk neutral: Straight line e.g. U(W) = 5W (E.g. Firms that diversify risks)

Example 4 With lottery 2, your wealth will be 169 with probability or 64 with probability , giving an expected payoff of 116.50. If I offer to swap this uncertain income with a certain amount, what is the minimum amount you would be willing to accept? A) < 116.5 B) Exactly 116.6 C) >116.5 A) Risk averse B) Risk neutral C) Risk loving

Certainty Equivalent CE: A certain amount of money that is equally preferred to the lottery. (E.g. decisions on whether or not to sell a business) Risk Averse: CE(l) < E(l) Risk Loving: CE(l) > E(l), must be compensated because you enjoy risk. Risk Neutral: CE(l) = E(l)

Risk Premium E(l) CE > o Amount willing to sacrifice for certainty, e.g. Home Insurance Premium

Measuring Risk Aversion (Risk Averse Diagram) Curvature of the utility function is related to risk aversion Arrow-Pratt coefficient of absolute risk aversion: ARA = - U(W) / U(W) Coefficient of relative risk aversion: RRA = -W [U(W) / U(W)]

Example U(W) = W U(W) = W = W1/2 U(W) = W-1/2 U(W) = -1/4 W -3/2

ARA = - U(W) / U(W) = -1/4W-3/2 / W1/2 = 1 / 2W RRA = RRA = -W [U(W) / U(W)] = (1 / 2W * W)

Risk Loving Diagram (Risk Premium E (l) CE < 0), Compensation to remove riskRisk Neutral Diagram (Straight linear line) (CE = E(l))

Comparing Lotteries Level of returns: Higher or lower Dispersion of returns: Less or more risky First order stochastically dominates: l1 (10, 20 ,30; 1/3, 1/3, 1/3) l2 (10, 20, 30; 1/8, 2/8, 5/8) Mean preserving spread l1 (50, 150; 1/2, 1/2) l2 (20, 180; 1/2, 1/2) Risk averse utility maximiser prefers l1

Lecture 10: Mitigating Risk

State-Contingent Income Space State of nature 1 (x1) and State of nature 2 (x2) Diagonal line (x1 = x2) is the line of certain income Level of expected utility is constant along the IC EU = 1u(x1) + 2u(x2) Probabilities are captured by the gradient of the IC MRS = - [(1) du(x1) / dx1 / (2) du(x2) / dx2] (Probaility 1 * marginal utility of x1 / Probaility 2 * marginal utility of x2) Along diagonal line: MRS = - (1 / 2) when x1 = x2

All convex to the origin ICs are risk averse people, concave for loving Risk averse people dislike risk and are willing to sacrifice money to acquire certainty Expected payoff of l1: E(l1) = 2x1 + 2x2 Hence: x2 = [E(l1) / 2] [/ 2]x1 Difference between CE and E(l1) is the risk premium

Risk Sharing (Diagram) David, risk averse, initial wealth 100 and utility function: u(x) = X David can invest 100 into a risky firm which yields profits: l = (225, 16; , ) EU = 1/2225 + 1/216 = 9.5 < 100 = 10 EU of wealth > EU of investment

Suppose David shares the project with Michael, who also has utility function: u(x) = x and wealth 100 Each invest 50 and share the profits Project yields (112.5, 8; , ) for each and still have 50 additional each EU = 112.5 + 50 + 8 + 50 = 10.18 > 100 = 10 David and Michael profit from investing risky project as partners

Risk Pooling (Probability Tree) Suppose there are n risk-averse agents, each can invest in an independent risk Suppose no agent is willing to invest in their risky project independently So they pool the risk, invest in all n projects and share the profits equally Essentially each agent invests in the average project, diversifying risk E.g. Invest in a portfolio of stocks of different firms

David and Michael each have initial wealth 100 and utility function u(x) = x Each have independent projects l = (225, 16; , ) Risk pool and invest 200 in both projects and share the profits Reduces risk and makes overall investment profitable for both Creates a moderate profit in between the two extreme outcomes

Un-pooled risk: Pay 100 get l = (225, 16; , ) Pooled risk: Pay 100, get lP = (225, 120.5, 16; , , ) EU = 225 + 120.5 + 16 = 10.2 > 100 = 10

Insurance Involves swapping part of a random income for a sure income A risk averse consumer values the random income being sacrificed at less than its expected value CE < E (l), certainty equivalent < expected payoff Risk neutral insurance company would value the random income being offered at close to its expected value, hence gains from trade being exploited

Suppose Julie a risk averse agent has a wealth w With probability , she suffers a loss L l = (w, w L; 1 , ) Julie can buy insurance, pays a premium, receives a payout in the event of loss

Let p be the premium for every unit of insurance purchased, X If things go well: x1 = w pX If things go bad: x2 = w L pX + X Julie chooses how much insurance to buy at price p

Full insurance: X = L If things go well: x1 = w pL If things go bad: x2 = w L pL + L = w pL If X < L she is partly insured

If p = of bad event, then optimal to fully insure (competitive insurance market; fair premium) (Lowest possible price insurance can be) If p > then part insurance (monopolistic insurance market) Profit margin is the gap between price and Diagram

Lecture 11: Violations of Expected Utility Theory (Powerpoint)

Lecture 12: Asymmetric Information 1 (Topic for rest of course)

Asymmetric Information So far assumed that all parties to a transaction have symmetric info Frequently information is asymmetric Unobserved Characteristics - Buying a second hand car, knowing whether an employees abilities Unobserved Action - Knowing how much effort a worker puts into their work - Knowing how much care an insured person takes to prevent a loss With symmetric information, markets tend to be efficient. With asymmetric info there is market failure with sub-optimal outcomes

Problems Arising From Asymmetric Information 1. Adverse Selection: Hidden information (Lectures 12-14) 2. Moral Hazard: Hidden Action (Lecture 15)

George Akerlof He demonstrated in a market where sellers have more information than buyers about quality can contract into an adverse selection of low-quality products Borrowers and lenders: Skyrocketing borrowing rates on 3rd world markets

Michael Spence The better informed take costly actions in an attempt to improve their market outcome by credibly transmitting information to the poorly informed (Signals) Education as a productivity signal in job markets

Joseph Stiglitz Screening applicants. Performed by insurance companies dividing customers into risk classes by offering a menu of contracts where higher deductibles can be exchanged for significantly lower premiums

Market for Lemons Suppose there are 1000 good cars (plums) 1000 bad (lemons) Assume very large number of purchasers for both (> 2000) Assume buyers and sellers are risk-neutral Buyers: - Maximum buyer is willing to pay for a lemon is 1000 - Maximum buyer is willing to pay for a plum is 2000 Sellers: - Minimum the seller is prepared to accept for a lemon is 800 - Minimum the seller is prepared to accept for a plum is 1600

Case A: Full information (Diagram) - Buyers know whether a lemon or plum - Buyers valuation of 1000 > 800, lemons will sell - Price at which they sell depends on overall supply and demand - Number of buyers > sellers, the lemon sells at 1000 - Buyers valuation of 2000 > 1600, plums will sell - Number of buyers > sellers, the plums will be bid up to sell at 2000 - Outcome is efficient; those who value most buy the car. No CS. PS of 200 for lemons, 400 for plums.

Case B: Imperfect but Symmetric Information (Diagram) - Buyers and sellers do not know whether the car is a lemon or plum - Expected value for Buyer: EVB = (1000) + (2000) = 1500 - Expected value for Seller: EVS = (800) + (1600) = 1200 - EVS < EVB so will sell - Number of cars < Number of buyers so common price bid up to 1500 - Outcome is efficient; those who value most buy the car. CS for plum buyers = 500. CS for lemon buyers = -500. PS perceived to be 300 by sellers.

Case C: Asymmetric Information - Buyers dont know whether a plum or lemon, sellers do - EVB = (1000) + (2000) = 1500 - Seller values lemons at 800 and plums at 1600 - Buyers willing to pay up to 1500 for a car, 1500 > 800 so lemons sell. - Number of buyers > Number of lemon, so price bid up to 1500 - 1500 < 1600 so plums dont sell - Outcome is inefficient: The better quality cars are not sold even though buyers value plums more than sellers do; surplus of 700 per sale for lemon sellers; CS = - 500 < 0

Case D: Asymmetric Information with Warranty - Buyers do not know, sellers do - Sellers offer a warranty, if the car proves to be low quality, seller gives 400 - EVB = (1400) + (2000) = 17000 - Sellers values lemons at 800 and plums at 1600 - Buyers willing to pay up to 1700 for a car which means plums and lemons both sell - Outcome is efficient: Plums and lemons end up with those who value them most; but CS = 300 > 0 for plum buyers, CS = - 300< 0 for lemon buyers. Lemon sellers PS = 500, Plum PS = 100 per sale

Solutions to Adverse Selection in the Second Hand Car Market Warranties shift the payoff structure to the expected value of buyers rises above plum sellers reservation prices - Can serve as a signal of quality (in practice, duration) Buying the second hand car from an official dealer - Signalling quality by manufacture Paying for a car inspection by an independent expert - Screening Legal protection - Minimum quality standards, certification. Direct regulation. These solutions restore efficiency so plums are sold, but they are costly - Full information still remains the best

Adverse Selection in the Labour Market Firms have less information about applicants abilities - Suppose there are high productivity (H) and low (L) people - In a competitive market with full information, wH > wL - If firms cannot distinguish between types, they offer a single wage (w) reflecting expected productivity - H paid less and L paid more than under full information - If w falls below Hs reservation wage, H drops out of labour market (self employment) Means L type may remain in labour market only. Solutions: Education as a costly signal, Screening in hiring

Adverse Selection in the Insurance Market - Insurance firms may have less information about the risk status than those buying insurance - Suppose population comprises of healthy (low risk) and sick (high risk) individuals - Under full information: An insurance would offer healthy people a lower premium than sick people for insurance against death, reflecting probabilities - Under asymmetric information: Cannot distinguish, therefore charges an average premium based on number of people - Disproportionately large number of sick people take out policy, therefore company doesnt break even - Solutions: Health screening, Universal Coverage (mandatory car insurance)

Adverse Selection in the Credit Market Banks offer loans to people with (low risk) and (high risk) Full information: Banks will charge a higher interest rate to higher risk group Asymmetric Information: Banks charge an average interest rate based on proportions of high and low risk individuals Bank might make a loss if disproportionately large number of high risk individuals/firms take out loans Explains skyrocketing borrowing rates in 3rd world

Lecture 13: Asymmetric Information 2: Spence Signalling Model

Spence Signalling Workers Consider a job market with two types of job applicants High productivity workers (H): MPH = 2 (exogenously given) Low productivity workers (L): MPL = 1 (exogenously given) Let q denote the proportion of type L and 1 q denote the proportion of H Assume employers are risk neutral Applicants can choose the level of education to attain (y) CH (y) < CL (y) for all education levels (for the same level of effort) Education is assumed to be unproductive, does not improve productivity

Case A: Full Information First Best Employer can costlessly discern productivity levels In competitive labour markets workers receive a real wage, w, equal to their marginal product: wH = MPH = 2, wL = MPL = 1 Wages are pinned down by known exogenous productivity levels, employer breaks even In this case, education is unproductive, so cannot improve MP and thus real wage It is thus optimal for yH = yL = 0, no education costs involved

Case B: Asymmetric Information No available credible signal Employers cant discern productivity level, only proportion of type L (q) No way for applicants to signal productivity (no education) Employers offer all applicants the same wage: Expected Productivity: - w = MPLq + MPH (1-q) = 1q + 2(1-q) = 2 q As q approaches 0, w approaches 2

If common wage is w = 2 q - For example if half the workers are L and half H: Then q = and w = 1.5 Will the workers accept this wage? - Low productivity workers receive a wage above MPL in this case - High productivity workers receive a wage below MPL in this case

If their reservation wage is higher than w = 2 q, they will not accept Unlikely that low productivity workers can do better High productivity workers are underpaid relative to their MP, if they have the opportunity to be self employed they will reap the full value of their productivity High quality workers may drop out of the market, adverse selection problem On average firm breaks even Assume they stay in the market for what follows

Case C: Asymmetric Information with Signalling Suppose applicants can acquire education level (y*) Suppose an employer believes an applicant with (y < y*) to be an L. With (y > y*) to be an H with probability 1 The cost of acquiring education varies between L and H types allows education to operate as a signal - i) Separating equilibrium: Where only H type invest in education y* thereby signalling their status - ii) Pooling equilibrium: All applicants invest in education level y* - iii) Pooling equilibrium 2: No applicants invest in education level y*

Separating Equilibrium Consider: CH(y) = y/2, CL(y) = y (Half the level of effort for H) Suppose education level y* is such that only high productivity types signal: (yH = y* and yL = 0) Employer can therefore distinguish and the real wage = marginal product When is this equilibrium? L applicants must prefer not to signal: Implies 1 > 2 y*, therefore y* > 1 WL = 1 < 2 q, therefore applicants are worse off than no signal scenario (B) H applicants must prefer to signal: Implies 2 y* / 2 > 1, therefore y* < 2 If type H applicants were to deviated, they would be perceived to be L and receive a wage equal to 1 Combining the two constraints, implies a separating equilibrium: 1 < y* < 2 Type H may or may not be better off relative to (B): 2 y* / 2 > 2 q

Key Results: - i) Applicant types are correctly identifies, thus the signal is informative and confirmed, workers receive their marginal product - ii) Separating equilibrium is not unique, changes in y* will not affect the outcome

Pooling Equilibrium (Wolf and Sheep) Both L and H send education signal Suppose education level y* is low enough that all applications signal Employer cannot distinguish between types, once again the wage received by all is w = 2 q (from which they deduct their education cost) When is this an equilibrium? - Any L type who chooses not to signal will get a real wage of 1 - Provided cost of y* is sufficiently low (effort), both will signal, even though not confirmed - For type L to prefer not to deviate: 2 q y* > 1 - For type H to prefer not to deviate: 2 q (y* / 2) > 1

Key Results: - i) Education signal is uninformative and hence doesnt improve information - ii) If all applicants are signalling this scenario Pareto dominated by the non-signalling outcome due to the costs of the signal. Costly uninformative signalling is socially undesirable - iii) Again, pooling equilibrium is not unique

Pooling Equilibrium 2 Neither L or H send an education signal Arises if y* is sufficiently high (seminar 10)

Welfare Outcomes Case A: Full information First Best: - wH = MPH = 2; wL = MPL = 1

Case B: Asymmetric Info: - wL = wH = 2 q - Cannot be ranked with A because L workers gain, H workers lose

Case C: Asymmetric Info with Signal: - i) Separating equilibrium - wH = MPH = 2; wL = MPL = 1. - But H types also pay CH(y). Second best relative to A, cannot be ranked relative to B.

- ii) Pooling equilibrium (all signal) - wH = w L = 2 q - H types pay CH(y) and L types pay CL(y) - Socially sub-optimal relative to (B) because of waste of expenditure

- iii) Pooling equilibrium (none signal) - Sets wH = wL = 2 q - Same outcome as B

Evidence of Signalling Extreme form of Spence model: Education no impact on productivity, only serves as a signal Unrealistic: More education likely to improve productivity Disentangle: Hard to disentangle the extent to which raises productivity Evidence: Part of returns due to signalling, part due to acquisition of skills

Sheepskin Effect - Increase in wage associated with obtaining a higher credential - Graduating university showed 5-6 times increase with high school credential - Empirical test between drop-outs and completers

- Discontinuous jump in the returns to education in years in which particular credential are earned - Hungerford and Solon 1987: Returns to 16th year 3 times higher than 15th What are credentials signalling? -Weiss 1988: Small effect of extra year of schooling in factor work. But high school graduates have far lower quit rates and absentee rates than non-graduates. - Hence: High school graduates may receive higher wages because they signal they will stay longer with the firm, fewer absences, raises productivity

Lecture 14: Insurance with Asymmetric Information

Insurance Markets Risk is costly to bear for risk averse agents in utility terms Mitigating Risk: Rish sharing, Risk pooling, Risk transferring (insurance). Risk is traded between parties w Strong case for insurance as a means of improving social welfare In a competitive insurance market where p = , people fully insure

Not Always Full Insurance: - Markets where not everyone is insured e.g. Health - Incomplete insurance where insurance exists (deductibles, caps on coverage, strict rules, denying coverage) - Non-existent insurance markets for some risks e.g. low earnings, bad choices

- i) Credit Constraints: People cannot afford insurance (e.g. buying health insurance which can be very expensive if left late) - ii) Non Diversifiable Risk: Cannot be insured as all face loss simultaneously (e.g. planet explodes, mass diseases, natural disasters) - iii) Adverse Selection: Individuals private information about their own riskiness causes insurers not to want to sell policies to people who want to buy (gains from trade not exhausted) (disproportionate risky customers) - iv) Moral Hazard (Hidden Action): Once insured, people take riskier actions than they otherwise would not. Makes policies prohibitively costly.

Adverse Selection in the Insurance Market Focus on Michael Rothschild and Josept Stiglitz (1976) Formal analysis on Asymmetric Information in insurance market (State Contingent Income State diagram) (Optimal Choice of Insurance Diagram) Fair odds line when premium = of bad thing happening

Michael Rothschild and Josept Stiglitz (1976)

Economic Environment: - Market where all potential buyers of insurance face two states of the world - i) No accident occurs: wealth is w - ii) An accident occurs: wealth is w L Two types of individuals - i) High risk: Probability of loss is H - ii) Low risk: Probability of loss is L (Where H > L) (Let denote the proportion that is high risk) Two types: High and Low Risk Diagram Example: H = 2/3 and L = 1/3

Therefore gradient of Hs IC across the 45 degree line is 1 - H / H = 1 2/3 / 2/3 = - 1/2 Gradient of Ls IC across the 45 degree line is 1 L / L = 1 1/3 / 1/3 = -2

Suppose thee is a large number of insurance companies, competitive market Insurance companies offer insurance contracts (a1, a2) - a1 is the premium payment and a2 is the net payout in the event of an accient (net of premium) - i) No accident occurs: Wealth: w a1 - ii) An accident occurs: Wealth: w L + a2

An insurance company will sell a policy if profits are not negative Perfect competition between insurance firms will ensure profits are 0 in equilibrium Rothschild-Stiglitz define an equilibrium as: -i) No insurance contracts make negative profits - ii) There are no potential insurance contracts that could be offered that would be more profitable than the contracts offered in equilibrium

Full Information Case (Diagram) - Consider the case where the insurer can distinguish between H and L - Insurance firms tailor the insurance contract to the individual on his type (aH1, aH2) and (aL1, aL2) - Perfect competition implies 0 expected profit for each contract type - i) A fair premium offered to each type High risk individuals pay a higher premium and hence end up with less income, L risk pay a low premium and end with more income

Asymmetric Information - Suppose insurer cannot distinguish between H and L - Two possible types of equilibria could arise: - Pooling Equilibrium: Both risk types buy the same policy - Separating Equilibrium: Each risk type buys a different policy (two policies offered and individuals select)

Candidate Pooling Equilibrium - Both risk types buy the same policy - Insurance firms earn 0 expected profit overall Policy must lie on the Aggregate Fair Odds Line - Probability of a loss (payout): H + (1 ) L = bar : average probability - Probability of no loss (payout): 1 bar - Both H and L are on a higher indifference curve on the policy (P) than at E

- i) P is a possible pooling equilibrium where expected profit is 0 - ii) There is a cross subsidy from L to H types, L pay the same premium but H make more claims Is it an equilibrium? - iii) No insurance contracts make negative profits - iv) No possibility for another firm to offer a better contract - v) P therefore cannot be an equilibrium Key Result: A pooling equilibrium cannot exist in a competitive market (Always an opportunity for B if P contracts are offered) - Asymmetric information prevents tailoring of policies - Pooled contract implies L-types cross subsidise loss making H types - Free entry in a competitive market - Firm could enter the market and skim the L types - Pool becomes unprofitable as only H types remain - Pooling policy disappears

Candidate Separating Equilibrium (Diagram) - C is the best policy a firm can off L types without attracting H types - Closer you go to 45 degree (x1 = x2) line the more insurance youre buying At the candidate separating equilibrium: - i) Policies C and H offered - ii) H individuals choose H, L choose C - iii) both policies break even since each lies on the fair odds line for the insured group - iv) Compared to full information case: High risk are fully insured. Low risk only partially insured (Ironic as they should be easier to insure) - v) Asymmetric information constrains what policies can be offered Is it equilibrium? - i) Policy D is preferred by both types (pooling policy) - ii) H and C might not be an equilibrium as it would be bettered by D. But D cant exist because it would be bettered by something else. - iii) However, if D lies above the AFO line, it is loss making so wont be offered ( high). Therefore separating equilibrium survives.

Key Result: Separating Equilibrium Depends on - If is low, there are few high risk individuals, so the L can cross-subsidise H; overall profit - Pooling policy D breaks the separating equilibrium. Policy D itself undermined by a cream-skimming policy. No equilibrium exists.

- If is high, there are many high risk individuals, so the L payments are insufficient to cross-subsidise H; overall loss - Pooling policy D not offered - Separating equilibrium survives

Lecture 15: Moral Hazard and The Principal Agent Model

Moral Hazard (Hidden Action) When one party can hide their action from another party E.g. How much effort a worker puts in, care taken by an insured person This is post-contractual information asymmetry - i) Implications for contract design - ii) Classic way of analysing moral hazard is through the principle agent model. The principal (e.g. owner) wants tot ensure the agent (e.g. manager) puts in the appropriate amount of effort

Example: Principal (Mr. Burns), Agent (Homer Simpson) - i) Stage 1: Contracting (wage) - ii) Stage 2: Effort Decision: Low effort eL or High effort eH - iii) Stage 3: Outcome (Good or Bad) even if eH, always some uncertainty but: - eL: Good outcome (L), Bad outcome (1 - L) - eH: Good outcome (H), Bad outcome (1 H) Fundamental Points: - Just by looking at the outcome doesnt reveal the amount of effort - Homers effort does effect the probability of the outcomes

Principal Agent Model: Assumptions The Principal (P) delegates management of the firm to am agent (A) The owner contracts with the manager before he manages the firm and exerts effort - w is the wage paid to A, e is the effort of the agent - A is risk-averse with utility: U(w, e) = u(w) e (u > 0, u x2 - Probability of high profits in influenced by the managers effort - H1 is the probability of high profit, if e = eH - L1 is the probability of high profit, if e = eL - H1 > L1

Contract Design Owner offers the manager a wage contract Could be a fixed wage or a contingent contract If the owner can observe the effort level of the manager then the owner can specify the effort level he requires and choose the wage accordingly If the owner cannot observe the effort level of the manager, then the owner cannot make choose the wage based on effort, but wage can depend on the profit of the firm, which is observable

What is the optimal contract in each case? - The owner aims to maximise his own profit - The contract must be sufficient to take the job - If effort is not observable, the contract should induce the agent to supply the effort level the manager wants

Framework Analysis Contract Design Either x1 or x2 (Not two goods like Edgeworth box) - w1 = w2 is the certainty line, wage is fixed Participation Constraints Expected utility is at least (Ubar) otherwise (A) would reject the job For e = eH EU (w, eH) = u(w1) H1 + u(w2)H2 eH > u(bar) For e = eL EU (w, eH) = u(w1) L1 + u(w2)L2 eL > u(bar) Owner aims to maximise his net profit so in equilibrium, the participation constraint of the manager will hold with equality (binding)

Owners Expected Utility Depends on whether the manager works hard or is lazy, since that in turn affects the probability of high profits

EU (if e = eH): EU(eH) = H1(x1 w1) + H2 (x2 w2) = (x1 w1)= EU / H1 (H2 / H1)*(x2 w2)

EU (if e = eL): EU(eL) = L1(x1 w1) + L2 (x2 w2) = (x1 w1)= EU / L1 (L2 / HL)*(x2 w2)

Full Information P must pay a higher wage to a hard-working A to compensate for the disutility of effort: w*H > w*L P must decide which he prefers, to pay w*H for e = eH or w*L for e = eL - Depends on the extent to which higher effort increases the chance of getting high profits - Depends on the disutility of effort, which determines how big the compensation need to be to induce high effort - The wage paid does not vary with profit level. P is risk neutral and A is risk averse (optimal risk sharing) requires the least risk averse person to bear the risk

Asymmetric Information P cannot observe As effort level It is no longer optimal for P to offer A the fixed wage w*H because then A would choose e = eL to increase his expected utility (moral hazard) Full information contract is not compatible with (A)s incentives

High effort makes high profits more likely, so there is an imperfect correlation between effort level and profit level P can offer a contingent contract where wage depends on the profit level of the firm: - w*1 if x = x1 and w*2 if x = x2 - w*1 > w*2 What determines w*1 and w*2

Suppose P would prefer A to exert high effort When designing the contract (w*1 and w*2), P maximises his payoff subject to two constrains: - Participation Constraint (PC): A must prefer to accept the job u (w*1) H1 + u(w*2) H2 eH > u(bar) - Inventive Compatibility Constraint (ICC): A must prefer to supply high effort than low effort u (w*1) H1 + u(w*2) H2 eH > u(w*1) L1 + u(w*2) L2 e Inducing diagram error (Blue is hard working, yellow is lazy) Agency Cost: [E(w) w*H] Cost to the economy of forcing people to behave

Trade Off Between Incentives and Risk Sharing P must expose A to risk, but A dislikes risk and so needs a risk premium as compensation Agency Cost: [E(w) w*H] Cost to the economy of forcing people to behave - Extra cost of implementing high effort under moral hazard - Difference in Ps expected payoff from FI and AI - Reduction of welfare of the less well informed party P without a corresponding increase in benefit of the better informed party A - Loss of efficiency arising from inefficient risk sharing necessitated by moral hazard Thus a trade off between the two

Monitoring Alternative to incentive contracts if for P to monitor As performance Choice between incurring monitoring costs or agency costs P may prefer to accept eL for a low non-contingent wage

Determinants of Sharp Incentives Performance related pay (common practice in certain sectors) Suppose employers offer a linear wage contract: W () = a + b (a is fixed, b is performance related What determines b? - = e + z (effort + noise) Optimal contract