Tourguide. Partial Equilibrium Models with Risk/Uncertainty Optimal Household s Behavior

Transcription

1 Tourguide Introduction General Remarks Expected Utility Theory Some Basic Issues Comparing different Degrees of Riskiness Attitudes towards Risk Measuring Risk Aversion The Firm s Behavior in the Presence of Risk General Equilibrium Models of Risk/Uncertainty Risk Sharing within a Group the Arrow-Lind Theorem Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

2 Household s Behavior When thinking about the optimal behavior of a household that faces risk, we can (conceptually) consider two cases: first, we can consider how the household would cope with risk in a static situation: Insurance decision and optimal portfolio choice second, we could ask how the household adjusts its dynamic behavior when facing risk Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

3 Optimal Saving Let us first of all turn to the effect of risk on the dynamic behavior. The most important dynamic/intertemporal choice which a household faces is its decision on how much to consume and how much to save. Having a framework which incorporates risk helps us to understand real world saving decisions and allows us to think about the impact of income shocks (such as unemployment), their welfare effects and their insurability. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

4 When thinking about the optimal consumption plan of the household we consider a simple two-period model. The household is characterized by a contemporaneous utility function u, which is defined over consumption c i in period i. Income of the household is y 1 and y 2 where the future income stream y 2 is risky. The household can borrow or save at a fixed interest rate i (=risk-free asset). Thus, the household does not face a combined savings and portfolio problem. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

5 By the intertemporal budget constraint it will be true that c 1 = y 1 s and c 2 = y 2 + (1 + i)s, where s denotes saving. Consider that the intertemporal utility U is just the sum of contemporaneous utilities. Thus, we write U = u[y 1 s] + u[y 2 + (1 + i)s] The optimal choice is hence only over savings s. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

6 Before deriving optimal behavior of the household let us turn to the risky income stream. Let f [y 2, β] denote the density function of the random variable y 2. β is some parameter of the function which can be used as a mean preserving spread. This is an important point (we will dwell on it later on) because we would like to understand the effect of risk on behavior. With this note that E(u[y 2 + (1 + i)s]) = ŷ 2 0 u[y 2 + (1 + i)s]f [y 2, β]dy 2. where ŷ 2 denotes the upper bound on tomorrow s income. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

7 The foc for optimal household savings reads de(u) ds ŷ2 := u [y 1 s ] + (1 + i) u [y 2 + (1 + i)s ]f [y 2, β]dy 2 = 0 0 This foc implicitly determines the optimal amount of saving (which we denote by s ). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

8 What we would like to know is how a change in the riskiness of say the lottery affects optimal savings. How can we measure this change? We argue that as β changes, the riskiness changes. Our measure is hence: ds dβ. Totally differentiating the foc gives ŷ2 u [y 1 s ]ds + ((1 + i) 2 u [y 2 + (1 + i)s ]f [y 2, β]dy 2 )ds 0 + ŷ2 ((1 + i) u [y 2 + (1 + i)s ]f β [y 2, β]dy 2 )dβ = 0 0 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

9 Hence, ds dβ = A 1 B 1 with ŷ2 A 1 := (1 + i) u [y 2 + (1 + i)s ]f β [y 2, β]dy 2 0 ŷ2 B 1 := u [y 1 s ] + ((1 + i) 2 u [y 2 + (1 + i)s ]f [y 2, β]dy 2 ) 0 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

10 In order to get some comparative static results, we have to sign this slope. Due to decreasing marginal utility, we have B 1 < 0. What about A 1? A 1 reflects how expected (tomorrow s) (marginal) utility changes as β changes, i.e. A 1 := deu dβ. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

11 Now the mean preserving spread comes into play. The mps property says that if β was a mps, the change in a function of the random variable under consideration (which is u in our case) is < 0 (ie negative) if u < 0 and vice versa. Thus, the optimal reaction to an increase in β (which is an increase in riskiness if β is a mps) depends on the third derivative of the utility function. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

12 In order to understand the effects of (increased) uncertainty on the optimal savings behavior, we have to understand the motives for the saving choice. First of all, consider a world with no uncertainty (in which the rate of interest is normalized to i = 0). The foc for optimal savings implies that (under certainty) u [y 1 s ] = u [y 2 + s ] hence the motivation for saving is smoothing out marginal utility. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

13 Suppose we add (simple) uncertainty to the second period income of the form that y 2 + z with prob 0.5 and y 2 z with prob 0.5 The household would like to overturn its saving decision if 0.5u [y 2 + z + s ] + 0.5u [y 2 z + s ] > u [y 2 + s ]. Note that this will only be the case if u is convex i.e. u > 0. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

14 Precautionary saving (i.e. saving due to income uncertainty) only occurs if u is convex. This is interesting and important since risk aversion ONLY affects (directly) the form of the utility function not of marginal utility. However, one can show that restrictions on the degree of absolute risk aversion imply u > 0 (Ask the TA). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

15 Decreasing absolute risk aversion implies u > 0. Only those household have a precautionary saving motive. How do we measure the strength of the precautionary saving motive between different households? Answer: the degree of prudence P = u u will do the job. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

16 Basically, the index of prudence is a normalized measure of the change of the curvature of marginal utility. Individuals whose P > 0 are called prudent and have a precautionary saving motive. This motive increases as individuals become more prudent. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

17 Portfolio Choice When thinking about risk and uncertainty, it is natural to think of future events or THE uncertain future. Hence, time and uncertainty are usually interlinked as was the case with the intertemporal saving problem. This interlinkage makes the the problems potentially quite complex. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

18 In order to cope with complexity we make simplifying assumptions. In the savings problem we had time AND uncertainty, but only a special sort of uncertainty. What we are going to discuss now is a static situation with uncertainty in the form of risky assets. This lies at the heart of the classical portfolio problem. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

19 Consider some household who wants to invest an exogenous amount of savings say s into two assets. There exits two assets: one yields a save return of 1 + r whereas the other yields return 1 + r where r is stochastic. When investing an amount α into the risky asset, the income of the household will be (s α)(1 + r) + α(1 + r) Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

20 The household s problem is to choose α such that his expected utility Eu is maximized. We will assume that utility is only defined over income, hence the problem reads max{e(u[s(1 + r) + α(r r)])} α Note that (r r) denotes the excess return of the risky asset. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

21 The foc for optimal α is given by E(u [s(1 + r) + α (r r)](r r)) = 0 which implicitly gives the demand of the household for the risky asset. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

22 In the optimization problem we did not consider any constraints on α, i.e. α is allowed to be positive or negative. What does α < 0 imply? It implies short sales of the household; the household sells the risky asset. With this, however, it is not straightforward to sign α. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

23 Consider it was true that α = 0. In this case u [s(1 + r)]e(r r) = 0 would hold. The household would not buy the risky asset if the expected excess return was zero. Moreover, we can say something about the expected marginal return of α for α = 0 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

24 It will be true that u [s(1 + r)]e(r r) which is the slope of expected utility at α = 0 determines the sign of α. Suppose that this was positive (u must be positive) and remember that marginal expected utility is decreasing. Thus, α > 0 if E(r r) > 0 and vice versa. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

25 Besides the rather trivial insight that the household would not invest if the expected return was zero, we also find the more intriguing insight that the household invests into the risky asset if the expected return was positive. The household would never opt for the safe haven no matter how risk avers he is. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

26 So how much of his wealth would a rational household invest into the risky asset (=stock market). To get an exact measure, we would have to specify the utility function AND the distribution of the risky return, and plugging into the foc. Luckily, we can derive an approximate value for the optimal portfolio choice. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

27 Using a second-order Taylor approximation of the foc around s(1 + r) gives E((r r){u [s(1 + r)] + α (r r)u [s(1 + r)]}) 0 which then can be rewritten to give (noting that s(1 + r) is NOT stochastic) u [s(1 + r)]e(r r) + α E((r r)) 2 u [s(1 + r)]] 0 which finally yields α u u E(r r) E(r r) 2 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

28 What we see is that the optimal amount invest into risky stocks is proportional to the revenue risk relation. The factor of proportionality is equal to the inverse of the degree of absolute risk aversion. If absolute risk aversion is high, the amount invested is low. Moreover, if profit per risk increases, the investment increases, too. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

29 Insurance Up to now we have considered situations in which the household faced dynamic and uncertain situations. This reflects somewhat the fact that (in the real world) uncertainty and time are intimately related. However, considering the choice of households who only face uncertainty is interesting, too. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

30 With this we can consider the choice of the household to shift its income over states. The device through which this is done is the insurance. An insurance buys the right to get a Euro conditional on the realization of some state of the world. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

31 Understanding insurances is important for a number of reasons. First of all, they are an important aspect of life: think of unemployment or health insurances. Second, the market is fragile and a close examination may help designing solutions against this fragility Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

32 Consider a very simple economy in which only 2 states of the world s 1, 2 can be realized. In state 1 the income of the household will be y whereas in state 2 the household faces a loss L leaving income at y L. Before the realization of the state of the world, the household can buy insurance cover q at a price p. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

33 One unit of q buys the household the right to get 1 Euro if state 2 was realized. Note that we look at a partial equilibrium model i.e. we do not ask who the counterparty of this insurance contract is. Thus, the household can (ex-ante) write a contract at a price pq (which has to be paid in any case) which generates additional income q if state 2 was realized. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

34 Let π i be the probability for state i. The expected income of the household is hence π 1 (y pq) + π 2 (y L pq + q) = π 1 (y pq) + π 2 (y L + (1 p)q) Thus, the insurance generates income pq with prob. π 1 and (1 p)q with prob. π 2 The insurance enables the household to shift income from state 1 to state 2 (which parallels the notion of the intertemporal budget constraint) Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

35 How much insurance will the household buy? q is chosen such that expected utility is maximized. Note that if q = L there would be full insurance (income is constant over all states). Expected utility is given by U = π 1 u[y pq] + π 2 u[y L + (1 p)q] where u[.] denotes the utility function. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

36 The first-order condition for optimal insurance choice is given by π 1 u [y pq ]( p) + π 2 u [y L + (1 p)q ](1 p) = 0 The marginal utility loss in state 1 must equal the marginal utility gain in state 2. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

37 Rearranging the foc gives π 1 p π 2 (1 p) = u [y L + (1 p)q ] u [y pq ] If the left hand side was 1 the optimal choice would imply equal marginal utilities in all states. This implies that income in all states have to be the same implying q = L, i.e. full coverage (for concave utility functions). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

38 Because π 1 = 1 π 2 (one state must be realized), the left hand side is 1 if p = π 2. This situation implies an actuarially fair premium. With a fair premium, the insurance costs (1 π 2 )pq equals what it pays π 2 (1 p)q. If the left hand side would exceed 1 (p > π 2 ), the household would like to have higher marginal utility in state 2 than in 1. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

39 With u < 0, this implies that state 2 income must be smaller than state 1 income. Thus, the household does not cover the loss L fully with an insurance, i.e. q < L. Note that all this only holds as long as preferences are state independent (which is basically a corollary of the independence axiom of expected utility). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

40 An important assumption in the preceding analysis was that the probability of state 2 (the loss) was a) exogenously given and b) common knowledge to both insurer and insured. Relaxing these assumption shows that these (informational) frictions have an important impact on the working of the market for insurance. Specifically these frictions may lead to a break down in competitive (private) markets for insurances. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

41 In the following we will use the standard model to analyze the effects of adverse selection. Adverse selection occurs if there (exogenously) exist individuals with different loss probabilities, but which cannot be identified by the insurer. The insurer faces a problem of asymmetric information. He knows that there exist high and low risk types, but does not know who is who Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

42 Adverse Selection Consider the same modeling structure as before except that the probability of a loss is just π and that there exist two types of risk. High risk types face π h and low risk types π l where π h > π l. Under full information, the insurer could offer two contracts one with p h = π h and the other with p l = π l. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

43 With no additional information on the types (except the knowledge that they exist) the insurer faces the following problem. If he would offer two contracts with p h = π h and p l = π l for full coverage qi = L (which would be equilibrium with symmetric information) the high risk types would buy the low risk contract. Hence they would pay (1 π h )π l q but would receive π h (1 π l )q Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

44 This contract would not be profitable for the insurer since payments to the insured would exceed revenues (in expectations). Thus, this type of contract cannot be supported in competitive insurance markets. The insurer has to change the type of contracts that he will offer. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

45 What can he do? Either he can offer one-for-all contracts with a kind of average premium pooling contract or he can offer cleverly designed contracts such that individuals self-select into the appropriate contract separating equilibrium. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

46 Pooling Contracts Let the fraction of high risk individuals in the population be denoted by 1 λ which is common knowledge. In this situation, the insurer might want to offer a pooled contract such that the fee for the contract equals the average risk π = λπ l + (1 λ)π h The optimal behavior of individuals can be described by the foc for optimal insurance choice. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

47 (Expected) utility of low risk households: (1 π l )u[y πq] + π l u[y L + (1 π)q] (Expected) utility of high risk households: (1 π h )u[y πq] + π h u[y L + (1 π)q] foc for low risk: (1 π l )u [y πq low ] π + π lu [y L + (1 π)q low ](1 π) = 0 foc for high risk: (1 π h )u [y πq high ] π + π hu [y L + (1 π)q high ](1 π) = 0 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

48 Using the foc it must be true that u [y piqlow ] u [y L + (1 π)q low ] < u [y πq high ] u [y L + (1 π)q high ] because π h > π l by definition. Thus, it must be true that q low q high. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

49 Moreover, note that both sides of the inequality are increasing in the respective q (with decreasing marginal utility). Hence it will be true that qhigh > q low. Intuitively, the contract will be relatively cheap for high risk individuals such that they choose a large insurance amount. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

50 Is this the end of the story? The insurer knows the foc of the households and knows that individuals who want to have large contracts are the high risk individuals. In this case the insurer would use this signal to sell the contract π h. Because the high-risk individual anticipates this behavior he will demand insurance qlow! This obviously violates the foc, but since it enables the high-risk household to hide its true type it will be profitable to do so. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

51 Thus, in the pooling contract all individuals behave as if they were the low-risk types. In a world with only one insurance, this could be an equilibrium. What about competitive insurance markets? If one insurance offered the pooling contract to all individuals there would be competing insurers that would cherry-pick on this contract. High-risk individuals strictly prefer the pooling contract, but the low-risk individuals could be lured out of these contracts as long as the competing contracts are only slightly better compared to the pooling contract. Since everybody in the economy would anticipate this, there cannot be a competitive equilibrium with pooling. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

52 As such, only equilibria with separating contracts are feasible. The insurance offers two types of contracts (p l, q l ) and (p h, q h ) where the insurance fixes the fee AND the amount of insurance. Individuals hence optimize under the additional restriction that the self-selection constraint must be fulfilled. The only thing the contracts have to fulfil is the self-selection constraint, i.e. low-risk types should prefer the low-risk contract and vice versa. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

53 Intuitively, the low-risk type will not act as if he were the high-risk type and buy the high-risk contract (ask the TA). The insurance only has to worry about high-risk individuals trying to act as being low risk individuals. They would not do so if (1 π h )u[y p h q h ] + π h u[y L + (1 p h )q h ] (1 π h )u[y p l q l ] + π h u[y L + (1 p l )q l ] Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

54 Consider the situation in which insurers charge the fair premium. The self-selection constraint modifies to (1 π h )u[y π h q h ] + π h u[y L + (1 π h )q h ] (1 π h )u[y π l q l ] + π h u[y L + (1 π l )q l ] Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

55 Without any constraints, households would choose q h = L and q l = L (i.e. full-insurance if types could be revealed). If the insurance offered such a contract, the self-selection constraint would not be fulfilled. Thus, the insurance has to adjust contracts: either decrease q h or decrease q l or both. (Note that increasing coverage is not an option since these contracts would not be bought by the respective risks). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

56 Let us consider these options in turn: first let the insurance decrease q h (starting from q h = L). This would decrease the lhs of the inequality and make the selection problem even more severe (i.e. the payoff of simulating being low-risk would increase). Let the insurance keep q h = L but lower the coverage for low risk contracts q l < L. This decreases the rhs of the inequality restoring the self selection constraint (which was violated with full coverage). To make high-risk individuals to reveal their true type, the low-risk contract must be made sufficiently unattractive. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

57 In the separating equilibrium, high risk types are as well-off as in the situation in which types are revealed. Low-risk types are made worse-off. Intuitively, the situation of the low-risk types must be made worse to scare off high-risk types to pretend to be low-risk. The friction decreases welfare. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

58 When having discussed the pooling equilibrium, we have thought about the stability of this equilibrium. We can do the same with the separating equilibrium: i.e. do competing insurances have an incentive to offer pooling contracts? The question is whether a pooling contracts exists that lures away low-risk types without making a loss. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

59 Intuitively, this will be more likely, the higher the fraction of low-risk types. With this the insurer can offer an attractive pooling contract. If there are enough low-risk types this contract would pay-off. Thus, there will be some threshold fraction λ of low-risk types in the economy. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

60 With λ increasing this threshold, the separating contract will be driven out of the market by the pooling contract. However, we have shown already that the pooling equilibrium will not exist and also be driven out of the market. Thus, if the fraction of low-risk types is large enough there will not exist an equilibrium under adverse selection (with competitive insurance). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

61 Moral hazard Up to now we have assumed that there is an information asymmetry between insurer and households where signalling was not possible. Moreover, we have assumed that Nature determines the risk type, i.e. the allocation of risk types was exogenous. What if households can endogenously choose which risk type to be? (In reality car drivers can drive careful or reckless.) Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

62 When an endogenous choice is possible, the market/the economy is characterized by moral hazard. Thus, the insurer has to offer incentive compatible contracts such that agents behave as expected. Intuitively, (as is true with car insurance) there will not be contracts with full cover. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

63 Consider a household that can act carefully (which costs a 1 ) or can act reckless (which is costless a 0 = 0). Acting carefully implies a probability of a loss of π 1 where acting reckless causes a loss with π 0 > π 1 With a fair premium, utility of a household reads (i {1, 2}) (1 π i )u[y a i π i q] + π i u[y a i L + (1 π i )q] Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

64 If the fee/premium that the insurer charges can be conditioned on the amount of care of the household the household would in either case choose full insurance q = L. With full coverage, maximized utility reads u[y a i π i L]. The individual will act carefully if u[y a 1 π 1 L] > u[y π 0 L], thus if a 1 π 1 L > π 0 L Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

65 This implies that π 0 > π 1 + a 1 L, i.e. the no-care insurance must be expensive enough such that the (average) caretaking costs are more than covered. In the following we assume that the symmetric equilibrium situation would result in caretaking (if this was not the case, hiding something which would not have been chosen is irrelevant). Consider the asymmetric information situation, ie the insurer cannot observe whether the household really invested a 1 in caretaking. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

66 If the insurer would offer a full cover contract at the fair price π 1 the households would buy this contract but would not act carefully. This contract incurs a loss which is even greater if the household could unilaterally choose the amount of cover. How does an incentive compatible contract look like? Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

67 The insurer would offer such an amount of cover q at price π 1 such that the utility of households is maximized (competition!) AND the household will (voluntarily!) act careful. Thus, the optimal insurance is the solution to the following problem max ((1 π 1 )u[y a 1 π 1 q] + π 1 u[y a 1 L + (1 π 1 )q]) q s.t. (1 π 1 )u[y a 1 π 1 q] + π 1 u[y a 1 L + (1 π 1 )q] }{{} U 1 [a 1,q] (1 π 0 )u[y π 1 q] + π 0 u[y L + (1 π 1 )q] }{{} U 0 [q] Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

68 Deriving the first-order condition for this problem (the amount of coverage which maximizes utility given the incentive compatibility constraint) can be derived using the Kuhn-Tucker-Lagrange approach. L = ((1 π 1 )u[y a 1 π 1 q] + π 1 u[y a 1 L + (1 π 1 )q]) + λ(u[a 1, q] U[q]) which implies the focs ( (1 π1 )u [y a 1 π 1 q ]π 1 + π 1 u [y a 1 L + (1 π 1 )q ](1 π 1 ) +λ( du 1[a 1, q ] dq du 0[q ] ) dq λ(u 1 [a 1, q ] U 0 [q ]) = 0 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

69 Suppose for the moment that the incentive compatibility constraint was not binding. Thus, λ = 0 and the individual would choose full cover q = L. But this implies that U 1 [a 1, q ] U 0 [q ] < 0. At the optimum, the incentive compatibility constraint must be binding, i.e. U[a 1, q ] U[q ] = 0 which is then the condition for the optimal contract. How will this contract look like? We know that if q = L it will be true that U 1 [a 1, q ] < U 0 [q ]. In which direction must q move such that equality is restored? Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

70 Note that du[a 1,L] dq = 0 and du[l] dq = π 0 π 1 > 0, hence the utility difference between careful and careless behavior decreases as q decreases (starting from q = L). The contract which implies incentive compatibility implies less than full cover, i.e. partial insurance. The economic intuition is that the careless individual should bear some of the cost of its action. But this requires less than full coverage. (Think of a car insurance) Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

71 The partial cover contract does not necessarily maximize utility (it is only constraint optimal). It could well be that the household would prefer the full cover contract at the higher price π 0. In the economy with asymmetric information and moral hazard the insurance can offer two contracts: a partial cover contract (which is relatively cheap) and the more expensive full cover contract (in which carelessness is taken into account). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

72 An important implication of the model with moral hazard is that there exists scope for governmental intervention. The government cannot observe the behavior of each individual agent (the insurance cannot so why should the government?). But the government can tax careless behavior or subsidize careful behavior, e.g. tax drivers who drive too fast. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

73 The Firm s Behavior in the Presence of Risk Tourguide Introduction General Remarks Expected Utility Theory Some Basic Issues Comparing different Degrees of Riskiness Attitudes towards Risk Measuring Risk Aversion The Firm s Behavior in the Presence of Risk General Equilibrium Models of Risk/Uncertainty Risk Sharing within a Group the Arrow-Lind Theorem Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

74 The Firm s Behavior in the Presence of Risk Up to now, we have focussed on the behavior of an individual household. We assumed that the household gets some exogenous income and faces risk which came in the form of an income risk (except for the portfolio choice). But where does income come from? Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

75 The Firm s Behavior in the Presence of Risk Income comes from production and hence from the decision of firms or to be precise the owners of the firms. What we would like to is to understand how uncertainty affects the production decision of the firms. In a first step, we look at a competitive market in which the price (i.e. the demand curve) is uncertain. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

76 The Firm s Behavior in the Presence of Risk A firm produces output x and sells it competitively at price p s. The price is state dependent because e.g. demand is uncertain. The profit of the firm is p s x c[x] where c[.] denotes the cost function. It is increasing in output. Firm s profit is distributed to the owners of the firm and these are characterized by a utility function u[y s ] where y s is their state dependent income. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

77 The Firm s Behavior in the Presence of Risk Let the income of the household only consist of profit income y s = p s x c[x]. How would the owner household like the firm to act (i.e. choose production)? Suppose (reference scenario) that the firm does not face any risk. Optimal output would then imply the foc u [y s ](p s c ) = 0. With positive marginal utility, we get the standard price equal marginal costs condition. Optimal output x is then implicitly defined. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

78 The Firm s Behavior in the Presence of Risk If the price was a random variable, the optimal output choice should maximize expected profit. The foc is then E(u [y s ](p s c )) = 0 Uncertainty implies that the firm/owner has to determine output before the price is revealed (readjustment is not possible). If the owner of the firm was risk neutral (which basically implies that u is a (positive) constant), the foc says that the firm should produce until marginal cost equal the expected prize, i.e. E(p s ) = c Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

79 The Firm s Behavior in the Presence of Risk If the owner are risk averse the decision does not turn out to be that easy. The foc consists of the expected value of the product of two random variables u [y s ] and p s. Moreover, these two random variables are stochastically dependent: if p s increases, u [y s ] decreases ( negative correlation) Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

80 The Firm s Behavior in the Presence of Risk To determine optimal firms behavior, we need the covariance. This is a measure of the correlation between two random variables. It is defined as Cov(X, Y ) := E((X E(X ))(Y E(Y ))), where X and Y are random variables. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

81 The Firm s Behavior in the Presence of Risk A large covariance between two random variables, hence, implies that the two move in the same direction. Having information about the realization of one reveals facts about the other. Think e.g. of intelligence and income (both of which are random variables). Observing a rich individual makes one probably think that she is also very intelligent. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

82 The Firm s Behavior in the Presence of Risk Using the definition of the covariance, we can write E((XY X E(Y ) E(X )Y + E(X )E(Y ))) which can be simplified to yield E(XY ) E(X )E(Y ) E(X )E(Y ) + E(X )E(Y ) Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

83 The Firm s Behavior in the Presence of Risk By the definition of the covariance we can show that Cov(u [y s ], (p s c )) = E(u [y s ](p s c )) E(u [y s ])E((p s c )) The foc claims that in an optimum, the first term on the rhs is zero. With the covariance being negative and the marginal utility being positive, it must be true that in an optimum E(p s ) c > 0. The firm owned by risk-averse household will produce less (c > 0) than the risk-neutral firm. There will be rationing due to uncertainty. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

84 The Firm s Behavior in the Presence of Risk The intuition for this rationing effect is the following. In the low price states (specifically in the states in which p s < c ), the firm incurs a loss on the marginal output (and vice versa). With c = E(p s ) the owner somewhat balances losses and gains. If the owner is risk-averse, however, he would rather tend to lower the loss situations (giving up potentially profitable situations). As such, he would like to produce less. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

85 The Firm s Behavior in the Presence of Risk The production function that we have considered have been linear. In a next step we would like to focus on technological uncertainty. Before we come to that however, we analyze the effects of price uncertainty with non-linear production, i.e. revenue is pf (x). Using this we get the foc E(u [y s ](p s f c )) = 0 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

86 The Firm s Behavior in the Presence of Risk Rewriting the foc using the definition of the covariance gives Cov(u [y s ], (p s f c )) = E(u [y s ])E((p s f c )) An increase in the price increases marginal revenue AND income, the covariance is negative leaving the results basically unchanged. The form of the production function does not change the rationing result. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

87 The Firm s Behavior in the Presence of Risk Now consider that the technology of the firm is the root of uncertainty. Let us assume that output is f s (x). As such, output produced by x varies stochastically. This has an important implication, we could observe shocks that increase output, but leave the marginal product unchanged (for example). Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

88 The Firm s Behavior in the Presence of Risk The foc now reads E(u [pf (x) s c(x)](pf s c )) = 0 which can again be rewritten using the definition of the covariance Cov(u [y s ], (pf s c )) = E(u [y s ])(pe(f ) c ) Again u is positive. As such, the production decision depends on the correlation between income and the marginal product. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

89 The Firm s Behavior in the Presence of Risk For reference, note that if the owner was risk neutral (u being constant), the correlation is zero (by definition). Thus, the foc implies that pe(f ) c = 0. Whether technological uncertainty increases or decreases output when moving to risk-averse ownership is ambiguous. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

90 The Firm s Behavior in the Presence of Risk If the technological shock is such that the marginal product increases AND output increases, the lhs will be negative which implies a decrease in production. Other situations are, however, possible as well: the shock affects marginal output or output only no output effect. The shock increases marginal productivity but decreases overall production output increases. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

91 The Firm s Behavior in the Presence of Risk We have assumed that producers can only engage in spot market transactions. They choose production and sell production instantaneously after the shock has occurred. What if the firm could choose to sell (part) of its output in a future market at price p f? Income in this case is y s = p s (x x f ) + p f x f c(x), where we consider the case of price uncertainty only. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

92 The Firm s Behavior in the Presence of Risk x is the amount of production and x f is the part of production which is sold (or bought!) on future markets. The foc read E(u [y s ](p s c )) = 0 E(u [y s ](p f p s )) = 0 Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

93 The Firm s Behavior in the Presence of Risk The second foc is the optimality condition for speculation if x is exogenously fixed at 0. Using the covariance definition, we can write Cov(u [y s ], (p f p s )) = E(u [y s ])E((p f p s )) If future traders were risk neutral (and there were no transaction costs), the futures price must be identical to the expected spot market price. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

94 The Firm s Behavior in the Presence of Risk In a situation with futures markets and risk-neutral future traders ( speculation ), risk-averse producers would sell their entire output on the futures market. As such, producers will get rid of of the risk they face and can generate state independent income streams. Moreover, since the income is state independent their optimal production behavior will replicate that of risk neutral producers output will increase. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221

95 General Equilibrium Models of Risk/Uncertainty Risk Sharing When thinking about the optimal insurance decision, we have focussed solely on the decision of an individual household. We assumed that there exists a risk-neutral insurer. The only restriction we considered was a zero profit condition. This risk-neutral insurer is, however, only a thought construct. In reality risk-averse individuals share the risk among themselves. Jörg Lingens (WWU Münster) Advanced Microeconomics October 25, / 221