1 Uncertainty Per Krusell & D. Krueger Lecture Notes Chapter 6 1 A Two-Period Example Suppose the economy lasts only two periods, t =0, 1. The uncertainty arises in the income (wage) of period 1. Not that this is an aggregate shock. Assume that there are n possiblestatesoftheworldinperiod1,i.e. ω {ω 1,ω 2,..., ω n }, with probabilities π i =Pr(ω = ω i ), i =1, 2,..., n. The consumer maximies expected lifetime utility: E [u (c 0,c 1i,n i )] = X n π iu (c 0,c 1i,n i ), i=1 where n i is working hours at time 1. Specifically, assume the utility function has the following form (time-separable and additive-separable), E [u (c 0,c 1i,n i )] = u (c 0 ) + β X n π i [u (c 1i ) + v (n i )], (1) i=1 where v 0 (n i ) < 0.
1.1 A Risk-free Asset There is a risk free asset (similar to a storage technology) denoted by a, whichis priced q at time 0, such that every unit of a purchased in period 0 pays 1 unit of goods in period 1, regardless of the state of the world. The consumer faces the following budget restrictioninperiod0,givenafixedtime0incomei: c 0 + aq = I. (2) At each realiation of the state of the world, his budget constraint in period 1 is c 1i = a + w i n i,i=1,..., n. (3) The consumer s problem is to choose (c 0,a,{c 1i,n i } n i=1) to maximie (1), subject to (2) and (3). The Lagrangian is L = u (I aq)+β X n π i [u (c 1i )+v(n i )] + X n λ i (a + w i n i c 1i ). i=1 i=1 2
The Euler equation implies u 0 (c 0 ) q = β X n i=1 π iu 0 (c 1i ), which says that on the margin, the marginal utility from purchasing one unit of asset (at price q) and reducing consumption at period 0 is equated to the discounted expected marginal utility from consuming one more unit of good in period 1 (Alternatively, you can simply reinterpret 1/q as the gross interest rate for your saving). Re-arranging the above equation, q = X µ n βu 0 π (c 1i ) i, i=1 u 0 (c 0 ) which is a very primitive version of asset pricing for the real risk-free asset (note that we need to specify the rest of the general equilibrium model to completely solve the asset price). 3
Example: Let u(c) = c1 σ 1 1 σ, where σ is the coefficient of relative risk aversion and is the inverse of the elasticity of intertemporal substitution (the higher σ, the less willing the consumer is to experience the fluctuations of consumption over time). let σ =1, then u(c) =ln(c). Wealsoassumev(n) =ln(1 n). Using the FOCs and the budget constraint at state i, we have c 1i = a + w i. 2 Note that there is no state-contingent transfer of wealth in the period 1, and thus period 1 consumption may fluctuate a lot, depending on the realiation of w i. The holding of asset a can be solved, again by FOCs and the budget constraints, q I aq = β X n π 2 i. i=1 a + w i Normaliing the supply of the asset to be unity, then a =1in equilibrium, and we can solve for the asset price from this equation. 4
1.2 Arrow Securities Instead of a risk-free asset yielding the same payout in each state, we consider Arrow securities (state-contingent claims): n assets are traded in period 0 (each priced q i ), and each unit of asset i purchased pays off 1 unit if the realied state is i, and0 otherwise. The budget constraint in period 0 is c 0 + X n q ia i = I. (4) i=1 At each realiation of the state of the world, his budget constraint in period 1 is c 1i = a i + w i n i,i=1,..., n. (5) 5 Will the consumer be better with this market structure? The market structure now allows the wealth transfer across periods to be state-specific: not only can the consumer reallocate his income between periods 0 and 1, but also move his wealth across states of the world.
6 Replacing a i, the (intertemporal) budget constraint becomes c 0 + X n q ic 1i = I + X n q iw i n i. i=1 i=1 The Euler equation implies u 0 (c 0 ) q i = βπ i u 0 (c 1i ). Note that the Euler equation holds for every realiation of the state of the world. Also the asset price i depends on the MRS of between consumption in period 0 and consumption in period 1 when state i occurs, q i = βπ iu 0 (c 1i ) u 0 (c 0 ) MRS(c 1i,c 0 ). If MRS(c 1i,c 0 ) is large, i.e., in order to increase one unit of date 1 good in state i, the consumer is willing to give up a large amount of date 0 good, and thus the consumer is willing to pay a high price for the security. The relative asset price across state equals to the marginal rates of substitution across
7 states q i = βπ iu 0 (c 1i ) q j βπ j u 0 (c 1j ) MRS(c 1i,c 1j ). Example: Given log utility function, the availability of Arrow securities implies c 1i = βπ i c 0, q i which says that consumption in each state is proportional to consumption in period 0, c 0 and is independent of the realiation of w i. This proportionality is a function of the cost of insurance: the higher q i (the consumer is willing to pay a higher price for the security i) inrelationtoπ i, the lower the wealth transfer into state i. In this way, agents areabletodiversifyriskefficiently.
2 Representation of Uncertainty 8 We start with the notion of an event s t. The realiations s t aredrawnfromasets: s t S, t. ThesetS = {η 1,,...,η N } of possible events is assumed to be finite and the same for all periods t. In the following, we use the notation s t =1instead of s t = η 1 andsoforth. An event history s t =(s t,s t 1,..., s 0 ) is a vector of length t +1keeping track of the realiations of all events up from period 0 to period t. Notice that s 0 = s 0, and we can write s t = s t,s t 1. Formally, S t = S S... S denoting the t +1-fold product of S, event history s t S t lies in the set of all possible event histories. Let π (s t ) denote the probability of a particular event history. Assume that π (s t ) 0 for all s t S t, for all t.
2.1 The Markov Process Note that the sets S t of possible events of length t become fairly big very rapidly, which poses computational problems when dealing with models with uncertainty. We assume that s t follows a first order discrete time, discrete state, and time homogeneous (not indexed by time) Markov process. Let π be the N N transition matrix: π 11 π 12... π 1N π 21 π 22... π 2N...... π N1 π N2... π NN Then, the conditional probability that the state in t +1is s t+1 = j S if the state in period t equals s t = i S, isgivenby. π s t+1 = j, s t s t = i, s t 1 = π [s t+1 = j s t = i] = π ij. By construction, the conditional probability π ij is time invariant and π (s t ) 0, thuswe have P N j=1 π ij =1.. 9
Suppose the probability distribution over states period t is given by the 1 N row vector P t =(p 1 t,...,p N t ),where P N i=1 pi t =1,forallt. Given today s probability distribution P t, the probability of being in state j tomorrow is p j t+1 = X N π ijp i i=1 t, i.e., the probabilities today are weighted by the transition probabilities starting out in state i. Given the initial probability distribution P 0, P 1 = P 0 π, P 2 = P 0 π 2,..., P t = P t 1 π =P 0 π t. A stationary (or invariant) distribution for π is a probability row vector P such that P = P π. (1 N) (N N) i.e. if you start today with a distribution over states P, then tomorrow you end up with the same distribution over states P. 10
A stationary distribution then satisfies PI = Pπ, where I is an identity matrix. Thus, we have P (I π) = 0 (1 N), i.e., That is, P is an eigenvector of π, associated with the eigenvalue λ =1. Theorem π has a unique invariant distribution (and is asymptotically stationary (i.e., P t converges in P t+1 = P t π, and the convergence does not depend on P 0 )) if π ij > 0, i, j. µ µ π11 π Consider π = 12 0.9 0.1 =, where state 1 represents for expansion and π 21 π 22 0.1 0.9 state 2 for recession. This transition mtrix shows that both expansions and recessions are highly persistent. We can show that the unique invariant distribution is P =(0.5, 0.5). 11
3 The Stochastic Neoclassical Growth Model 12 Consider a neoclassical (constant returns to scale) aggregate production function subject to a TFP shock has the form F t (k t, 1) = t f (k t ), where t is a technology shock. f is assumed to have the usual properties, i.e. has constant returns to scale, positive but declining marginal products, and satisfies the Inada conditions. Assume that the technology shock has unconditional mean 0 and follows a N-state Markov chain. Let Z = {η 1,,...,η N } be the state space of the Markov chain. Let π =(π ij ) be the N N transition matrix and P the stationary distribution. An event history t =( t, t 1,..., 0 ) is a vector of length t +1keeping track of the realiations of all events up from period 0 to period t. Notice that 0 = 0, and we can write t = t, t 1. Let π ( t ) denote the probability of a particular event history. Assume that π ( t ) 0 for all t Z t,forallt.
3.1 Social Planner s Problem The social planner s problem maximies X X β t π t u c " # X t t = E β t u (c t ). t=0 t Z t t=0 The feasibility constraint requires that the consumer chooses a consumption and investment amount that is feasible at each (t, t ): c t t + k t+1 t t f k t t 1 +(1 δ)k t t 1. 13 The Euler equation implies u 0 c t t = X t+1, t t u 0 c t+1 t+1, t t+1 f 0 k t+1 t +(1 δ) t+1 Z t+1 βπ = E t [u 0 (c t+1 ) R t+1 ], where π ( t+1, t t )= π( t+1, t ) π( t ),andr t+1 t+1 f 0 (k t+1 ( t )) + (1 δ) is the marginal return on capital realied for each t+1. (6)
Using the assumption that the technology shock follows a N-state Markov process, the planner s problem can be expressed in recursive form NX V (k, i ) V i (k) =max k 0 u [ if(k) k 0 +(1 δ)k]+β π ij V j (k 0 ). The Euler equation is given by NX u 0 [ i f(k) k 0 +(1 δ)k] =β π ij u 0 [ j f(k 0 ) g(k 0, j )+(1 δ)k 0 ] [ j f 0 (k 0 )+(1 δ)]. j=1 Thesolutiontothisproblemisthepolicyfunctionk 0 = g(k, ). The Euler equation is a nonlinear, stochastic difference equation. In general, we will not be able to solve it analytically, so numerical methods or lineariation techniques will be necessary. j=1 14
15 3.2 Competitive equilibrium In the social planner s problem, the planner allocates {c t ( t ),k t+1 ( t )} according to the history of states t. Note that a commodity with a particular history of realiations of the random sequence c t ( t ) is a commodity different from c t ( t ) resulting from another history of realiations of the random sequence. How to implement these state contingent commodities? A complete markets structure will allow contracts between parties to specify and enforce the delivery of physical good in different amounts for different realiations of the random sequence for a different price. The Arrow-Debreu economy specifies the full set of state contingent commodities at date 0 and assumes that these contracts can be perfectly enforced.
3.2.1 Arrow-Debreu Economy The definition of the Arrow-Debreu (A-D) equilibrium is sequences {c t ( t ),k t+1 ( t ), p t ( t ),r t ( t ),w t ( t )} such that (1) Given prices {p t ( t ),r t ( t ),w t ( t )}, the representative consumer choose {c t ( t ), k t+1 ( t )} by solving max {c t ( t ),k t+1 ( t )} X X s.t t=0 X t=0 t Z t p X t Z t p X X β t π t u c t t t Z t t c t t + k t+1 t t=0 (2) The FOCs of the firm s problem hold t [ r t t +(1 δ) k t t 1 + w t t n t t ]. r t t = t f k kt t 1,w t t = t f n kt t 1. 16
17 (3) Market clearing c t t + k t+1 t t f k t t 1 +(1 δ)k t t 1, t, t. 3.2.2 Sequential Markets Economy Suppose there are m securities with asset j paying off r ij consumption goods in period t +1if the current state is i. Let the portfolio a =(a 1,..., a m ) 0. The following matrix shows the payoff of each asset for every realiation of t+1 : 1 2. N a 1 a 2... a m r 11 r 12... r 1m r 21 r 22... r 2m...... r N1 r N2... r Nm R
Given the portfolio a and the payoff matrix R, the returns on the portfolio across the N states are br N 1 = R N m a m 1, where each component b R i = P m j=0 r ija j is the amount of consumption goods obtained in state i from holding portfolio a. What conditions should be imposed on R so that any arbitrary returns R b can be generated? The restrictions are (1) m N; (2) rank (m) = N. In this case, we say that the markets are complete, i.e., the asset structure of an economy spans the set of states (By spanning we mean that a combination of assets can be used to transfer any amount of wealth from one state to another). 18
19 Recall that Arrow security i pays off 1 unit of good if the realied state is i, and0 otherwise. Since each Arrow security is linearly independent, N Arrow securities will be able to span the entire state returns space. In this case, the markets are complete. If rank (m) <N, i.e. the number of linearly independent securities is less than the number of states, then the set of assets does not span the entire state returns space and we will have incomplete markets. In this case, Pareto optimality will be in general not achievable.
20 3.3 A Two-agent and Two-period Example Suppose there are two periods, t =0, 1, and there are two agents i =1, 2, each with the following expected utility U i = u i (c i 0)+β NX π j u i c i j. We assume that Agent 1 is risk neutral and agent two is risk averse: u 1 (x) =x, and u 2 (x) is strictly concave. The purpose for this specification is that since agent 1 is willing to absorb as much as risk, we can examine under what market structure agent 1 is able to provide full insurancetoagent2. Each agent is endowed with ω 0 units of consumption goods in period 0, and one unit of labor in period 1 (which will be supplied inelastically since leisure is not valued. j=1
Consumption goods are produced in period 1 with a constant-returns-to-scale technology. The production at state j is y j = j K α µ L 2 1 α, where j is the technology shock, j Z. K and L denote the aggregate supply of capital and labor services in period 1, respectively. Given that state j is realied, the rental rate and wage rate are r j = j αk α 1, µ 1 α = j K α, 2 w j where we have used the choice of labor services in equilibrium L =2. The agents decide how much savings 21
3.3.1 A Single Asset Suppose capital is the only asset that is traded. With K denoting the aggregate capital stock, a i denotes the capital stock held by agent i, and therefore the asset market (again, this is similar to a storage technology that pays one unit of good at date 1 for one unit of storage at date 0) clearing requires that The budget constraints for each agent are a 1 + a 2 = K. c i 0 + a i = ω 0, c i j = a i r j + w j, j =1,..., N. 22 Given that agent 1 is risk neutral, the maximied utility function and the constraints are linear in this case. By the arbitrage condition (ero net expected return on the portfolio), NX 1+β π j r j a i =0. j=0
23 Then we must have 1=β NX π j r j = β j=0 NX π j j αk α 1 = αβk α 1, where P N j=0 π j j, the optimal choice of K by agent 1 is j=0 K =(αβ) 1 1 α. Since agent 1 is risk neutral, only the average value of the shock matters. The Euler equation for agent 2 is u 0 2(ω 0 a 2 )=β NX π j u 0 2 a 2 r j + w j rj. j=0 Given K chosen by agent 1, r j and w j are determined. Then a2 can be solved, which is independent of j. Agent 2 will face a stochastic consumption for period 1, c 2 j = a2 r j +w j, because r j and w j are stochastic. This implies that agent 1 has not provided full insurance to agent 2.
3.3.2 Arrow Securities Now there are N state-contingent claims that spans the entire state returns space. Let a j denote the Arrow security j (similar to equities) that pays off one unit of good if the realied state is j and ero otherwise. Let q j denote the price of a j. The budget constraint of each Agent i is 24 c i 0 + NX q j a i j = ω 0, j=0 c i j = a i j + w j, j =1,...,N. Total investment in all N Arrow securities must equal to the aggregate capital stock NX q j a 1 j + aj 2 = K. (7) j=0 For each security j (or for each state j that occurs) we require that a 1 j + a 2 j = r j K,
where a 1 j + a2 j is the total returns for security j (or when state j occurs) which must be equal to total payout from the share of production to capital service at state j, r j K. Multiply both sides by q j and then sum up over j s, NX X N q j a 1 j + a 2 j = K q j r j. j=0 j=0 By (7), we have NX q j r j =1, j=0 which is the no-arbitrage condition in this environment. The left hand side is the total marginal returns for an agent to sell the portfolio ({a j }) and use the proceeds to invest in capital. And the right hand side is the cost of a marginal unit of capital investment. If P N j=0 q jr j > 1, all agents have an incentive to sell an infinite amount of units of such a portfolio (q j ), and using the proceeds from this sale to finance an unbounded physical capital investment. But clearly there will be no demand for this portfolio. 25
26 Using the FOCs of agent 1 s problem, we have q j = βπ j, and total capital chosen remains to be the same as before K =(αβ) 1 1 α. Agent 2 s problem yields the Euler equation Using the result q j = βπ j,wehave u 0 2(c 2 0)q j = βπ j u 0 2 c 2 j, for all j. u 0 2(c 2 0)=u 0 2 c 2 j, for all j. which implies agent 2 is fully insured from agent 1, and agent 1 bears all the risk. Note that in general the asset price depends on MRS of the agent, reflecting how much the agent is willing to purchase insurance. But the risk-neutral agent is willing to bear all the risk so that the risk-averse agent is able to diversify all the risk re-arranging his portfolio (purchase more a 2 j when state j yields a low productivity).