Macroeconomics Summary

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1 Macroeconomics Summary Rebecca Sela March 0, 2006 In general, every model consists of preferences, endowments, technology, and demography. When defining a competitive equilibrium, one should be sure to specify all of the conditions for household maximization, the government budget condition, and for market clearing. Asset Pricing In asset pricing, we generally use representative agent models, in which agents are identical or they can be aggregated. These models can either be partial equilibrium models using Euler equations or general equilibrium models using Lucas Trees.. Partial Equilibrium and No Arbitrage Pricing Suppose prices are exogenous and the representative agent wishes to maximize E t j=0 βj uc t+j, where c t is consumption. The agent can choose wealth, A t, in the form of bonds, L t, which are claims to one unit of the good at time t + at an interest rate R t, and a number of shares N t, each of which pays a dividend in each period. Let p t be the share price and y t be the dividend paid on the shares owned at the beginning of the period. Then, the consumer s Euler equations are: u c t R t = E t βu c t u c t p t = E t βp t+ + y t+ u c t+ To solve asset pricing problems in full details: Write the maximization problem.

2 Write the Bellman equation. Derive the Euler equation from the first order conditions and the envelope theorem. Compute asset prices from the social planner s allocation and the Euler equation. For arbitrage-pricing arguments, one should know what to buy and sell in order to make money whether one price is too high or too low. No-arbitrage arguments involve only prices, never probabilities. No-arbitrage pricing can be based on: Computing the household budget constraint possibly in multiple periods to ensure that the budget set is bounded. Using the firm s zero profit conditions. Finding two prices and relating them to ensure that there is no way to make a sure profit. This is done by finding a redundant price pinned down by all the others. For example, if there is only one asset, there can be only one intertemporal price..2 Lucas Tree Economy Suppose the representative agent wishes to maximize E t j=0 βj uc t+j, where c t is consumption. The agent can choose wealth, A t, in the form of risk-free bonds, L t, which are claims to one unit of the good at time t + at an interest rate R t, and a number of shares trees, each of which pays a non-storable dividend fruit in each period. Let p t be the share price and y t be the dividend paid on the shares owned at the beginning of the period. Because this is general equilibrium, the price now depends on the current state. We assume that y t is the function of a Markov state, s t, on a continuous state space with: F s, s = P s t+ s s t = s = s Generally, we normalize that each agent has one tree N t =. Then, the budget constraint is: c t + L t R t + p t N t A t fz, sdz A t+ = L t + p t+ + y t+ N t We may also have borrowing constraints L t L or N t N to ensure solvency. 2

3 The agents Euler equations are: u c t R t = E t βu c t u c t p t = E t βp t+ + y t+ u c t+ We also have the transversality conditions: lim t β k u c t+k L t+k k R t+k = 0 lim tβ k u c t+k p t+k N t+k k = 0 These ensure that the wealth does not increase or decrease unboundedly in the limit, which pins down the level of consumption. If R t = R > is constant, then βr u c t = E t u c t+, and marginal utility follows a Markov process. If utility is quadratic, then marginal utility is linear and c t+ = constant + βr c t + ɛ t+. In general, we may use the second Euler equation and the fact that EAB = EAEB + CovA, B to write: p t = E t βp t+ + y t+ u c t+ u c t = βe t p t+ + y t+ E t u c t+ u c t + βcov t p t+ + y t+, u c t+ u c t The covariance term is zero if agents are risk neutral so that marginal utility is constant or there is no uncertainty. We can solve the social planner s problem with the representative agent to find the equilibrium allocation the representative agent consumes all the fruit in each period. Then, we use the equilibrium allocation to compute prices: = E tβu y t+ R t u y t u y t p t = E t βp t+ + y t+ u y t+ = E t βp t+ u y t+ + E t βy t+ u y t+ =... = E t β j u y t+j y t+j + E t lim k βk u y t+k p t+k where the limit term is zero by the transversality condition and the equilibrium condition that each agent continues to hold one tree. This yields the price of the asset as a function of the 3

4 current state: p t = E t β j u y t+j y t+j u y t This yields a stochastic process for prices that depends on the stochastic process for dividends. Timing affects prices in this economy. We generally assume that one receives the dividends from trees bought last period; that is, trees are traded after dividends are paid. If this were not true, then prices and budget constraint would be: u c t+ p t = βe t u p t+ + d t c t p t N t + c t N t y t + N t p t However, the general conclusions would not change. This model is in contrast to the real business cycles model, which depends on capital and includes persistence in the technology shocks..3 Term Structure of Interest Rates In general, to price assets, there are two options: Use the prices that are already known from the economy in a no-arbitrage argument. Add the new asset into the budget constraint and solve the general equilibrium model again to determine its price. Definition The term structure describes the relationship between a risk-free bond s time to maturity and its per-period yield. Let R jt be the risk-free gross return between t and t + j. Let L jt be the quantity of these bonds. Then, the budget constraint becomes: c t + L jt R jt + p t N t A t A t+ = L t + j=2 L jt R j,t+ + p t+ + y t+ N t 4

5 where A 0 may be given and we impose some borrowing constraint. Since the dividends can be described by a Markov process, we have a value function: V A t, s t = max{uc t + βe t V A t+, s t+ } where the maximization is taken over the constraints above. The Lagrangian yields the first order conditions with respect to c t, L jt : u c t λ t = 0 βe t V A t+, s t+ A t+ λ t = 0 L jt R jt V A t+, s t+ = u c t+ u c t+ = βe t R jt u c t R j,t+ Since the social planner s problem yields an equilibrium allocation of c t = y t, this yields the term structure of interest rates: u y t+ u = βe t R jt u = β j s t+j E t y t u s t R j,t+ In general, R jt is a function of y t = y 0,..., y t. If y t is a Markov process, the interest rate term structure depends only on y t, not on t or the history. We may also calculate the per-period interest rate: R jt = u s t /j β E t u s t+j Suppose s t is independent and identically distributed. Then: R j s t = u s t β j Eu s = R s t Definition 2 In pure expectations theory, premium. R 2t = β j R t E t R,t+. In this case, there is no risk In general, we may split the expectation of the product to compute: u s t+ u s t+ = βe t R 2t u E t + βcov t s t R,t+ u s t, R Pure expectations theory holds if the second term the risk premium is 0. This will occur if agents are risk neutral or if there is no uncertainty. For an IID process, the second term becomes β 2 Eu s u s t Covu s t+, u s t+, which is always negative. In this case, a two-period bond must pay more interest than a one-period bond because gaining knowledge of the second period might change decisions. If there is bad news next period, then u s t+ will be higher, which will make the subsequent interest rate higher. The expression above again reduces to R 2t = β R t. This covariance is negative even if agents are risk-loving. 5

6 .4 No-Arbitrage Pricing Suppose {s t } is Markov. Then, the pricing kernel is: Q j s j, s = β j u s j u s f j s j, s where f j z, s = f z, s j f j s j, sds j is the j-step ahead transition function. For example, consider an insurance policy that pays one good next period if y t+ α. This could be added to the budget constraint to find the price. Instead, we use a no-arbitrage argument to say that q α s = α Q s, sds. We may also express Q in terms of primitives though this goes beyond a no-arbitrage argument, since no-arbitrage arguments never depend on probabilities, only on prices: q α s = α α = = E Q s, sds β u s u s fs, sds β u s t+ u s t s t+ α, s t P s t+ α s t If agents are risk neutral, this reduces to q α s = βp s α, since the ratio of marginal utilities is constant at. If agents are risk averse, u s u s > when s α s, so that individuals will pay more for insurance if the current period is better than α..5 Modigliani and Miller Theorem Suppose we wish to sell off a tree or a firm using B bonds that pay r each period as long as rb Y t ; otherwise, they pay Yt B and N shares that pay dividends N Y t rb in each period or 0 if rb Y t. Suppose that the equilibrium prices for the two assets are p B t N, B, r, s t and p N t N, B, r, s t. Then, the value of the tree for a particular structure is: Γ = Bp B t N, B, r, s t + Np N t N, B, r, s t We wish to maximize the value with respect to N, B, r, assuming that markets are complete with Arrow-Debreu trading. Theorem 3 Modigliani-Miller. Γ depends only on s t, not on B, N, r. 6

7 Proof. Using a no arbitrage argument, we are able to price the assets according to their payment streams, using equilibrium pricing kernels: p B t = rq j s t+j, s t ds t+j p N t = Yt+j rb Q j s t+j, s t ds t+j N Then, the value of the firm is: Bp B t + Np N t = = Br + N Yt+j rb N Y t+j Q j s t+j, s t ds t+j Q j s t+j, s t ds t+j which does not depend on B, N, r. Note that this is a very simple model of an economy; this theorem need not hold in other models. Also, if it is certain that Y t rb, then the bond price depends only on r, while the stock price depends on all three other variables. In the IID case with log utility, p B t = E t rβ j u y t+j u y t = reu y u β j y t β = ry t E y β = rpy t E y p N yt+j rb t = E t N = β j y t y t+j E t β j rb/y t+j y t N = py t rbe/y N In general, holding an asset leads to: 7

8 Capital gains or losses: These are measured by p t+ p t. Interest or dividends: These are measured by d t+ p t. In this model, shares, trees, and bonds all have the same capital gains. Only some have risk, and the risk and therefore the risk premium varies. For bonds, the total return is pb t+ p B t As the amount paid out in bonds called the leverage increases, the shares become more risky, and therefore must have a better return. + r r B t. 2 Stochastic Growth Model 2. Time 0 Trading In the stochastic growth model, we have: A discrete, finite state space. A representative household with preferences ordered by t=0 βt s t S t π ts t uc t s t, l t s t ; notice that decisions may depend on the entire history. Technology defined by c t s t + x t s t A t s t F k t s t, n t s t and k t+ s t = δk t s t + x t s t, where x t s t is investment, A t s t is the exogenous productivity level, and k t s t is this period s capital stock, which was decided in the previous period Endowments of k 0 and of n t s t + l t s t = in each period. Production firms which hire n t s t, k P t s t in order to produce and sell c t s t, x t s t. Investment firms which own the capital and rent it to the production firms. Markets for: Labor services: The household sells and the production firms buy at price w 0 t s t Capital services: Production firms buy and investment firms sell at price r 0 t s t Goods for either consumption or investment: Households buy, production firms sell, and investment firms either buy or sell depending on whether investment is positive or negative at price q 0 t s t. Initial capital: Households sell, investment firms buy at price p k0 8

9 Firm Ownership: This turns out not to matter. Time 0, state-contingent trading. Step : Solving the social planner s problem. L = β t t= s t S t π t s t uc t s t, n t s t + µ t s t A t s t F k t s t, n t s t + δk t s t c t s t k t+ s t Because they are multiplied by β t π t s t, the µ t s t are in terms of time t and state s t prices instead of time 0 prices. The optimal allocation must solve: u l c t s t, n t s t u c c t s t, n t s t = F n k t s t, n t s t A t s t u c c t s t, n t s t π t s t = β s t+ s t u c c t+ s t+, n t+ s t+ πs t+ F k k t+ s t, n t+ s t+ A t+ s t+ + δ where the first equations described the labor-leisure trade-off and the second equation described the intertemporal trade-off. Step 2: Solving the optimization problems of households and the two types of firms. Note that there are no risks in any of these equations because all the decisions are state-contingent at time 0. Households maximize t= βt s t S t π ts t uc t s t, n t s t subject to t=0 s t S t q0 t s t c t s t t=0 s t S t w0 t s t n t s t + k 0 p k0. This leads to the first order conditions if the solution is in the interior: β t π t s t u c c t s t, n t s t = ηq 0 t s t β t π t s t u l c t s t, n t s t = ηw 0 t s t Production firms maximize t=0 s tq0 t s t c t s t x t s t rt 0 s t kt P s t wt 0 s t n 0 t s t subject to c t s t + x t s t A t s t F kt P s t, n t s t. Under constant returns to scale, F k, n = nf k n = nfˆk. Then, the firms maximize t=0 s t n ts t q 0 t s t A t s t fˆk P s t r 0 t s t ˆk P s t w 0 t s t, and the constraint is no longer needed. The optimal capital-labor ratio solves: f ˆk P s t = r 0 t st q 0 t st A ts t. Given prices, this completely determines the coefficient on n ts t. If the coefficient is positive, the firm would try to set n t to be infinite. If the coefficient is negative, the firm would not produce. Thus, to ensure equilibrium with positive production, we must have the FOC s: w 0 t s t = q 0 t s t A t s t F n k t s t, n t s t r 0 t s t = q 0 t s t A t s t F k k t s t, n t s t 9

10 This does not determine n t but the rest of the economy will. Also, since there is free entry into the market, the maximum must be at 0. Investment firms maximize p k0 ko+ I t=0 s t S tr0 t s t kt I s t qt 0 s t kt+ I st δkt I s t by choosing k0 I, ki t+ st. We rewrite this to consolidate the k t : p k0 +r0s δq0s 0 0 k0+ I rt 0 s t + δqt 0 s t kt I s t qt s 0 t kt I s t t=0 s t S t s t S t Consolidating shows that the coefficient on k I t s t is s t s t r0 t s t +q 0 t s t δ q 0 t st, which must be zero to ensure that the economy is in equilibrium. This yields the first order conditions: p k0 = r 0 0s 0 + δq 0 0 q 0 t s t = s t s t r 0 t s t + q 0 t s t δ Step 3: We substitute the optimal allocation into the first order conditions to compute the relative equilibrium prices: q 0 0s 0 = qt 0 s t = β t π t s t u c s t wt 0 s t = u ls t u c s t q0 t s t = u ls t u c s 0 βt π t s t r 0 t s t = β t π t s t u c s t A t s t F k k t s t, n t s t p k0 = r 0 0s 0 + δq 0 0 In this model, if we re-opened markets at time t < τ, the new prices for time t goods, q t τ s τ would simply have a new numeraire: q t τ s τ = q0 τ s t q 0 t st = βτ t u cs τ u c s t π τ s τ s t w t τ s τ = w0 τ s t q 0 t st r t τ s τ = r0 τ s t q 0 t st The implied wealth of the household at time t is: Υs t = τ=t s τ s t q t τ s t c τ s τ n τ s τ w t τ s τ 0

11 which is future contingent claims less future obligations. Note that τ=t s τ s t qt τ s t c τ s τ = rts t t + δk t s t, so that non-labor wealth depends on the current value of capital even though the households don t own the capital in this model. 2.2 Stochastic Growth with Sequential Trading Suppose there is a pricing kernel, Q t s t+ s t, together with wage and rental rates, w t s t, r t s t. Households may choose ã, which are one-period contingent claims. Then, the household budget constraint and first order conditions become: c t s t + s t+ ã t+ s t+ s t Qs t+ s t w t s t ñ t s t + ã t s t s t Q t s t+ s t = β u cs t+ u c s t π ts t+ s t w t s t = u ls t u c s t The production firms now maximize c t s t + x t s t r t s t k P t s t w t s t ñ t s t. For the production firms to be in equilibrium, we have the zero profit conditions: r t s t = A t s t F k s t w t s t = A t s t F n s t Investment firms assuming free entry, except that if they enter they are committed for two periods maximize k I t+ st + s t+ s t r t+s t+ + δ k I t+ st Qs t+ s t. This yields the zero profit condition: = s t+ r t+ s t+ + δ Qs t+ s t We conjecture that: Q t s t+ s t = q t t+s t+ w t s t = w t ts t r t s t = rts t t ã t+ s t+ s t = rt+ t+ + δk t+ s t ã 0 = k 0 r0s δ and we could check this by verifying that they lead to the same values of the choice variables. In this model, trading in labor and capital happens during the period, so their trades are not state-contingent. However, assets are state-contingent. We may also form the problem recursively, if we assume that:

12 s t follows a Markov process with transition probability πs t+ s t. A t s t = AA t s t, s t is Markov. That allows s t to be a growth rate. Then, the current state of the economy is X = s, A, K, where s is the current state, A is last period s productivity, and K is the capital stock at the beginning of the period. The household s state vector is X, a, where a is the wealth of the household at the beginning of the period which does not matter to the economy if households are homogeneous. The social planner s value function is: vs, A, K = max uc, N + β s S πs sev K, S, s where the maximization is subject to the laws of motion: K + C AsF K, N + δk A = As s πs s This yields the optimal policy functions, defined by: C = Ω C X N = Ω N X K = Ω K X The only state variable that households can choose is a though they also choose their individual consumption and labor supply. Let a be the vector of arrow securities for next period. The households have the value function: Ja, X = max c,n,a uc, n + β x Jax, x ĤX X subject to the constraints, c + X QX XaX wxn + a and that ax is greater than some borrowing constraint. In equilibrium, a will depend on the value of capital. This yields the optimal policy functions for the household: c = σ c a, X n = σ n a, X ax = σ a X, a, X 2

13 Production firms deal only in the spot market, and maximize c + x rxk wxn subject to c + x AsF k, n. As before, this leads to the zero profit conditions: rx = AsF k k, n wx = AsF n k, n Investment firms have stochastic future profits; however, pricing kernels completely remove the uncertainty, since there are complete markets. That is, they must maximize k + X QX XrX + δ. This yields the zero profit condition: = X QX XrX + δ This requires that the price today which is normalized to equals the payoff tomorrow, discounted back by the pricing kernel. In equilibrium, markets clear, which requires equilibrium in the credit market: or, equivalently in the goods market: ax = rx + δk c + x = AsF K, N Substituting from the credit market, we check that the household budget constraint holds: c + X QX XrX + δk = wxn + rx + δk c + K = wxn + rx + δk Solving for K, substituting in equilibrium conditions K = k, N = n, C = c and decision rules, we find that: K = wxn + rx + δk c = AsF n k, nn + AsF k k, n + δk c = AsF K, N + δk C = AsF K, σ n a, X + δk σ C a, X = AsF K, σ n rx + δk, X + δk σ c rx + δk, X = GX This yields a law of motion for K. Definition 4 A recursive competitive equilibrium consists of: 3

14 given prices, r, w, Q and perceived laws of motion for the state variables, ĤX X, the decision rules, σ c, σ n, σ a X, a, X and the value function, Ja, X, solve the household s problem, both types of firms are maximizing profits, which implies that rx = AsF K K, N = AsF k K, σ n rx + δk, X wx = AsF N K, N = AsF n K, σ n rx + δk, X = X QX XrX + δ Note that these expressions depend only on the current state variables. The laws of motion of capital, rental rate, consumption, and labor satisfy GX above. People have rational expectations about the state: πx X = ˆπX X. 3 Government Finance 3. Lump Sum Taxation For these models, we assume that the government spends {g t } in each period, which yields no utility for consumers. This is an exogenous stochastic process with g t < y t for all t. The resource constraint is y t = c t + g t. The only tax is a lump sum tax, τ t. We assume that the government can borrow with -period state-contingent bonds, b t s t+. The bond pays off one unit of the consumption good next period if s t+ is realized. In this model, the state is described by s t = y t, g t, which we assume is Markov. The government budget constraint in state s t is: g t + b t s t = τ t + Qs t+ s t bs t+ ds t+ where Q is the one period ahead pricing kernel. To find a competitive equilibrium in this economy, we recall that lump-sum taxes are nondistortionary since they don t affect the Euler equations, so we apply the welfare theorems and solve the social planner s problem of maximizing t=0 E 0β t uc t, subject to c t y t g t. By non-satiability, c t = y t g t. We then compute the representative agent s Euler equations and evaluate them at this allocation to compute prices and the equilibrium interest rate: p t = E t β j u y t+j g t+j u y t+j y t g t R jt = β j E t u y t+j g t+j u y t g t 4

15 The level of government debt does not affect the interest rate this is part of Ricardian equivalence. The pricing kernel is: If we have time 0 trading, then the prices are: Q j s t+j s t = β j u y t+j g t+j u f j s t+j, s t y t g t q 0 t+js t+j = Q s t+j s t+j...q s s 0 = β j u y t+j g t+j u y 0 g 0 which is the price of one good in period t + j if history s t+j = s 0,..., s t,...s t+j, s t+j occurs. Note that f j is not used, since the entire history is specified in time 0 trading. Government bonds are specified by: b t s t s t = τ t s t g t s t + Q s t+ s t b t s t+ s t ds t+ where taxes and bond holdings may depend on the entire history but government spending depends only on s t and the price kernel depends only on s t, s t+ because the process is Markov. To measure fiscal sustainability, we must discount the budget constraint at time t + by Q s t+ s t ; since the future state is unknown, we integrate over all possible s t+. This becomes: Q s t+ s t b t s t+ s t ds t+ = Q s t+ s t τ t+ g t+ ds t+ Q s t+2 s t+ b t+ s t+2 s t+ ds t+2 ds t+ After continued substitution, this leads to the budget constraint: b t s t s t = τ t g t + qt+jτ t t+j s t+j ds t+j s t + + lim k Q j s t+j s t gs t+j ds t+j q t t+k+ s t+k+ s t+k b t+k s t+k+ s t+k ds t+k+ s t where we define xs t+j ds t+j s t =... xs t+j ds t+j...ds t+2 ds t+, which integrates over all possible futures that start with s t. Notice that τ t depends on the entire history, so that the associated prices and integrals must depend on the entire history as well, but g t is Markov, so we can use a normal integral with a j-step pricing kernel. In equilibrium, the limit term must be non-positive, since, otherwise, individuals would be accumulating assets which they would rather consume. If the limit is negative, then the government is accumulating assets. This is not ruled out by equilibrium conditions, but we generally assume that it does not happen. 5

16 The household budget constraint is: c t s t +τ t s t + c t+j s t+j +τ t+j s t+j qt+js t t+j ds t+j s t ps t +ys t N t s t +b t s t s t +limit Notice that intermediate bonds vanish in the present value budget constraint; initial trees and initial bonds describe a consumer s initial wealth. The limiting term must be non-positive by the transversality conditions and non-negative if we do not allow Ponzi schemes. Proposition 5 Ricardian equivalence. If the government satisfies its budget constraint, consumers utility will not be affected by how the government finances itself through borrowing and taxes. That is, equilibrium prices and consumption depend only on {y t, g t } Proof. In equilibrium, N t s t = ps t = E t β j u y t+j g t+j u y t+j y t g t b t ds t s t = b t s t s t We may replace b t s t s t from the government budget constraint to find that the household budget constraint is: c t s t + c t+j s t+j qt+jds t t+j s t ys t gs t + ys t+j gs t+j Q j s t+j s t ds t+j which does not depend on taxes or bonds. One could try to measure fiscal sustainability by checking if debt is financed by future surpluses: b t s t = τ t g t + Eτ t g t R jt where R jt is the j-step ahead, risk-free interest rate. In a Lucas tree economy, the requirement above is not right, because covariance terms should be included, or there should be integrals over pricing kernels. For a policy to be sustainable, the bonds must satisfy this equilibrium expression: b t s t s t = τ t s t g t s t + E t β j u y t+j g t+j u τ t+j g t+j y t g t = τ t s t g t s t + E t τ t+j g t+j + Cov β j u y t+j g t+j R jt u, τ t+j g t+j y t g t 6

17 Since τ t+j is not uniquely determined, the covariance term can be manipulated to be non-zero. If there are non-altruistic generations, then we instead of restricted Ricardian equivalence, in which people are only indifferent between policies that have the same net present value during their lifetimes. 3.. Taxes and Stochastic Growth Based on Bohn, 995 Suppose preferences are given by uc = c γ γ, and that the endowment obeys y t y t = ỹ t, and ỹ t is independent and identically distributed. We assume that gt y t = c is constant. We consider two tax policies. In the first, τ 0 > 0 and τ t = 0 for all future t. In this case, τ 0 may be greater than the endowment, but the government can then lend back to the people, so this is feasible. In this case, τ 0 is the present value of all government expenditures. In the second policy, the government holds a constant debt-to-gdp ratio: b = bt/rt y t. We assume that all debt is one-period and risk free not state contingent. The required debt will then imply the taxes for each period. Note that any constant debt policy is sustainable; however, discounting by the risk-free interest rate won t work: lim j E t b t+j = lim E t by t+j R,t+j R j+,t j R j+,t j = lim E t R by t ỹ t+i j R j Eỹ = by t lim j As long as R > Eỹ, this limit will be 0 otherwise, it will be finite or infinite, either of which is a Ponzi scheme. R j i= 7

18 In either case, we compute the interest rate: = β j E t u y t+j g t+j R jt u y t g t = β j E t cy t+j cy t γ = β j E t = β j j i= = R t j so that risk-free interest rates are deterministic. j i= ỹ γ t+i E t ỹ γ t+i A constant debt-to-gdp ratio leads to a positive covariance that is, a pro-cyclical policy, since for a lower value of ỹ t, u y t+ g t+ u y t g t is higher while the debt is lower since GDP is also lower and taxes must be higher. To check if a policy is sustainable, we should NOT discount future cash flows using the interest rates R t, but rather by the pricing kernel, qt+j t. If you use the interest rates, this policy may seem unsustainable. Using the pricing kernel, we must check that lim j qt+j t st+j b t+j s t+j+ = 0. This pricing kernel depends on both R t and the covariance term. With a countercyclical policy, the covariance term is negative, which means that taxes seem higher than they need to be. Note that all of this depends on taxes being lump sum. If taxes are distortionary, all of this may change. In this economy, the interest rate depends on all of the primitives: Eỹ scales everything. If γ = V arỹ = 0, then R = β. If γ = 0, then R = β even as the variance increases. If V arỹ = 0, then the growth rate is constant but people want to smooth consumption, which means that the interest rate must be higher in equilibrium to encourage saving. If both are non-zero, things get more complicated. 8

19 3.2 Optimal Distortionary Taxation All of these strategies assume government commitment. Suppose {g t } t=0 is given. The government imposes capital and labor taxes and may choose the sequences {τt n, τt k } t=0. Preferences are given by t=0 βt uc t, l t. The technology is c t + g t + k t+ = F k t, n t + δk t, where F has constant returns to scale. Households own the capital. The endowment is k 0 and l t + n t =. Households have the one-period budget constraint: c t + k t+ + b t+ R t τ n t w t n t + τ k t r t k t + δk t + b t where b t are one period bonds, bought in time t that pay off in time t since there is a representative agent, all of the borrowing is between the household and the government. This gives the first order conditions: u l c t, n t u c c t, n t u c c t, n t βu c c t+, n t+ = τ n t w t = τ k t+r t+ + δ = R t Firms have profit: F k t, n t r t k t w t n t = F k k t, n t r t k t + F n k t, n t w t n t by the Euler equation for constant returns to scale. In equilibrium, to ensure zero profits and finite production, we must have F k k t, n t = r t and F n k t, n t = w t. The government budget constraint is g t + b t = τ k t r t k t + τ n t w t n t + b t+ R t In a competitive equilibrium with distortionary taxes, households are maximizing their utility, taking prices and government decisions as given, firms are maximizing profits, the government is obeying its budget constraint, taking household actions and all prices as given, and there are no Ponzi schemes in borrowing. 9

20 Notice that this does not uniquely determine an allocation, because there are many possible taxation schemes, and their distortionary nature means that there is no longer Ricardian equivalence. Definition 6 The Ramsey allocation is the competitive equilibrium with distortionary taxes that maximizes consumer utility. In this context, the Ramsey problem is to maximize t=0 βt uc t, n t, subject first to feasibility conditions both budget and resource constraints: τt k r t k t + τt n w t n t + b t+ b t g t R t = 0 F k t, n t + δk t c t g t k t+ = 0 l n + n t = τ n t w t n t + τ k t r t k t + δk t + b t c t k t+ b t+ R t = 0 Note that the last budget constraint is actually redundant; if we sum the household budget constraint with the government budget constraint and use the firm s zero profit conditions, we recover the feasibility constraint. This means we can remove any one of them. This would no longer be true if the budget constraints were in present value terms; then we would have to keep the one-period feasibility constraint but could remove either of the others. We also replace l t = n t. Because the allocation must be implementable, it must also satisfy all the first order conditions: u l c t, n t u c c t, n t τt n w t = 0 u c c t, n t βu c c t+, n t+ τt+r k t+ + δ = 0 R t τt+r k t+ + δ = 0 F k k t, n t r t = 0 F n k t, n t w t = 0 Note that the government s only choice variables are τ k t, τ n t, but we maximize over c t, n t, k t+, r t, w t, τ k t, τ n t, b t+, R t, and l t many of which are implied once others are known. In an open economy, interest rates and rental rates are exogenously determined by the world economy. Furthermore, the household and government budget constraints no longer imply the usual resource constraint, since government and household bond holdings do not have to sum to 0. Furthermore, household and firm capital now satisfy: c t + k t+ + g t + k H t+ k F t+ + b G t b H t = F t + bg t+ bh t+ R t δ + r t k H t k t + δk t 20

21 3.2. Chamley Approach Define the after-tax wage and rental rates by: Then, the Euler equations become: r t = τ k t r t w t = τ n t w t u l c t, n t u c c t, n t w t = 0 u c c t, n t βu c c t+, n t+ r t+ + δ = 0 R t r t+ + δ = 0 and the tax revenues and government budget constraint become: τ k t r t k t + τ n t w t n t = r t r t k t + w t wn t = F k t, n t r t k t w t n t F k t, n t r t k t w t n t + b t+ R t b t g t = 0 Substituting in for r t, w t using the zero profit conditions and for the taxes leaves the following constraints: We then maximize the Lagrangian: L = t=0 F k t, n t r t k t w t n t + b t+ b t g t R t = 0 F k t, n t + δk t c t g t k t+ = 0 u l c t, n t u c c t, n t τ n t w t = 0 u l c t, n t u c c t, n t w t = 0 u c c t, n t βu c c t+, n t+ r t+ + δ = 0 R t r t+ + δ = 0 β t ut + ψ t F t r t k t w t n t + b t+ b t g t + θ t F t + δk t c t g t k t+ R t +µ t u l t u c t w t + µ 2t u c t βu c t + r t + δ + µ 3t R t r t+ + δ The first order condition with respect to capital is: β t θ t + β t+ ψ t+ F k t + r t + θ t+ F k t + + δ = 0 With no taxation, this would simply be: β t θ t + β t+ θ t+ F k t + + δ = 0 2

22 The coefficient on the extra constraint measures the additional value of the taxes that the government collects off of extra capital. Suppose g t becomes constant for all t > T and that a steady state exists. Then, we must have θ t θ and ψ t ψ, which yields the first order condition: θ = βψr r + θr + δ In the steady state, the second Euler equation tells us that: u c = βu c r + δ r β = δθ = βψr r + θr r 0 = βψ + θr r By non-satiation, θ > 0. Furthermore, β > 0, ψ 0. This means that r = r in the steady state, and the steady state capital tax must be 0 for the optimal allocation. Zero capital taxation means that intertemporal substitution is not changed in the steady state. It is also related to the persistence of capital though it would still hold if δ = Primal Approach To use the primal approach:. Obtain the first order conditions of the households and firms, and use these to solve for the prices and taxes as functions of the allocations. 2. Use these equations to eliminate prices and taxes from the household s present value budget constraint. This yields the implementability constraint. 3. Solve the Ramsey problem by maximizing utility subject to the resource and implementability constraints. 4. Use this allocation to back out prices and taxes. This always uses present value budget constraints, which is equivalent to the previous approach, as long as b t is not restricted. This imposes the first order conditions from the household and firm before optimization, which removes prices, turning the problem into a social planner problem, with the additional implementability condition. 22

23 The household s present value budget constraint and first order conditions are: qt 0 c t = t=0 q 0 t = qt 0 τt n w t n t + τ0 k + δk 0 + b 0 t=0 t i=0 q 0 t = β t u ct u c 0 τ n t w t = u lt u c t q 0 t τ n t w t = β t u lt u c 0 R i, q 0 0 = All of the capital and bond terms after time 0 cancel by the no-arbitrage conditions. For example, combining the budget constraints across two periods: c t + k t+ + b t+ = w t n t + r t k t + δh t + b t R t c t + R t c t+ + b t+ R t + b t+2 R t+ = w t n t + w t+n t+ R t + rt + δ R t k t+ + r t k t + δk t + b t The coefficient on k t+ must be 0 to keep the budget constraint bounded. Using similar logic leads to only k 0 appearing in the present value budget constraint. As before, the firm has zero-profit conditions: These equations eliminate prices and taxes. r t = F k t w t = F n t Since k 0 is given, τ0 k acts like a lump-sum tax and is not distortionary. Thus, it would be optimal but trivial to choose τ0 kk 0 = t=0 q0 t g t. Instead, we require that τ0 k τ k 0. τ 0 k will always be maximized in an optimal solution. Substituting for prices and wages in the household budget constraint yields the implementability constraint: t=0 β t u ct u 0 t c t = β t u c tc t u l tn t A = 0 t=0 t=0 u l t u c 0 βt n t + τ k 0 + δk 0 + b 0 A = u c 0k 0 τ k 0 + δ + b 0 23

24 Then, the Lagrangian for the Ramsey problem is: L = β t ut + β t θ t F t δk t c t g t k t+ + φ β t u c tc t u l tn t A = t=0 t=0 β t V c t, n t, φ + θ t F t + δk t c t g t k t+ φa t=0 V c t, n t, φ = uc t, n t + φu c tc t u l tn t The first order conditions with respect to c t, n t, k t+, c 0, n 0 are: V c t = θ t V n t = θ t F n t θ t = βθ t+ F k t + + δ V c 0 = θ 0 + φa c V n 0 = θ 0 F n 0 + φa n which shows that the optimal allocation is characterized by: V c t = βv c t + F k t + + δ V n t = V c tf n t We can back out prices and taxes from this. V n 0 = φa c V c 0F n 0 + φa n If a steady state exists, the tax on capital satisfies: β = τ k F k + δ However, V c = βv c F k + δ, which means that we must have β = F k + δ, and the steady state capital must still be 0. t= Consumption Taxes If there is a consumption tax instead of a capital tax, the budget constraint and implementability conditions become: qt 0 + τt 0 c t = qt 0 τt n n t w t + r 0 + δk 0 + b 0 t=0 t=0 β t u c tc t u l tn t A = 0 t=0 A = u c 0 F k 0 + δk 0 + b 0 + τ0 c 24

25 If τ0 c = 0, then this reduces to the problem with τ t k = 0; more generally, for any τk 0, we can find τc 0 that will lead to the same A the value will depend on k 0, b 0, δ. Now, we write prices as: qt 0 = β t u ct + τ c 0 u c 0 + τt c R t = u c t βu c t + + τt+ c + τt c = r t+ + δ w t = u lt + τt c u c t τn t Comparing steady states, we see that, for any steady state in c, n, g, k, the capital tax is given by uc βu c = τt+ k F k + δ, which means that the capital tax is also constant. For a consumption tax, instead, we must have uc +τt+ c βu c +τt c = F k + δ, which means that the ratio of consumption taxes must be constant in the steady state, and + τ c t is constant. If τ k is positive in the steady state, this corresponds to + τ c t > 0, so that the tax rate would increase forever, which is not feasible. This is another reason that the steady state capital tax is 0. In the steady state, w, u l u c are also constant, so τ n t must also grow at a constant rate. This would lead to a labor subsidy to offset the growing consumption tax. In contrast, in the optimal steady state, τ c is constant but need not be 0. In fact, since there are two Euler equations for the households, and two of labor, capital, and consumption taxes can be used to achieve the same outcome. If human capital accrues the way capital does that is, it disappears in present value, then the steady state labor tax is 0 as well, which means that the government would have to tax a lot initially Optimal Taxation Under Uncertainty This follows Lucas and Stokey. In this economy, the technology is c t s t +g t s t = n t s t, the endowment is n t s t +l t s t =. We assume that there are complete credit markets, so that the government may have statecontingent debt. This gets tougher if all debt is risk-free. Taxes are also state-contingent and government spending depends on the state but is exogenous. This yields the government 25

26 budget constraint: g t s t + b t s t s t = τ n t s t w t s t n t s t + s t+ b t+ s t+ s t p t s t+ s t We set q 0 t+ st+ = q 0 t s t p t s t+ s t. Firms maximize c t s t + g t s t w t s t n t s t subject to c t s t + g t s t n t s t. This yields the zero profit condition w t s t =. The household present value budget constraint and first order conditions are: t=0 s t S t q 0 t s t c t s t = b 0 + t=0 s t S t qt 0 s t = β t π t s t u cs t u 0 s 0 u l s t u c s t = τt n s t τ n t s t n t s t q 0 t s t Combining all of these conditions yields the implementability condition: β t π t s t u c s t c t s t u l s t n t s t b 0 u c s 0 = 0 t=0 This gives the social planner s Lagrangian: L = t=0 s t S t φu c s 0 b 0 us t + φ u c s t c t s t u l s t n t + θt s t n t s t c t s t g t s t The sign on the Lagrange multipliers for the household budget constraint suggests that the constraint is actually Spending Income. Since the household will spend all its income anyway, this will make the multipliers non-negative for the social planner. The government budget constraint is of the form T axes Speing as usual. Then, the first order conditions with respect to c t s t, n t s t, c 0 s 0, n 0 s 0 are: u c s t + φu c s t + φu cc s t c t s t u lc s t n t s t θ t s t = 0 u l s t φu l s t + φ u lc s t c t s t + u ll s t n t s t + θ t s t = 0 u c s 0 + φu c s 0 + φu cc s 0 c o s 0 u lc s 0 n t s 0 θ 0 s 0 φu cc s 0 b 0 = 0 u l s 0 φu l s 0 + φ u lc s 0 c t s 0 + u ll s 0 n t s 0 + θ 0 s 0 + φu cl s 0 b 0 = 0 which characterizes the optimal solutions. Substituting for θ t s t, we find, for all t : + φu c s t + φu cc s t c t s t u lc s t n t s t = φ + u l s t + φu cl s t c t s t u ll s t n t s t 26

27 The only unknowns in this equation are φ and c t s t, since we may write n t s t in terms of c t s t and the known g t s t. The Laffer curve plots revenues versus tax rates. If the government spending lies completely above the Laffer curve, it is infeasible. In other cases, the government spending line intersects the curve in two places. If there are multiple solutions for the tax rate, then check them in the resource constraint. One might lead to negative consumption. That solution might minimize consumer utility instead of maximizing it. This curvature occurs because the problem is no longer concave, so that there can be more than one solution. Suppose g t s t = g t+k s t+k. Since φ is constant, the equation above implies that c t s t = c t+k s t+k. Thus, whenever the government expenditure is the same, the allocation and therefore the prices and taxes should be the same as well. This means there is no history dependence in the allocation. This depends on having complete markets. For example, if government spending is constant across time, then c t, n t, τt n across time. If b 0 = 0, then the tax rate must satisfy τ n = g n. are constant If we substitute: u l s t u c s t = τ n t s t n t s t g t s t = c t s t into the implementability constraint, we find that: β t π t s t u c s t nt s t g t s t τt n s t n t s t + b 0 u c 0 s t S t = 0 β t π t s t u c s t τ n t s t n t s t g t s t u c s 0 b 0 t=0 s t S t = 0 t=0 The inner term looks like the individual terms of the government budget constraint. Multiplying the first order conditions by c t s t and n t s t respectively and summing, we find: + φu c s t c t s t u l s t n t s t + φc t s t 2 u cc s t 2c t s t n t s t u lc s t + n t s t 2 u ll s t θ t s t c t s t n t s t = 0 Since utility is strictly concave, Q = c t s t 2 u cc s t 2c t s t n t s t u lc s t + n t s t 2 u ll s t < 0. Furthermore, θ t s t > 0 by non-satiation and φ > 0 as long as the present value of government expenditures exceeds its initial assets, b 0. Suppose g t is zero except in period T when it is g t = g > 0; this is a perfectly foreseen expense. Again, when t T, consumption, labor and taxes will all be constant, both before 27

28 and after the expense. Furthermore, because of the sign restrictions in the equation above and because c t = n t since g t = 0, we must have: c t s t u c s t n t s t u l s t > 0 for all t T. This means that u ls t u cs t < and therefore τ n t s t > 0. This means that taxes are collected every period, which smooths the taxes needed for the later government spending. At time T, c t s t < n t s t, and θ t s t c t s t n t s t > 0 while φq < 0. This makes the sign of c t s t u c s t n t s t u l s t indeterminate. Depending on the level of government spending and the particular utility function, taxes at time T could be positive, negative, or 0. This shows that, in this scenario, the government taxes before it must spend, goes into debt for the spending, and then uses future taxes to pay the interest on the debt as the debt s present value goes to 0 it never actually pays off the debt, because then it would be optimal to stop taxing. Suppose g t = 0 for all t T and g T > 0 with probability α. As before, the allocation and taxes will be the same for all t T, as well as in period T if g T = 0. Again, the tax rate will be positive for those periods. In this case, the government taxes in each period and then buys Arrow securities at time T that will pay off if g T > 0 and will go further into debt if g T = 0. In either case, the government will pay interest on the same debt with future taxes. Again, after T, the past doesn t matter, even without the process being Markov. Complete markets allow the debt to depend on the outcomes of g t. This is what allows the paths of taxes and consumption not to depend on history. If markets are not complete, debt may become history dependent. Once we have found the optimal consumption, labor and taxation paths, we may read off the path of debt, b t+ s t+ s t, from {τ t n t g t } and the interest rates. We may compute debt backward, based on b 0, τ, g, τ 2, g 2,..., or forward, with the equation: b t s t s t = τt n s t n t s t g t s t + τt+js n t+j n t+j s t+j g t+j s t+j qt+js t t+j s t+j s t = β j π t+j s t+j s t u cst+j u j=0 c s t u ls t+j ct+js t+j u c s t+j + g t+js t+j g t+js t+j s t+j s t where the second equality follows by substituting in the equilibrium expressions for prices, taxes, and labor. From this expression, we see that {g t+j } determines the future path of consumption and therefore the value of future debt. Furthermore, g t s t depends only on the current state, so that if s t is Markov, then debt must be Markov as well. In this case, bs t s t depends only on s t, and there is no history dependence. In a stochastic world, there is no steady state. Instead we may do asymptotics Zhu, 992. Given the household first order condition: u c s t = βe t u c s t+ τ k t+s t+ r t s t + δ 28

29 and the Ramsey allocation, {c t s t, n t s t, k t+ s t } s t,t, we may derive pricing kernels, ps t+ s t, wages, rents, taxes, and bond holdings. However, there is indeterminacy with respect to capital taxes, τ k t s t and bond holdings, b t+ s t+ s t ; given any solution, we may find an alternative. Let {ɛ t s t } be any stochastic process with E t u c s t+ ɛ t s t r t+ s t+ = 0. Define: For this new allocation: ˆτ 0 k = τ0 k ˆτ t+s k t+ = τt+s k t+ + ɛ t+ s t+ ˆbt+ s t+ s t = b t+ s t+ s t + ɛ t+ s t+ r t+ s t+ k t+ s t Government capital tax revenue and maturing government debt are unchanged. Government debt carried from time t to t + have the same present value at time t, evaluated at the original prices. The household intertemporal Euler equation continues to hold. 4 Labor Markets: the Lucas-Prescott Island Model Suppose we have: a continuum of agents, a large number of spatially separated islands, each with technology θfn, where f is constant across islands and satisfies the Inada conditions, θ <... < θ m, with πθ θ with P θ θ k θ = k i= πθ, θ i a decreasing function of θ so that good and bad times tend to be persistent, risk-neutral preferences, t=0 βt c t, and a moving technology so that, if a worker wants to move, it takes one period of foregone wages. Let βv u be the value of moving it is discounted by β because of the lost wages during moving. In equilibrium, this will be equal across all individuals since everyone has the same information and therefore would pick the same optimal islandss. βv u includes all the information about the distribution of populations and productivity across islands. Notice that after a productivity change, it takes one period for the labor force to change. It is also possible that productivity changes in two consecutive periods, so that people move in and then out. 29

30 Let x be the population on the island at the beginning of a period and θ the current productivity level on the island. Though we assume all workers know the productivity levels on all the islands, only one s own island appears in the value function. Let wθ, x = θf nθ, x be the wage on the island; this depends on productivity and on the number of employed workers, nθ, x note that nx, θ x, since the number of workers can t exceed the labor force at the beginning of the period. Assuming that V u is known, the value function is: V θ, x = max{βv u, wθ, x + βev θ, x θ, x} The first option corresponds to moving, while the second option corresponds to working this period and staying on the island. We consider 3 cases: n < x: In this case, people are leaving the island, so V θ, x = βv u. However, not everyone leaves because of the Inada conditions. Thus, we must have wθ, x+βev θ, x θ, x = βv u. n = x: Since everyone is working, we know that the wage must be wθ, x = θf x. x > x: Since people are moving in, we must have EV θ, x θ, x = βv u. This means that the value function is V x, θ = θf x + βv u. x = x : This determines the population next period, so that V x, θ = θf x + βev θ, x θ. This yields the value function: V θ, x = max{βv u, θf x + β min{v u, EV θ, x θ}} We then use the three cases to determine the equilibrium flows between islands: n < x: In this case, x = n. We choose n so that people are indifferent between moving and staying: βv u = θf n + βev θ, n θ We define x + θ as the n that solves this equation for a given θ. n = x, x > x: Note that, no matter the current value of x, x will be the same; otherwise, more people would move in. This means that βv u = βev θ, x θ. This equation determines x θ = x. n = x, x = x: In this case, there is no movement. This will only happen if x θ x x + θ. This defines a range of inactivity in which wages are high enough that no one moves out but not so high that people move in. 30

31 Holding V u fixed, x θ < x + θ and both curves are upward-sloping in θ since more productivity will lead to a better wage with the same number of workers. For any x, the population will move to the closer end of the range of inactivity. If x + θ x θ m, then any x [x θ m, x + θ ] is in the range of inactivity. Thus, once the population reaches this interval, it will never move again. This is an equilibrium with no labor movement. On the other hand, if x + θ x θ m, then the ergodic set will be x {x θ i, x + θ i }, with x + θ x x + θ m since once x is inside this range, it will never go below x + θ. Since V u is fixed in equilibrium, this is a finite number of possible points. Note that not all transitions of θ will lead of a movement, if [x θ i, x + θ i ] and [x θ j, x + θ j ] overlap. In either case, the stationary equilibrium is an equilibrium across all of the islands; an individual island s productivity and therefore population keep changing, but the distribution of productivity and population is constant across all islands. βv u determines the welfare of people on the islands the the population and on the nonoptimal islands?. If βv u increases, the population on non-optimal islands increases. Given V u, we may compute the transition function: Γθ, x θ, x = πθ θix = x + θ, x > x + θ +πθ θix = x θ, x < x θ +πθ θix = x, x θ x x + θ Then,the distribution of population across islands can be found from the transition function: Ψ t+ θ, x ; V u = Γθ, x θ, xψ t θ, x; V u θ x We iterate on Ψ t until we find the stationary distribution, Ψx, θ; V u. Then, the population must be X = xψθ, x; V u x X θ,...,θ m Each value of V u can be mapped to some x. Thus, given the actual population size, we may determine V u in equilibrium. Remember that if Γ is a transition function, the stationary distribution satisfies q Γ = q. Because we have made minimal assumptions about f, profits θfn θf nn need not be 0. We could give the profits out in an arbitrary way, since agents are risk-neutral and have no disutility of labor. 3