Government The government faces an exogenous sequence {g t } t=0

Part 6 1. Borrowing Constraints II 1.1. Borrowing Constraints and the Ricardian Equivalence Equivalence between current taxes and current deficits? Basic paper on the Ricardian Equivalence: Barro, JPE, 1974. This material is from Sargent and Ljungqvist, Chapter 10. Wereturntotheanalysisofgovernmentintheeconomy,butherewerelaxthe assumption of a balanced budget. In this analysis we concentrate on lump-sum taxation, and here the notation will be τ t. (equivalent to φ t before). Government The government faces an exogenous sequence {g t } of real expenditures and has an initial debt b 0. The sequence of budget constraints for the government is g t + Rb t = τ t + b t+1, t 0, where R is the risk-free rate on a one-period bond. We impose the transversality condition lim t R t b t =0. A government policy is denoted by {g t,τ t,b t+1 }. There is no optimization problem for the government. The only requirements on this policy is that it satisfies the constraints. The intertemporal budget constraint of the government can be obtained by substituting the expressions for b t,t=1, 2,... from the constraints for periods t =1, 2,..., into the constraint for period 0. The resulting expression, using the transversality condition, is b 0 = 1 R R t (τ t g t ). Households The number of identical households in this economy is normalized to one. Each chooses {c t,a t+1 } so as to maximize P β t u(c t ), 0 <β<1,

where βr =1and u satisfies the standard conditions, and in particular lim c 0 u 0 (c) =, subject to c t + a t+1 = Ra t + y t τ t, t 0, a given value of initial assets a 0, an exogenous income sequence {y t }, P R t y t <, and a borrowing constraint yet to be chosen. To close the model we have two possibilities: (a) We can assume that this is a small open economy facing an international capital market, where borrowing and lending can be carried out at the rate R. In the world markets, the constraints on the government and household sector are the corresponding transversality conditions, which imply the natural borrowing constraints. (b) It can be a closed economy (or a large open economy), but then the assumption that R =1/β has to be changed. Let s adopt the Option (a) first, and later address Option (b). Definition of an equilibrium in this economy: Given b 0 and a 0, an equilibrium consists in a household plan {c t,a t+1 } and a government policy {g t,τ t,b t+1 } such that the intertemporal budget constraint of the government is satisfied, and, given P R t τ t, the household plan solves its optimization problem. The Ricardian Equivalence Proposition Assume that both households and the government are constrained by the corresponding ª transversality conditions. Given an initial condition (b 0,a 0 ), let c 0 t,a 0 t+1 and ª g 0 t,τ 0 t,b 0 t+1 be an equilibrium. Consider another tax policy satisfying R t τ 1 t = R t τ 0 t. The Ricardian Equivalence Proposition is that then, c 0 t,a 1 t+1ª and g 0 t,τ 1 t,b 1 t+1ª is also an equilibrium. (i.e., the two tax policies are equivalent.) 2

We can show that thee proposition holds as follows. Under the natural borrowing constraint, the households will choose the consumption plan subject to the intertemporal budget constraint a 0 = 1 R t (y t τ t )+ 1 R t c t, R R or, a 0 = 1 R t (c t y t )+ 1 R t τ t. R R Given that the consumption plan depends only on the present value of taxes, and that the alternative policy does not change it, the household does not change the consumption plan. Then, the budget constraints c t + a t+1 = Ra t + y t τ t, are used to construct the new a t+1 series, where a t+1 is adjusted minus-one to one with the new taxes. When τ 1 t >τ 0 t,a 1 t+1 <a 0 t+1bythesameamount,andthe opposite at times when taxes were reduced. The new sequence for government debt is constructed similarly. From g t + Rb t = τ t + b t+1, t 0, τ 1 t >τ 0 t,b 1 t+1 <b 0 t+1by the same amount, and vice-versa. Extensions of the basic setup We check in each of the following extensions whether the Ricardian Equivalence holds. 1. A closed economy (Option (b) above): In this case, the interest rates are not constant. Using the equilibrium condition in the financial market a t = b t, adding up the budget constraints of household and government c t + a t+1 = R t a t + y t τ t, g t + R t b t = τ t + b t+1, 3

yields y t = c t + g t. The Ricardian Equivalence proposition should be extended as follows. Given the initial conditions (b 0,a 0,R 0 ), let c 0 t,a 0 t+1,r 0 t+1ª and g 0 t,τ 0 t,b 0 t+1ª be an equilibrium. If we have another tax policy satisfying where à ty j=1 R 1 j! 1 τ 1 t = à ty j=1 R 0 j! 1 τ 0 j, j=1 0Y R j 1, then c 0 t,a 1 t+1,rt+1ª 0 and g 0 t,τ 1 t,bt+1ª 1 is also an equilib- rium. To show that the interest rate sequence is not altered we use the Euler equations: u 0 (c t )=βu 0 (c t+1 )R t+1, t 0. In the initial equilibrium, c 0 t = y t g 0 t, t 0 holds. The policy change does not alter this equation. Hence, interest rates do not change either. The Ricardian Equivalence holds. 2. Ad-hoc borrowing constraint. Back to the open economy (In the closed economy with a representative agent y t = c t + g t will always hold). If the condition a t 0 prevails (or other constraint tougher than the natural one), the Ricardian Equivalence will in general not hold. In this case, not only the present value of taxes matters, but also the current liquidity situation. If initially, the borrowing constraint holds, i.e., a 0 t+1 =0is the optimal choice, c 0 t = y t τ 0 t + Ra 0 t, τ 1 t >τ 0 t, requires that c 1 t <c 0 t. However, if the agent starts with positive assets, the Ricardian Equivalence will hold for tax changes which do not lead to a corner where a t+1 =0. 3. Finite horizon for the individual. 4

The proposition is based on the assumption that T in P T R t τ t is the same for households and the government. If T h <T g then the policy change may not be neutral. Barro (1974) argued that planning horizons of households may practically be, if parents take into account the utility of their children, and use bequests B as a way to transferring resources to the next generation. If we define V (B t,y t ) as the maximized utility of a one-period-lived agent of generation t, given the bequest received and current exogenous income, then V (B t+1,y t+1 ) is the maximized utility of the next generation. When parents take into account the (discounted) welfare of their children, then their utility function is u(c t )+βv (B t+1,y t+1 ), 0 <β<1. By definition then, subject to V (B t,y t )=Max{u(c t )+βv (B t+1,y t+1 )}, c t + B t+1 y t τ t + RB t, and B t+1 0. Negative bequests are ruled out. In a recursive formulation, this problem is equivalent to the infinite-horizon version above with B t = a t, and the no-borrowing condition a t 0. Hence, the Ricardian Equivalence will hold only if bequests are positive, and the tax changes do not lead to the corner solution of no bequests. This issue can be analyzed more in detail in an overlapping-generations model, as in Barro (1974). 4. Distortionary taxes, uncertainty. 1.2. Borrowing constraints and labor supply Consider the problem of a household with the utility function P P βt c 1 σ t (1 n t ) 1 λ / (1 σ) if σ 6= 1, βt ln c t +ln(1 n t ) if σ =1. and the sequence of budget constraints R t b t + c t w t n t + b t+1, 5

where b t is the household s debt at the end of period t. Concavity of the utility function requires that (1 σ)(1 λ) is positive. Thehouseholdtakespricesasgiven. If the borrowing constraint is of the natural type, or, alternatively, the transversality condition holds, the intertemporal budget constraint is à ty! 1 à ty! 1 c t = w t n t R 0 b 0, with 0Y 1. R 1 τ The first-order conditions for utility maximization are c σ t (1 n t ) 1 λ = βc σ t+1 (1 n t+1 ) 1 λ R t+1, 1 λ c t = w t. 1 σ 1 n t The solution proceeds by solving the labor condition for n t and n t+1, substituting the expressions into both the Euler equation and the intertemporal budget constraint, in which now only consumption and wages appear. The remaining task is then the main one of determining the path for consumption. The point here: 1. If future wages increase, c t will jump upwards given the household s ability to borrow. 2. Then, for any given w t,higherc t should be accompanied by lower n t (wealth affect of a higher future wages). Let us introduce now the ad-hoc borrowing constraint b t+1 0. When the profile of future wages increases, if the borrowing constraint binds, c t cannot go up as in the previous case. 1. When the borrowing constraint binds, c t is lower than when borrowing is possible, and then n t will be higher. 2. When the borrowing constraint binds at least from the previous period, c t = w t n t. Substituting this into the labor supply condition yields µ 1 λ 1 σ wt n t 1 n t = w t 6 µ 1 λ 1 σ nt 1 n t =1.

Hours worked are constant, regardless of wage movements. Empirical implications of borrowing constraints for labor supply elasticity. 3. If the borrowing constraints binds this period, but did not bind last period, then c t = w t n t Rb t, b t < 0, µ 1 λ wt n t Rb t w t =. 1 σ 1 n t Higher future wage does not affect hours, but a current higher wage increases hours. 1.3. Borrowing constraints in general equilibrium For borrowing to take place in general equilibrium we need differenciated households. Here we assume different rates of time preference. Exogenous output Assume an economy with two groups of agents of the same size, which are identical in all respects, except that the rate of time preference of one group is higherthantherateoftimepreferenceoftheother. Theutilityfunctionofthe two groups are ˆβ t u(ĉ t ), β t u( c t ), where ˆβ < β, and u has the standard properties. Each household in both groups receive the same endowment y, andallstart period 0 with no assets, i.e., ˆb0 = b 0 =0. The budget constraints are, correspondingly, ĉ t y + ˆb t+1 R tˆbt, c t y + b t+1 R t bt. Equilibrium in this economy requires ĉ t + c t =2y ˆbt + b t =0. 7

n What is the equilibrium path R t+1, ĉ t, c t, ˆb t+1, b o t+1? n Definition: A competitive equilibrium consists of sequences of allocations ĉ t, c t, ˆb t+1, b o t+1 and returns {R t+1 } that solve the impatient and patient problems and all markets clear, i.e., ĉ t + c t =2y, ˆbt + b t =0, for all t 0. We turn now to compute and interpret this equilitbrium path. The intertemporal budget constraints for both households are where ĉ à t ty! = y 0Y =1. This implies that 1 à ty! = ĉ à t ty! = c à t ty!, c à t ty!. (1.1) To proceed we use the Euler equations, assuming that u(c) =lnc. The Euler equations for the two groups are Similarly, ĉ t = ˆβR t ĉ t 1, ĉ t = ˆβR tˆβrt 1 ĉ t 2,... ĉ t = ˆβ t à ty! ĉ 0. (1.2) à ty! c t = β t c 0. (1.3) 8

Using the Euler equations into (1.1): Ã ty! ˆβ t ĉ 0 Ã ty! = Ã ty! β t c 0 Ã ty! ˆβ t ĉ 0 = β t c 0 ĉ 0 = 1 ˆβ 1 β c 0 φ c 0. ˆβ < β φ>1 ĉ 0 > c 0. Using ĉ 0 = φ c 0 and the resource constraint for period 0 yields c 0 = 2 1+φ y<y ĉ 0 = 2φ 1+φ y>y. Hence, in period 0 the impatient borrows and the patient saves. The next step is to solve for the interest rates using the solution for the starting consumption levels, the Euler equations, and the resource constraints. Let s start with R 1. ĉ 1 = ˆβR 1 ĉ 0 = ˆβR 2φ 1 1+φ y, 2y ĉ 1 = βr 1 c 0 = βr 2 1 1+φ y. Adding up the two equations and dividing by y yields µ 2φ 2=R 1 1+φ ˆβ + 2 1+φ β 1 R 1 = φ 1+φ ˆβ + 1 1+φ β. 9

Hence, the market discount rate is a weighted average of the two discount rates. To compute the future interest rates we need to show (left for the homework) that the interest discount factors are the following weighted averages of the two subjective discount factors: Ã ty! 1 = φ 1+φ ˆβ t + 1 1+φ β t, φ > 1. We can express this equation as ty = 1+φ φˆβ t t. (1.4) + β and given that ty t+1 = Y 1 R t+1, 1 = φˆβ t+1 + βt+1 R t+1 φˆβ t + β t = φˆβ t φˆβ t + β ˆβ β t t + φˆβ t t β. + β The market discount rate is always a weighted average of the two discount rates, but, given that ˆβ < β, the weight of the higher discount rate declines over time, and φˆβ t φ(ˆβ/ β) t lim t φˆβ t t =lim t + β φ(ˆβ/ β) t +1 =0. Hence, at the limit, the interest rate is set by the patient s rate of time preference. 1/R = β The profiles of consumption for the impatient and patient can be computed using the Euler equations in (1.2) and (1.3) and the discount factors in (1.4). It can be shown (left for the homework) that lim t ĉt =0, lim t c t =2y. 10

zero, and the consumption of the saver increases over time, converging to total output. Thebehaviorofthedebtcanbecomputedasfollows: Ã ty! 1 ˆbt+1 = ˆb1 =ĉ 0 y,... Ã ty! ˆbt+1 =ĉ t y + R t (ĉ t 1 y)+r t 1 R t (ĉ t 2 y)... + (ĉ 0 y), Ã tx iy! 1 (ĉ i y). i=0 Using (1.4), solving the finite geometric sums, and rearranging, we get β ˆbt+1 = y ³ ³ φ(ˆβ/ β) t +1 1 β ³1 (ˆβ/ β) t+1. t, ˆb t+1 ˆb = β ³ 1 β y. Interpretation: Given that at the limit R =1/ β, this equation can be written as 1 ˆb = R 1 y. This is the natural borrowing constraint, i.e., the debt which leaves no consumption. The periodical interest payment is (R 1) = y. The transversality condition holds given that the debt converges to a constant, and the interest rate is positive. Imposing an ad-hoc borrowing constraint Assume now the borrowing constraint b t+1 0. This render the previous solution unfeasible. Conjecture: If the impatient doesn t have any assets, the equilibrium interest rates are now R t+1 =1/ β, t 0 11

Under this conjecture, R t+1 < 1/ˆβ, t 0, and hence the impatient would like to borrow permanently. However, given the borrowing constraint, ĉ t = y, t 0. The patient, in contrast, is not constrained, but because of the lack of saving or borrowing by the impatient, he becomes similar to the representative agent in a standard economy. With the logarithmic utility function, the Euler equation of the patient is c t+1 = βr t+1 c t. Equilibrium in the output market implies that c t = y for all t. Hence, 1= βr t+1, R t+1 =1/ β. The interest rate is set according to the patient, who is the only agent free to trade in the financial market. In the present case this equality holds immediately, and not asymptotically as in the previous case because b t 0. If b t φ>0, there would be a gradual convergence. The conjecture does represent an equilibrium because output and debt markets clear, and the first-order condition of the patient holds and the impatient does his best given the constraint. Two basic directions for extending this framework: 1. Adding leisure to the utility function, i.e., making income endogenous, and 2. Introducing capital accumulation (shortly discussed). 1. Endogenous labor Consider the same framework as above, but now the utility and production functions are: u(c t, 1 n t )=lnc t + ϕ ln(1 n t ), ϕ > 0, y t = wn t, w > 0. 12

The budget constraints are now ĉ t wˆn t + ˆb t+1 R tˆbt, c t wñ t + b t+1 R t bt. and the equilibrium conditions are ĉ t + c t = w(ˆn t +ñ t ), ˆbt + b t =0, t 0. TheEulerequationsarethesameaswheny t = y, but now there are additional conditions regarding the consumption-leisure choice: ϕĉ t 1 ˆn t = w wˆn t = w ϕĉ t, ϕ c t 1 ñ t = w wñ t = w ϕ c t. Case (a): Only the transversality condition applies. Substituting the conditions for labor into the budget constraints, to get rid of the n s, and rearranging, yields ĉ t = 1 ³ˆbt+1 R tˆbt + w c t = 1 ³ bt+1 R t bt + w. It is left as an exercise to show that the model can be solved similarly as in the case of exogenous output, and that consumption and leisure of the impatient go over time to zero. Hence, at the limit, the debt is b = w/(1/ β 1). For the patient, consumption and leisure at the limit are determined as follows: Ifthereisaninteriorsolution, w = c = w(1 + ñ ), ϕ c 1 ñ ñ = 1 ϕ, 13,

c = w(1 + 1 ϕ )= 2 w. This requires that ϕ<1. If ϕ 1, ñ =0, c = w. Case (b): The borrowing constraint b t+1 Ω, t 0, holds, and ˆb 0 = Ω, b0 = Ω. Assumption (i): Ω < w for all {R R t 1 t}. Note that the equilibrium sequence {R t } is perfectly foreseen at time 0. Conjecture: The equilibrium sequence of interest rates is R t = R =1/ β for all t 0. The key point here: This conjecture implies that the borrowing constraint on the impatient binds. The impatient wishes to borrow more at all times, but given the borrowing constraint, the consumption of these agents is ĉ t = wˆn t (R 1)Ω. Substituting this expression into the consumption-leisure condition we get w = ϕ [wˆn t (R 1)Ω] ˆn t = 1 1 ˆn t + ϕ (R 1) w Ω, ĉ t = 1 w 1 (R 1)Ω. Consumption and labor of the impatient are constant for all t 0, and the assumption Ω < w implies that ĉ R t 1 t > 0 and 0 < ˆn t < 1. The labor supply of the impatient has peculiar properties created by the wealth effect from interest rate payments which disappears when Ω =0. The interest payments have a positive effect, due to the liquidity constraint. The wage has a negative effect, given that as it increases, less effort is required to pay the interest. For the patient, the conjectured interest rates implies that c t is constant over time. The consumption and labor supply of the patient are then given by w = c = b t+1 R b t + wñ t, ϕ c 1 ñ t wñ t = w ϕ c. 14

Substituting the condition for labor into the budget constraint yields c = 1 ³ bt+1 R b t + w. Given that in equilibrium, b t+1 = b t = Ω, the budget constraint becomes c = 1 [w +(R 1)Ω]. Substituting this expression for consumption expression into the labor condition yields ñ t =1 ϕ c w ñ t = 1 ϕ (R 1)Ω. w Forthesaver,thewagehasapositiveeffect and the interest rate has a negative one. Note that the resource constraint ĉ t + c t = w(ˆn t +ñ t ) is also satisfied (Walras 2 Law). Total consumption is w, which is equal to total output. For an interior solution for the saver s labor, 1 ϕ (R 1)Ω 0. w With a borrowing constraint, the condition for an interior solution depends not only on ϕ, but also on R, w, Ω. (***) Summary: The conjecture we started from satisfies the equilibrium conditions. Additionally, given the starting point where the impatient s debt equals the constraint, the economy is from t =0onwards at a steady state with a binding borrowing constraint on the impatient. Variable productivity Using the previous case of a steady state with a binding borrowing constraint as a starting point, we consider now productivity varying around the level w. Specifically, assume the exogenous sequence {w t }, w t [w ε, w + ε], where ε is sufficiently small (in particular ε<w), so that the borrowing constraint on the impatient will always be binding. 15

Additional assumptions: ii. Either w t > Ω for all t, or w t < Ω for all t, ii. w t w = w as t, iii. The economy at time 0 is at the steady state derived above. This requires that R 0 = R 1 = R =1/ β, (***) and w 0 = w 1 = w. Additional assumption: iv. Ω < wt, for all t 0. Note that all future values of w R t 1 t and R t are known at time 0. This replaces assumption (i). Under the assumption that the borrowing constraint on the impatient always binds, we generalize the solution of the model above by computing the equilibrium sequence {R t+1 }. The impatient s consumption and labor at any t 0 are given by ĉ t = w tˆn t (R t 1)Ω, w t = ϕ [w tˆn t (R t 1)Ω] 1 ˆn t, (R t 1)Ω, w t ĉ t = 1 w t 1 (R t 1)Ω. ˆn t = 1 + ϕ Note that assumption (iv) implies that ĉ t > 0 and 0 < ˆn t < 1. For the patient, the budget constraint and first-order conditions are c t = b t+1 R t bt + w t ñ t, c t+1 = c βr t+1, t w t = ϕ c t w t ñ t = w t ϕ c t. 1 ñ t Substituting the condition for labor into the budget constraint, to get rid of ñ t, yields c t = b t+1 R t bt + w t ϕ c t, c t = 1 ³ bt+1 R t bt + w t. 16

In equilibrium, b t+1 = b t = Ω, and hence this budget constraint becomes c t = 1 [(R t 1)Ω + w t ]. (1.5) Substituting (1.5) into the labor condition ñ t =1 ϕ ct w t ñ t = 1 ϕ (R t 1)Ω. w t yields We turn now to solve for the interest rate sequence. Using the patient s Euler equations and (1.5) for c t and c t+1 we get (R t+1 1)Ω + w t+1 (R t 1)Ω + w t = βr t+1. Tosolveforthecurrentinterestrate(R t+1 )weneedthevalueoflastperiod s interest rate and so on backwards. Given assumption (iii) that the economy in period 0 is at a steady state we should have (R 1 1)Ω + w 1 (R 0 1)Ω + w = βr 1 (R 1)Ω + w (R 1)Ω + w =1. Then, (R 2 1)Ω + w 2 (R 1)Ω + w = βr 2 R 2 = w 2 Ω β (w Ω), and so on for R 3,R 4... In general, R t is predetermined at time t and R t+1 = w t+1 Ω. β (w t Ω)+Ω ³ βrt 1 A current productivity shock reduces the interest rate and a future productivity shock increases it. Summary: 17

ˆn t = 1 + Ω ϕ (R t 1), w t (R t 1), w t ĉ t = 1 w t Ω 1 (R t 1), ñ t = 1 Ω ϕ c t = 1 w t + Ω 1 (R t 1). RBC with this structure. A productivity shock increases hours worked of high income people and reduce hours worked of low income people. Aggregate labor remains constant. Effects of relaxing the borrowing constraint, i.e., increasing Ω (still satisfying assumption iv Ω < w t R t 1 ). Who is worse off? Consider now a steady state with the natural borrowing constraint: Ω = w R 1. Thesolutioninthiscaseis ˆn t = 1 + ϕ w (R t 1) w t (R 1) 1 ñ t = 1 ϕ w (R t 1) w t (R 1) 1 ϕ ĉ t = 1 w t 1 (R t 1) (R 1) w 0 c t = 1 w t + 1 (R t 1) (R 1) w 2 w 2. Introducing capital accumulation The incorporation of a production technology operated with capital, but without exogenous labor, was addressed by Becker (QJE, 1980). He showed that at the limit only the most patient agent will hold all the capital, and hence, the marginal productivity of capital (the interest rate) will be equalized to his rate 18

of time preference. All the other agents will receive wages only. By not having leisure in the utility function, leisure is zero at all times. In this case however, as capital is the only asset which cannot be negative, the model is equivalent to one with the ad-hoc borrowing constraint of zero borrowing. Aiyagari (1994). Additional capital accumulation of capital due to a borrowing constraint. Precautionary saving. Krusell and Smith (JPE 1998) extended the Aiyagari model in two respects: i. Incorporated an aggregate productivity shock, so that they could study the precautionary savings over the business cycle, and more importantly, in the present context, ii. incorporated three levels of type preference, with a slow but stochastic evolution over time. Two main results here: First, agents who are currently impatient tend to be borrowing constrained, and hence consume their current income. This explains the excess sensitivity of consumption to income found in the data. Second, wealth distribution is much more unequal than when the only source of inequality are individual income shocks. The degree of inequality is governed primarily by the persistence in the rates of time preference. 19