Government

1. Government • Barro published a series of papers developing a theory of government, specifically, the effects of government purchases and looking at the impact of financing methods. • First paper examines the differences between financing methods. Later papers look at a theory of how a government might choose to finance itself and how government expenditures impact the economy. • Financing discussions is referred to as Ricardian Equivalence.

2. “Are Government Bonds Net Wealth?” Barro JPE 1974 Model Setup: People live for two periods and are identical. N people exist in each generation. People work only while young (period 1), earning w. People provide one unit of labor supply. People have static expectations. Definitions: A = Assets, r = Real Interest Rate c = Consumption, w = Real Wage

3. Superscripts: y = Young, o = Old Subscripts: Generations 1 and 2. Budget Constraint of Generation 1 Currently Old: Bequest from the previous generation. Bequest to succeeding generation. Budget Constraint for Generation 2 Person Currently Young: Budget Constraint for Generation 2 Person Currently Old:

4. Bequest Motive: motivated by the desire to achieve a stipulated welfare level for the household’s immediate descendants. Solution to Optimization Problems: Generation 1: Generation 2:

5. Additional Relationships: Supply of capital arises from competitive optimality conditions for capital and labor where marginal products equal factor prices. Demand for assets arises from bequests motive and the need to finance second period consumption. Output is paid out to capital owners and to labor. Output is used for capital accumulation and consumption.

6. Government Debt: government issues one-period bonds paying interest rB. Bonds are given to generation 1 old persons and financed with taxes to pay interest levied on generation 2 persons while young. Bonds are retired by taxes levied on generation 2 persons while old. Budget Constraint of Generation 1 Who Are Old: Generation 2 Budget Constraints:

7. Generation 2 Overall Budget Constraint: Barro then argues that the old will simply save the bonds, passing them on to the younger generation in order to maintain their targeted welfare level. The crucial condition is that there be operative bequests. If not, the old will view their windfall as an expanded opportunity set, spending the bonds. Social Security: if old persons in generation 1 receive S with payments levied on generation 2, currently young. Same results arise – generation 1 passes on higher bequests to cover payments incurred by generation 2.

8. Inheritance Taxes: suppose now that inheritances are taxed at the proportionate rate . That is, Pre-Tax Transfer: After-Tax Transfer: Barro argues that a higher tax rate will lower bequests but that, as long as the bequest motive is operative, the results of the previous models go through. • Ricardian Equivalence may fail due to: • imperfect capital markets. • myopia • inoperative bequests

9. “On the Determination of the Public Debt” – Barro JPE 1979 Objective: offers a theory of public debt issue. Government Budget Constraint: Definitions: G = Real Government Expenditures, r = real Interest Rate,  = Lump-Sum Taxes, b = Real Stock of Government Debt. Government budget constraint implies that

10. Collection Costs: Minimize Optimality: Transitory changes in government expenditures, such as during wars, would be financed by bond issue to avoid the costs of raising taxes and thus keeping the tax-income ratio constant.

11. The longer that government expenditures are above trend, the more they are financed by current taxation. The longer that income is above trend, the less transitory taxation occurs (or, equivalently, the more permanent taxation there will be).

12. “Output Effects of Government Purchases” – Barro JPE 1981 Objective: develops a theory of government expenditures. Households: Government:

13. Households and Government: Transitory G raises the productivity of the economy, raising output supply. An increase in permanent government expenditures is equivalent to an increase in the present value of taxes paid by the household. People work more as a result. Aggregate System:

14. r Ys r* Yd Y* Y

15. Output Effects of Transitory Change in G Ys r Yd Y

16. Barro argues that an increase in G will raise the real interest rate, reducing private demand for goods, as government takes a larger share of the economy’s output - The real interest rate rises in wars.

17. Output Effects of Permanent Government Purchases Ys r Yd Y

18. The reduction in permanent income is substantial enough to induce an increase in labor supply sufficient to keep the real interest rate approximately constant.

19. Government Debt and Taxes – Sargent Chap. XIII max subject to

20. Taxes are AR(1) or a martingale if time preference and the discount rate are equal. Assume that government expenditures obey Sargent derives

21. Taxes are a function of all future deviations of government expenditures from their mean. If Then

22. Suppose that b=1: there are only transitory changes in government expenditures. Shocks (transitory changes in government expenditures) are financed primarily by debt but there will also be a permanent increase in taxes.

23. Blanchard and Fischer, Chap. 3, Sec. 3.2-3.5 A Fully-Funded Social Security System In a non-altruistic OG Model, in equilibrium we have In a fully-funded system, the government raises revenue from the young, paying it to the old.

24. If the young pay dt, and if the government invests this as capital, paying bt=(1+rt)dt-1 to the old whose contribution was invested in period t-1, the equilibrium conditions for consumption and saving are This is the same equilibrium as before. To see this eliminate st from the first equation using the second. Implication: social security has no effect on saving or capital accumulation.

25. A Pay-As-You-Go System Here bt+1=(1+n)dt and the government pays a rate of return of n because there will be more people alive in the future to make contributions to the system. Blanchard and Fischer show that savings will decline and the capital stock will fall in the case where the steady state is unique.

26. Bequests and Unfunded Social Security: With operative bequests, transfers to the old, financed by taxes on the young, will be offset by additional bequests from the old to the young. • A Model of Perpetual Youth • Builds in the uncertainty of dying that faces every agent • It will be assumed for simplicity that the probability of dying is independent of age. • p denotes the probability of dying • Time until death, X, obeys the exponential density

27. p can be thought of as an index of the individual’s horizon: as p goes to zero, the horizon for the household becomes infinite. At each instant of time, a group of individuals, referred to as a cohort, is born, and it decrease in time at the exponential rate p, A normalization is convenient: each cohort is of size p so that, at ant time t, the size of the population is unity. Insurance is assumed to be available (there are no bequests) that works as follows: agent receives a payment from the insurance company and, if the agent dies, all of their wealth goes to the insurance company.

28. Individual Optimization: The last condition is the no-Ponzi-game requirement. Optimality requires that

29. Aggregate Behavior: Consumption Aggregate System: The parameter >0 captures the fact that income eventually declines with age. Note that human wealth is discounted at a rate above the real interest rate.

30. Relevance to Ricardian Equivalence: The government discounts its activities at the rate r. If we impose the No-Ponzi-Game condition Then the government’s budget constraint is The government uses a discount rate below that used by the public.

31. This is because the individuals faces the possibility that they will not be around when the taxes are levied in the future (1-e-ps is the probability that they won’t be alive at time s in the future). Therefore, Ricardian Equivalence does not hold unless p and  are zero. Note to Students: Skip Section 3.5.