On The Determination of the Public Debt

1. On The Determination of the Public Debt Robert Barro 1979

2. Overview • Accepts that the Ricardian Invariance Therom is a valid first-order assumption • This paper introduces a discussion of second order conditions to examine the effects of ‘excess burden’ of taxation • Several typical features of public debt analysis are dismissed by the Ricardian Therom • Thus, the paper will focus on less common issues dominated by first order effects

3. Hypothesis • That there is a positive effect of temporary increases in government spending on debt issue • The negative effect of temporary increases in income • The growth rate of debt will be independent of the debt-income ratio and would only be slightly effected by the level of government expenditure

4. Summary of Results • Used data on US post-WWI public debt issue • Finds that the empirics agree with the proposed hypothesis • Debt issue since WWI seems to be explained by a small number of variables

5. Model • Model Characteristics • Applies only to large nations with exogenous populations • Government must finance through either current taxation or public debt issue • Variables • Gt - Volume of Real government expenditure in period t • t - Real tax revenue generated in each period • Yt - Aggregate real income • bt - real stock of public debt outstanding at the end of t • P - Price level and is assumed to be constant • r - Real, constant, rate of return on public and private debts

6. Budget Constraint • In each period • Budget constraint at date t

7. Determination of Burden • Collection Cost in period t • Zt - The real cost incurred in period t. • Present value of Collection Costs

8. Optimal Tax Levels • Optimization requires that 1… are chosen to minimize the present value of revenue-raising costs • This requires that the marginal cost of raising taxes be the same in all periods • This implies that /Y is equal in all periods

9. Constant Income and Government Expenditures • When Y is constant over time the constancy of /Y implies constancy of . • If G is constant as well then  is determined immediately from Equation 2 • Combining with Equation 1 dictates that the budget always be balanced and thus steady state value of debt is determined only by its initial value and not as a function of G, Y, r, etc

10. Constant Rate of Growth of Income and Government Expenditure • If Yt = Y0(1+p)t than in order for the present value of future income to be finite r > p • It is assumed that Gt = G0(1+)t thus if G/Y<1 is true ≤ p < r • Thus p =  is the only equality that provides finite, steady state growth of G/Y

11. Introducing Taxes • The tax-income ratio remains constant, thus taxes grow with income and t=0(1+p)t • Combining this with the initial budget constraint leads to a formula for the current budget deficit:

12. Transitory Income and Government Expenditure • Assume G1=(1+)G0(1+p) and that Y1=(1+u)Y0(1+p) • The equation for the determination of taxes in all periods is as follows

13. Transitory Income and Government Expenditure • The longer a “transitory” period of government spending is expected to last the higher the current taxation will be • At the same time the longer a “transitory” period of government income the lower the current taxes

14. Transitory Income and Government Expenditure • Growth of Budget Deficit in transitory periods: • The deficit grows dependent upon the departure of the current government spending from normal and the proportional departure of income from normal

15. Changes in Prices • Price changes are treated exogenously • Future prices increase to P1 and remain static • Equation 1 is now modified to be: • The primary effect is that changes in the price level, or inflation rate, do not change the growth rate of the nominal debt

16. Changes in Prices II • If prices are assumed to change at a constant rate Pt=P0(1+)t • Equation 1’ remains almost the same with the exception that the growth rate of nominal debt increases by  • This changes Equation 7 to the following: • As a result, when inflation is included nominal debt grows by p+ 

17. Changes in Rate of Return • If r is not equal to r0 the analysis remains the same as long as debt is measured at market rather than par values • Basic result is that increasing r above the average of previous rates reduces the growth rate of debt in terms of par values

18. Empirical Analysis • Bt is the stock of nominal debt at the end of the calendar year t • B¯t is the average amount of debt outstanding • t is the average anticipated rate of inflation • Pt is the average price level • Gt is real federal government expenditure • Yt is aggregate real income (GNP) • Y¯t is the level of normal income

19. Variables Continued • 0: Equal to p as long as the growth of Y and G are equal • 1: Equal to unity • 2: Equals the [(1+p)/(1+r)]k term in equation 8 • 3: Equals the [(1+p)/(1+r)]n term in equation 8

20. The Data • Data comes from US public debt information post 1917 • B is measured as the outstanding stock of federal debt at the end of each calender year • These values are not adjusted for changes in rates of return •  is constructed based on the estimated GNP deflator from Barro 1978 • Uses this for the sample 1922-1976 with a dummy for pre-1941

21. Table 1

22. Empirical Results

23. Table 3

24. Conclusions • Areas of future research: incorporation of currency issue, applications of optimal taxation to public debt determination, and a treatment of uncertainty about future spending • Empirically a fix for the anticipated inflation problem is needed