On The Determination of the Public Debt

Robert Barro 1979

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

Hypothesis

1. That there is a positive effect of temporary increases in government spending on debt issue

2. The negative effect of temporary increases in income

3. 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

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

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

Budget Constraint

In each period

Budget constraint at date t

Determination of Burden

Collection Cost in period t Zt - The real cost incurred in period t.

Present value of Collection Costs

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

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

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

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:

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

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

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

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

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+

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

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

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

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

Table 1

Empirical Results

Table 3

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