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### 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

- 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

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

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

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