# A2 Summers (1981)

In an important paper, Summers (1981) argued that human wealth is related to the interest rate; the higher the interest rate, the lower the present value of future labor earnings and the lower the level of wealth. Using a two-period model, whereby the agent works in the first period alone to study the capital income tax issue, does not fully capture the influence of the interest rate on wealth. Further, including wealth in a consumption function along with an interest rate variable, as Boskin did, would also tend to confound the different effects.

The interest elasticity of saving that takes into account the effect of the interest rate on human wealth could be much higher than previously thought. Using a calibrated computer simulation model, Summers simulated the model and calculated an interest elasticity with an order of magnitude larger than Boskin’s empirical estimates.

Consider the two-period life cycle model studied earlier, but suppose that the consumer also receives labor income in the second period. Her or his two budget constraints are

where Yj is labor earnings in period j (j = 1,2). If the utility function is given as

where p = 1/(1 + p) and p is the rate of time preference, it is straightforward to show that the consumption function is given by

where W = (Y_{1}+RY_{2}) is the present value of labor income, or human wealth, and R = 1/(1+r).

Then, saving is given by

Saving responds to the interest rate when human wealth is not held constant in accordance with

Not only would the elasticity of saving be positive in this case, it is also possible for it to be greater than one in magnitude. It will also vary with income.

In this example, the elasticity of saving is given as

The denominator must be positive for saving to be positive. If the following condition (8.A7) holds, the elasticity is greater than one in magnitude.

I

For example, interpreting the formula on an annual basis, suppose r = 0.02 and p = 0.01. This requires that Y_{2}/Y_{1} > 1.029. However, if r = 0.025 and p = 0.01, then Y_{2}/Y_{1} > 1.039. This requires the wage profile to increase by at least 3.9 %, which may be empirically unlikely if we interpret the inequality as involving an annual comparison.

However, if we interpret the two-period example with each period being 25-30 years long (a generation), an annual interest rate of 2.5 % would have a doubling time of about one generation, 28 years. Thus, if we set r = 1 in the two-period example and p = 0.5, then Y_{2}/Y_{1} > 2. Wages would have to at least double over a single generation in order for the condition to be satisfied. This seems empirically reasonable. See Batina and Ihori (2000) for further discussions on this topic.

**Questions**

8.1 Assume the Cobb-Douglas utility function

where U is utility, c is consumption, and L is labor supply. Suppose wage is 60 and consumption tax is 0.2. What is the optimal value of L?

- 8.2 Suppose the initial tax rate is 0.1 and the corresponding excess burden is 10. If the tax rate rises to 0.3, how much is the excess burden at the new equilibrium?
- 8.3 Say whether the following statements are true or false and explain the reasons.
- (a) The excess burden increases with the income effect.
- (b) Changes in the consumption tax rate and labor income tax for the same amount of tax revenue have no effect on the real economy.

## References

Batina, R., & Ihori, T. (2000). *Consumption tax policy and the taxation of capital income. *New York: Oxford University Press.

Boskin, M. (1978). Taxation, saving and the rate of interest. *Journal of Political Economy, 86,* S3- S27.

Doi, T. (2016), *Incidence of corporate income tax and optimal capital structure: A dynamic analysis.* RIETI DP 16-E-022.

Rosen, H. S. (2014). *Public finance.* New York: MacGraw-Hill.

Stigliz, J. E. (2015). *Economics of the public sector.* New York: W. W. Norton & Company. Summers, L. H. (1981). Capital taxation and accumulation in a life cycle growth model. *American Economic Review, 71,* 553-544.

Varian, H. R. (2014). *Intermediate microeconomics: A modern approach.* New York: Norton.