Appendix: Derivation of Equation (1.5)
The liquidity constrained consumers consume a constant fraction of their current income. Hence, their expected rate of growth of aggregate consumption equals the expected rate of growth of real income:
For the forward looking consumers the Euler equation holds:
Through some simple algebraic manipulations equation (A 2) can be transformed into the following expression:
It should be noted, that the step from equation (A 2) to (A 3) is not mathematically rigorous. In particular, this transformation violates Jensen’s inequality. However, (A 3) represents a reasonable approximation, and provides the intermediate link to an estimable expression.
By the same reasoning (A 3) can be approximated by the expression:
where a drift term (θ) has been included in the equation, to represent the long run rise in aggregate consumption. The parameter σ, which is equal to 1/α, is the intertemporal elasticity of substitution. This expression is the same as equation (1.4) in the main text.
The above equation is equation (1.5) in the main text.
If the real interest rate is assumed constant, or the elasticity of substitution (σ) is zero, equation (A 5) simplifies to the following expression,
This version of the model was also estimated.
Appendix: Data Sources
The data on real and nominal consumption came from OECD Quarterly National Accounts (QNA). Disposable income data were obtained from national sources; the U.S. and Canadian data came from the DRI data tape, the Japanese data came from the Nikkei Telecom Japan News and Retrieval Service tape, the French and U.K. data from quarterly national accounts tapes while the Swedish data was obtained from the authorities. For Japan, France, Canada and Sweden the data was only obtainable in nominal terms; it was deflated using the implicit deflator for total consumption from the OECD QNA. The consumption data for Japan, and the disposable income data for both Japan and Sweden were seasonally unadjusted, and were adjusted using the smooth command in RATS with options exponential trend and multiplicative seasonals.
Interest rates and real share prices were obtained from the International Finance Statistics tape. The Treasury Bill rate (line 60c) was used for all countries except Japan and France, where the data did not exist so the money market rate (line 60b) was used. Real interest rates were obtained using the implied deflator for non-durable and service consumption. Nominal share prices were obtained from IFS, line 62, and deflated by the consumer price index, line 64. Population figures were also obtained from the IFS tape, line 99z. The mid-year values were linearly interpolated to give a quarterly series.
Bernanke, B., “Adjustment Costs, Durables and Aggregate Consumption,” Journal of Monetary Economics, No. 15 (January 1985), pp. 41–68.
Campbell, J., and G. Mankiw, “Permanent Income, Current Income and Consumption,” National Bureau of Economic Research Working Paper No. 2436 (November 1987).
Campbell, J., and G. Mankiw, “Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence,” National Bureau of Economic Research Working Paper No. 2924 (April 1989).
Cochrane, J., “The Sensitivity of Tests of the Intertemporal Allocation of Consumption to Near-Rational Alternatives,” The American Economic Review, Vol. 79, No. 3 (June 1989), pp. 319–37.
DeLong, J.B., and L. Summers, “The Changing Cyclical Variability of Economic Activity in the United States,” in R.J. Gorden (ed.), The American Business Cycle: Continuity and Change (Chicago: University of Chicago Press, 1986).
Davidson, J., D. Hendry, F. Srba, and S. Yeo, “Econometric Modeling of the Aggregate Time Series Relationship between Consumers’ Expenditure and Income in the United Kingdom,” The Economic Journal, No. 88 (December 1978), pp. 661–92.
Eichenbaum, M., L. Hansen, and K. Singleton, “A Time Series Analysis of Representative Agent Models of Consumption and Leisure Choice Under Uncertainty,” National Bureau of Economic Research Working Paper No. 1981 (1986).
Epstein, L., and S. Zin, “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns. I: A Theoretical Framework,” Working Paper No. 8715 (Toronto: University of Toronto, 1987).
Flavin, M., “The Adjustment of Consumption to Changing Expectations About Future Income,” Journal of Political Economy 89 (October 1981), pp. 978–1009.
Hall, R., “Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence,” Journal of Political Economy, No. 86 (October 1978), pp. 971–87.
Hall, R., and F.S. Mishkin, “The Sensitivity of Consumption to Transitory Income: Estimates from Panel Data on Households,” Econometrica, No. 50 (1982), pp. 461–81.
Hansen, L., and K. Singleton, “Stochastic Consumption, Risk Aversion and the Temporary Behavior of Asset Returns,” Journal of Political Economy, No. 91 (April 1983), pp. 249–65.
Hansen, L., and K. Singleton, “Generalized Instrumental Variables Estimation of Nonlinear Rational Expectation Models,” Econometrica, No. 50 (1982) pp. 1269–86.
Hayashi, F., “The Permanent Income Hypothesis: Estimation and Testing by Instrumental Variables,” Journal of Political Economy, No. 90 (October 1982), pp. 895–16.
Hendry, D., “Econometric Modelings The Consumption Function in Retrospect,” Scottish Journal of the Political Economy, No. 30 (November 1983), pp. 193–20.
Jappelli, T., and M. Pagano, “Consumption and Capital Market Imperfections: An International Comparison,” CEPR Discussion Paper No. 244, June 1988.
Mankiw, G., J. Rottenberg, and L. Summers, “Intertemporal Substitution in Macroeconomies,” Quarterly Journal of Economics, No. 100 (February 1985), pp. 225–51.
Mishkin, F.S., “The Real Interest Rate: A Multi-Country Empirical Study,” Canadian Journal of Economics, No. 17 (May 1984), pp. 283–311.
Obstfeld, M., “How Integrated are World Capital Markets? Some New Tests,” National Bureau of Economic Research WP No. 2075 (November 1986).
Organization for Economic Cooperation and Development, Structural Adjustment and Economic Performance, OECD Paris, France, 1987.
Working, H., “Note on the Correlation of First Differences of Averages of A Random Chain,” Econometrica, No. 28 (1960), pp. 916–18.
Zeldes, S., “Consumption and Liquidity Constraints: An Empirical Investigation,” Journal of Political Economy, Vol. 97 (1989), pp. 305–46.
Miss Koujianou is a graduate student at Stanford University. She contributed to this paper while working as a summer intern in the Research Department. The authors would like to thank Eswar Prasad for considerable technical assistance.
Carroll and Summers (1989) argue that the observed correlation between consumption and income growth indicates that consumption smoothing takes place over periods of several years rather than over a lifetime. The relevance of the Euler equation approach, however, is not invalidated by this criticism.
Other work on liquidity constraints includes Zeldes (1986) and Hall and Mishkin (1982) on the micro-economic side, and Campbell and Mankiw (1987 and 1989) and Jappelli and Pagano (1988) on the macro-economic side. The other main explanation for the failure of the simple version of the forward-looking model is that consumption is not separable in the utility function. Works in this vein include Mankiw, Rottenberg and Summers (1985) and Eichenbaum, Hansen and Singleton (1987) on labor supply and Bernanke (1985) on durable goods and Aschauer (1985) on government purchases. The general results from this literature are not encouraging, and are not explored here.
The Appendix contains a complete derivation of equation (1.4). A “drift” term has been included, to represent the growth in consumption over time caused by factors such as technological change. This type of effect is implicitly included in most tests of the theory, such as Hall (1978).
Given the assumed utility function the coefficient σ can also be identified with the reciprocal of the coefficient of risk aversion (a). Less restrictive assumptions about preferences would break this correspondence. Since the equation deals with changes in consumption over time, the intertemporal elasticity appears a better characterization (Hall (1988)). For a discussion of a wider class of utility functions see Epstein and Zin (1987).
This test is constructed by exploiting instrumental variable techniques; the excess of instruments over parameters generates a set of overidentifying restrictions of the orthogonality conditions implied by the expectations operator. The instruments can be any predetermined variables; they do not have to be exogenous in the model.
See note on data sources.
First lags are inadmissable as instruments because consumption and income are measured as quarterly averages rather than at points in time. According to the permanent income hypothesis, consumption is a random walk; Working (1960) shows that averaging a random walk induces serial correlation between the contemporaneous value and the first lag, but not earlier lags, making first lags invalid instruments. See also Hall (1988) and Campbell and Mankiw (1987 and 1989).
This finding contrasts with the positive results reported by Hall (1978), who found that stock price changes have explanatory power. However, he included first lags in his set of instruments.
For a discussion on long-run trends in disposable income and consumption between countries, and their implications for permanent income/life cycle models see Caroll and Summers (1989).
For a survey of such reforms see OECD (1989).
Another possible reason for this lack of precision is that different seasonal adjustment procedures were used on the consumption and disposable income data. While the consumption data was seasonally adjusted, the income data was not and was seasonally adjusted by the authors.
Alternative instrument sets produced both correctly and incorrectly signed estimates, none of which were significant.
The regressions were performed every two years. The estimating model assumes real interest rates to be constant.
Carroll and Summers (1989) note that, in the long run, the rate of growth of consumption in different economies is closely linked to the rate of growth of disposable income. They argue that permanent income should be thought of as a medium term concept. Using this interpretation the ratio of discount rates can be seen as a drift term, similar in function to the constant in the ‘random walk’ theory of consumption.
For evidence that real interest rates are not equated across countries in the floating rate period, see Mishkin (1984).
Single equation estimation was also carried out. These results give useful insights as to whether equation (2.2a) holds between specific countries, however they are not reported for the sake of brevity.
Attempts to estimate the national and international equations as a system were also unsuccessful.
This is often represented as a theoretical restriction. However, Carroll and Summers (1989) point out that this empirical regularity across countries in fact represents a rejection of the model in which life-time consumption is smoothed.
Davidson et al (1978), the seminal piece in this literature, states “Unfortunately, much existing economic analysis … leaves many important decisions in formulating an operational model to ad hoc considerations (e.g., functional form, dynamic specification, error structure, treatment of seasonality, etc.) Nevertheless, economic theory does furnish some helpful postulates about behavior in steady state environments…” (see Davidson et al, 1978, page 662). This piece was written before the adoption of Euler equations in consumption research. However later papers in the same tradition (Hendry (1982)) have a similar flavor.
This model assumes that the real interest rate is constant. It is the main version of the model estimated by Campbell and Mankiw (1989). The empirical work in this section uses data on total consumption and income, rather than per capita data, since this appears more appropriate for forecasting equations.
Tests using different sets of instruments generally gave similar results.
The sample was 1971:1-1988:1; evidence of a shift in the liquidity parameter over this period was discussed above. The long sample was chosen because of the number of regressors/instruments (seven). The earlier results also indicate that the model fits the data quite well even over the full sample.
If first lags of the instruments are included, the restrictions are rejected for all countries. Since the instruments are inappropriate, the importance of this result is difficult to determine.
It calls into question arguments that rules of thumb, such as consuming current disposable income, are used by many consumers because the utility losses associated with such rules are small (Cochrane (1989)). This hypothesis implies that financial deregulation should make little difference to consumption patterns.