# Consumption smoothing

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Consumption smoothing is the economic concept used to express the desire of people to have a stable path of consumption. Since Milton Friedman's permanent income theory (1956) and Modigliani and Brumberg (1954) life-cycle model, the idea that agents prefer a stable path of consumption has been widely accepted.[1][2] This idea came to replace the perception that people had a marginal propensity to consume and therefore current consumption was tied to current income.

Friedman's theory argues that consumption is linked to the permanent income of agents. Thus, when income is affected by transitory shocks, for example, agents' consumption should not change, since they can use savings or borrowing to adjust. This theory assumes that agents are able to finance consumption with earnings that are not yet generated, and thus assumes perfect capital markets. Empirical evidence shows that liquidity constraint is one of the main reasons why it is difficult to observe consumption smoothing in the data.

## Model

Robert Hall (1978) formalized Friedman's idea.[3] By taking into account the diminishing returns to consumption, and therefore, assuming a concave utility function, he showed that agents optimally would choose to keep a stable path of consumption.

With (cf. Hall's paper)

${\displaystyle E_{t}}$ being the mathematical expectation conditional on all information available in ${\displaystyle t}$
${\displaystyle \delta =1/\beta -1}$ being the agent's rate of time preference
${\displaystyle r_{t}=R_{t}-1\geq \delta }$ being the real rate of interest in ${\displaystyle t}$
${\displaystyle u}$ being the strictly concave one-period utility function
${\displaystyle c_{t}}$ being the consumption in ${\displaystyle t}$
${\displaystyle y_{t}=w_{t}}$ being the earnings in ${\displaystyle t}$
${\displaystyle A_{t}}$ being the assets, apart from human capital, in ${\displaystyle t}$.

agents choose the consumption path that maximizes:

${\displaystyle E_{0}\sum _{t=0}^{\infty }\beta ^{t}\left[u(c_{t})\right]}$

Subject to a sequence of budget constraints:

${\displaystyle A_{t+1}=R_{t+1}(A_{t}+y_{t}-c_{t})}$

The first order necessary condition in this case will be:

${\displaystyle \beta E_{t}R_{t+1}{\frac {u^{\prime }(c_{t+1})}{u^{\prime }(c_{t})}}=1}$

By assuming that ${\displaystyle R_{t+1}=R=\beta ^{-1}}$ we obtain, for the previous equation:

${\displaystyle E_{t}u^{\prime }(c_{t+1})=u^{\prime }(c_{t})}$

Which, due to the concavity of the utility function, implies:

${\displaystyle E_{t}[c_{t+1}]=c_{t}}$

Thus, rational agents would expect to achieve the same consumption in every period.

Hall also showed that for a quadratic utility function, the optimal consumption is equal to:

${\displaystyle c_{t}=\left[{\frac {r}{1+r}}\right]\left[E_{t}\sum _{i=0}^{\infty }\left({\frac {1}{1+r}}\right)^{i}y_{t+i}+A_{t}\right]}$

This expression shows that agents choose to consume a fraction of their present discounted value of their human and financial wealth.

## Criticism

Although Hall's proof, ${\displaystyle \beta E_{t}R_{t+1}{\frac {u^{\prime }(c_{t+1})}{u^{\prime }(c_{t})}}=1}$, is extremely short, just nine lines, Hall's corollary 4, ${\displaystyle c_{t+1}=c_{t}}$, can be found in Flavin.[4] Wu [5] has shown that changes in consumption being zero is probably the result of a misspecification error.

There are many ways to write a sum using summation notation so it is necessary to determine whether the correct range for the lower and upper index numbers is being applied. Writing the same consumption stated in Flavin, for period t+1, in a different way for the summation of the expected future incomes, it is possible to show that changes in savings is a function of income growth. Wu's proof is shown below applying the same definitions and techniques found in Sargent.[6]

A consumer maximizes

${\displaystyle \sum _{i=0}^{\infty }b^{t}\left[u_{0}+u_{1}c_{t}-{\frac {u_{2}}{2}}c_{t}^{2}\right],}$ subject to
${\displaystyle A_{t}=R[A_{t}+y_{t}-c_{t}]}$, where ${\displaystyle y_{t}}$, under a stochastic process, is ${\displaystyle E_{t}y_{t}}$
where ${\displaystyle c}$ is consumption, ${\displaystyle A}$ is non-human assets, ${\displaystyle y}$ is labor income, ${\displaystyle R}$ is gross rate of return (all at the beginning of period). ${\displaystyle E}$ is expectation, ${\displaystyle t}$ is time.

Under the "Euler equation approach," optimal consumption for period t is given by

${\displaystyle c_{t}=(1-R^{-2}b^{-1})A_{t}-{\frac {u_{1}}{u_{2}}}{\frac {(R^{-1}b^{-1}L^{-1})}{1-R}}+{\frac {(1-R^{-2}b^{-1})}{1-L^{-1}R^{-1}}}E_{t}y_{t}}$, where,${\displaystyle L}$ is the lag operator

Repeating Euler optimization, one should find that consumption, ${\displaystyle c_{t+1},c_{t+2},...,c_{t+n}}$, should be given by

${\displaystyle c_{t+1}=(1-R^{-2}b^{-1})A_{t+1}-{\frac {u_{1}}{u_{2}}}{\frac {(R^{-1}b^{-1}L^{-1})}{1-R}}+{\frac {(1-R^{-2}b^{-1})}{1-L^{-1}R^{-1}}}E_{t+1}y_{t+1}}$
${\displaystyle c_{t+2}=(1-R^{-2}b^{-1})A_{t+2}-{\frac {u_{1}}{u_{2}}}{\frac {(R^{-1}b^{-1}L^{-1})}{1-R}}+{\frac {(1-R^{-2}b^{-1})}{1-L^{-1}R^{-1}}}E_{t+2}y_{t+2}}$
. . .
${\displaystyle c_{t+n}=(1-R^{-2}b^{-1})A_{t+1}-{\frac {u_{1}}{u_{2}}}{\frac {(R^{-1}b^{-1}L^{-1})}{1-R}}+{\frac {(1-R^{-2}b^{-1})}{1-L^{-1}R^{-1}}}E_{t+n}y_{t+n}}$

Assuming ${\displaystyle Rb=1}$,

${\displaystyle c_{t+n}=(1-R^{-1})\left[A_{t+n}+{\frac {E_{t+n}y_{t+n}}{1-L^{-1}R^{-1}}}\right]}$

Since ${\displaystyle {\frac {1}{1-RL}}}$ can be expanded as

${\displaystyle {\frac {1}{1-RL}}={\frac {-(RL)^{-1}}{1-(RL)^{-1}}}=-{\frac {1}{R}}L^{-1}-\left({\frac {1}{R}}\right)^{2}L^{-2}-\left({\frac {1}{R}}\right)^{3}L^{-3}-...}$

and by definition of lag operator

${\displaystyle L^{-n}y_{t}=y_{t+n}}$

implies

${\displaystyle {\frac {(RL)^{-1}}{1-(RL)^{-1}}}y_{t}={\frac {1}{R}}L^{-1}y_{t}+\left({\frac {1}{R}}\right)^{2}L^{-2}y_{t}+\left({\frac {1}{R}}\right)^{3}L^{-3}y_{t}+...=\sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j}y_{t+j}}$

(Sargent, pp. 178–179)

thus for any n

${\displaystyle {\frac {E_{t+n}y_{t+n}}{1-L^{-1}R^{-1}}}=\sum _{j=n}^{\infty }\left({\frac {1}{R}}\right)^{j-n}E_{t+n}y_{t+j}}$

then

${\displaystyle c_{t+n}=(1-R^{-1})\left[A_{t+n}+\sum _{j=n}^{\infty }\left({\frac {1}{R}}\right)^{j-n}E_{t+n}y_{t+j}\right]}$

and for n = 1

${\displaystyle c_{t+1}=(1-R^{-1})\left[A_{t+1}+\sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j-1}E_{t+1}y_{t+j}\right]}$

## Flavin's Misspecification and Critical Assumption

Flavin stated that

${\displaystyle c_{t+1}=(1-R^{-1})\left[A_{t+1}+\sum _{j=0}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t+1}y_{t+j+1}\right]}$

Comparing the last two equations, one could easily reach the false conclusion that these equations yield identical results regarding the summations when written in the long form, i.e.,

${\displaystyle \sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j-1}E_{t+1}y_{t+j}=\sum _{j=0}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t+1}y_{t+j+1}=y_{t+1}+\left({\frac {1}{R}}\right)E_{t+1}y_{t+2}+\left({\frac {1}{R}}\right)^{2}E_{t+1}y_{t+3}+...}$

Since we know that a formula can yield different structural formulas but not all structural formulas will yield the same result. One should reasonably consider that, even though the number of incomes goes to infinity, as the consumer ages, there is a loss of income going forward one period. In contrast, in Flavin’s equation, the number of incomes remains constant. This can be seen by the lower limit of the index of the summations, which, for period t+1, vary from 1 to infinity, while in Flavin the lower limit of the index remains from zero to infinity. It is relatively straightforward to show how the difference of two summations with the same number of incomes may equal to zero. The untenable (and implicit) assumption that a consumer won't lose any labor income while he/she ages is the reason why one must always check the range of the summation.

Further, in Flavin’s approach,

${\displaystyle c_{t+1}-c_{t}=(1-R^{-1})\left[A_{t+1}-A_{t}+\sum _{j=0}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t+1}y_{t+j+1}-\sum _{j=0}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t}y_{t+j}\right]=(1-R^{-1})\sum _{j=0}^{\infty }\left({\frac {1}{R}}\right)^{j}(E_{t+1}-E_{t})y_{t+j+1}}$

It is relevant to point out that, Flavin had to make the critical assumption, "if expectations of future income are rational, the expectation of next period's revision in expectation, ${\displaystyle (E_{t+1}-E_{t})y_{t+j+1}}$, is zero to reach the conclusion that ${\displaystyle E_{t}c_{t+1}=c_{t}}$."

Clearly, one needs to question whether Flavin’s assumption is really necessary; whether it holds true from period t to period t+1 or for any period t+n and under what conditions.

## Restatement and Wu's Change in Savings and Income Growth Result

Applying the corrected ${\displaystyle c_{t+1}}$, the change in consumption is,

${\displaystyle c_{t+1}-c_{t}=(1-R^{-1})\left[A_{t+1}-A_{t}+\sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j-1}E_{t+1}y_{t+j}-\sum _{j=0}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t}y_{t+j}\right]}$

and since,

${\displaystyle \sum _{j=0}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t}y_{t+j}=E_{t}y_{t}+\sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t}y_{t+j}=y_{t}+\sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t}y_{t+j}}$

If we assume that,

${\displaystyle \sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j-1}E_{t+1}y_{t+j}-\sum _{j=1}^{\infty }\left({\frac {1}{R}}\right)^{j}E_{t}y_{t+j}=0}$

and applying the definition of total income or "measured" income,

${\displaystyle y_{mt}=\left(1-{\frac {1}{R}}\right)A_{t}+y_{t}}$

the change in consumption can be written as

${\displaystyle c_{t+1}-c_{t}=y_{mt+1}-y_{t+1}-y_{mt}+y_{t}-{\frac {(R-1)}{R}}y_{t}}$

thus

change in savings = ${\displaystyle (y_{mt+1}-c_{t+1})-(y_{mt}-c_{t})=\left(y_{t+1}-{\frac {y_{t}}{R}}\right)}$

One of the advantages of Wu’s result,[7] change in savings is a function of income growth, is that, even if the difference in the sum of expected future incomes assumption were not hold true, change in savings would still be dependent on income growth. In fact, one can relax both assumptions that change in expectations of income and the sum of expected future incomes to be zero and still be able to reach this new result. Note that Wu’s result is not obvious, one cannot derive income growth from Flavin’s approach. Moreover, Wu and Flavin have opposite conclusions.

## Implications

- In Adam Smith’s Wealth of Nations (1776),[8] Smith thought that “it is not the actual greatness of national wealth, but its continual increase, which occasions a rise in the wages of labour.” It is a misconception that higher savings will lead to a “better” economy because change in savings is a function of income growth. Furthermore, since growth is dynamic, targeting a saving rate level is irrelevant.
- In Keynes’ General Theory (1936), “a decline in income due to a decline in the level of employment, if it goes far, may even cause consumption to exceed income . . ..[p. 98]” Wu’s result shows that Keynes’ theories can be mathematically derived from Modigliani’s Life Cycle Hypothesis and Friedman’s Permanent Income Hypothesis, where income growth is dynamic leading to the disequilibrium model in Keynes’ [9] saving and dissaving. Similarly, Keynes’ fiscal stimulus policy follows the logic of employment, income growth and change in savings. In other words, trade imbalance, currency, technology, labor cost, tax, and other factors affecting income growth may help explain why countries have periods of accelerated and then slower change in savings, e.g., Japan and China.
- Similarly, in Lucas critique (1976),[10] “. . . any change in policy will systematically alter the structure of econometric models.” Arguably, Lucas critique should also apply to the first difference in such models. Thus, in consumption theory, policies affecting income growth will determine change in savings.
- Clower's [11] Dual Decision Hypothesis (1965) may help explain why consumption is continually re-evaluated as income changes, leading to a 2 step decision making. That is, the difference between expected and actual income may cause errors in optimal consumption, which may require corrections on consumption.
- Modigliani and Brumberg (1952) [12] hypothesized that “in the long run the proportion of aggregate income saved depends not on the level of income as such but, rather, on the rate of growth of income . . ..” Wu has expanded this relationship by showing the relationship between savings and growth may not always hold true, i.e., depending on savings at the intercept of zero growth, consumers may have positive growth and negative savings and vice versa. That helps explain why the U.S. saving rate is near negative even though growth is positive.

## Empirical Evidence

Robert Hall (1978) estimated the Euler equation in order to find evidence of a random walk in consumption. The data used are US National Income and Product Accounts (NIPA) quarterly from 1948 to 1977. For the analysis the author does not consider the consumption of durable goods. Although Hall argues that he finds some evidence of consumption smoothing, he does so using a modified version. There are also some econometric concerns about his findings.

Wilcox (1989) argue that liquidity constraint is the reason why consumption smoothing does not show up in the data.[13] Zeldes (1989) follows the same argument and finds that a poor household's consumption is correlated with contemporaneous income, while a rich household's consumption is not.[14]

## References

1. ^ Friedman, Milton (1956). "A Theory of the Consumption Function." Princeton N. J.: Princeton University Press.
2. ^ Modigliani, F. & Brumberg, R. (1954): 'Utility analysis and the consumption function: An interpretation of cross-section data'. In: Kurihara, K.K (ed.): Post-Keynesian Economics
3. ^ Hall, Robert (1978). "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence." Journal of Political Economy, vol. 86, pp. 971-988.
4. ^ Flavin, M.A. (1977, published 1981), "The Adjustment of Consumption to Changing expectations about Future Income," Journal of Political Economy, vol. 89n, no. 5.
5. ^ Wu, C. K. (2016), "New Result in Theory of Consumption: Changes in Savings and Income Growth - Nineteen Years Later," Journal of Economics Library, Vol. 3, No. 1, pp. 77-81, http://www.kspjournals.org/index.php/JEL/article/view/684
6. ^ Sargent, T. J. (1987), "Macroeconomic Theory," 2nd Edition, Academic Press.
7. ^ Wu, C. K. (1997), "New Result in Theory of Consumption: Changes in Savings and Income Growth," http://ideas.repec.org/p/wpa/wuwpma/9706007.html
8. ^ Smith, A. (1776), “The Wealth of Nations,” U. of Chicago Press
9. ^ Keynes, J.M. (1936), "The General Theory of Employment, Interest, and Money," Hartcourt Brace Jovanovich.
10. ^ Lucas, Robert (1976). "Econometric Policy Evaluation: A Critique". In Brunner, K.; Meltzer, A. The Phillips Curve and Labor Markets. Carnegie-Rochester Conference Series on Public Policy 1. New York: American Elsevier. pp. 19–46. ISBN 0-444-11007-0.
11. ^ Clower, R.W. (1965), "The Keynesian Counterrevolution: A Theoretical Appraisal,", The Theory of Interest Rates, ed. F.H. Hahn and F.P.R. Brechling, Macmillan, p. 103-25.
12. ^ Modigliani, F. and Brumbert, R. (unpublished manuscript 1952, published 1979), "Utility Analysis and Aggregate Consumption Functions: An Attempt at Integration," Collected Papers of Franco Modigliani, ed. A. Abel, Vol. 2, MIT Press.
13. ^ Wilcox, James A. (1989). "Liquidity Constraints on Consumption: The Real Effects of Real Lending Policies." Federal Reserve Bank of San Francisco Economic Review, pp. 39-52.
14. ^ Zeldes, Stephen P. (1989). "Consumption and Liquidity Constraints: An Empirical Investigation." Journal of Political Economy, University of Chicago Press, vol. 97(2), pp. 305-46