# Consumption smoothing

Consumption smoothing

Consumption smoothing is the economic concept used to express the desire of people for having 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 for example, when income is affected by transitory shocks, 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 earning that are not yet generated, thus, it assumes perfect capital markets. Empirical evidence shows that liquidity constraint is one of the main reasons of why it is so difficult to observe consumption smoothing in the data.

## Model

Robert Hall (1978) formalized Friedman 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.

Agent's choose the consumption path that maximize:

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

Subject to a sequence of budget constraints:

At + 1 = Rt + 1(At + ytct)

The first order necessary condition in this case will be:

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

By assuming that Rt + 1 = R = β − 1 we obtain, for the previous equation:

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

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

Et[ct + 1] = ct

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

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

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

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

## 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 to find some evidence of consumption smoothing it is done for a modified version. Also there is some econometric concerns around his finding.

Wilcox (1989) argue that liquidity constraint is the reason for the consumption smoothing not to hold in the data.[4] Zeldes (1989) follows the same argument and finds that poor household's consumption is correlated with contemporaneous income while rich household is not.[5]

## 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. ^ 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.
5. ^ 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

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