An approximate consumption function

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Abstract

This paper proposes an approximation to the consumption function. The approximation is based on the analytic properties of the consumption function in the buffer-stock model. In such model, the consumption function is increasing and concave and its derivative is bounded from above and below. We compare the approximation with the consumption function obtained using the endogenous grid-points algorithm and show that using the former or the latter for estimating the Euler equation leads to very similar results.

Introduction

The availability of large scale data-sets makes it attractive to estimate realistic models of consumers behavior. Most often, however, the models do not feature a closed form solution. Therefore, to pin down structural parameters, one should be able to nest estimation with the model solution. The challenge is often one of dimensionality and CPU time. This makes approximations an attractive option. Using approximations involves errors, but there is not agreement on the effect of such errors on estimation. To the extent that the errors are small, one can still rely on approximations. Furthermore, approximations might turn to be useful in the specification search.

Approximating the consumption function has been a common exercise among economists since long time. The use of perturbation methods in precautionary saving models dates back to Leland (1968). Only recently, however, Feigenbaum (2005) has investigated the accuracy of second, third and higher approximations to the consumption function and provided some warnings on the use of perturbation methods.

This paper does not use perturbation methods in that departing from the literature, but provides a complementary approach. Feigenbaum (2008) shows that the perturbation approximation is valid for large values of wealth, while our approach performs better around the target wealth for the consumer problem and zero. We employ a class C function to approximate the consumption function in the buffer stock model of saving. The approximation is derived for the Carroll's (1992) incarnation of the buffer stock model, but equally applies to the Deaton's (1991) version of such model. It relies on the monotonicity of the consumption function, on concavity, and on the fact that the consumption function is bounded from above and below and so is its derivative.

The paper is organized as follows. Notation is lied down in Section 2. Section 3 discusses the endogenous grid-points algorithm. The approximation is derived and described in Section 4. Section 5 compares the approximate consumption function with the solution obtained using the endogenous grid-points algorithm in five economies, and Section 6 explores the factors affecting the shape the approximate consumption function. Gains and losses from using the approximate consumption function are discussed in Section 7, while Section 8 concludes.

Section snippets

The notation

Consumers are infinitely lived.1 They maximize E0t=0βtu(Ct)with respect to consumption, Ct, under the dynamic

Standard solution methods and the endogenous grid-points algorithm

Problem (2) has not a closed form solution but can be given a recursive characterization. The solution is found by value or policy function iteration or using projection methods.2

A common solution strategy amounts to iterate the Euler equation for consumption. This means defining a grid for mt, i.e. {μ1,μ2,,μI}, discretizing the distribution of permanent and transitory

The approximate consumption function

To derive our approximate consumption function, we exploit the analytic properties of the marginal propensity to consume out of cash-on-hand (MPC) in the precautionary saving model. Carroll and Kimball (1996) show that the MPC is decreasing and Carroll (2004) that is bounded from above and below, namely thatlimm0c(m)=κ¯limmc(m)=κ̲where c(m) is the MPC, κ¯>κ̲>0 andκ¯=1-R-1(Rβp)1/ρκ̲=1-R-1(Rβ)1/ρThis suggests to approximate the MPC with the following family of functions:(1+e-ba)(κ¯-κ̲)1+eb(m

Five examples

This section compares the solution obtained with the endogenous grid-points algorithm with the approximate consumption function in five economies.

Our first economy assumes G=1.03, R=1.04, ρ=2, β=0.96, p=0.005, σθ=σψ=0.1. These are the values used in Carroll (2004). In this parametrization of the model, a and b are equal to 0.8982 and to 1.0941.

Fig. 5 displays the actual and the approximate consumption function. The approximate is very close to the actual consumption function in the 0–2 range of

The choice of a and b

This section explores how the choice of a and b depends on the deep parameters of the consumer problem. Table 3 computes a and b in several experiments for the discount factor, the relative risk aversion, the growth and the interest factor, the probability of unemployment and the standard deviation of the logarithm of permanent and transitory income shocks.

Letting the discount factor to vary between 0.95 and 0.99, we find a to decrease from 0.99 to 0.48 and b to increase from 1.05 to 1.40. To

Gains and losses from using the approximate consumption function

In order to quantify gains and losses from using the approximate consumption function, three exercises are performed. First, we simulate a length 100 series using the approximate and the actual consumption function in our five economies, and compare the run-times between the two methods. We perform the test on a Intel(R) Core(TM)2 CPU U7700 1.33 GHz computer with RAM equal to 1.99 GB running Windows XP. The results show that the approximate consumption function delivers appreciable gains in

Conclusions

This paper has provided an approximate consumption function for the buffer stock model of saving. The approximation exploits the asymptotic behavior of the consumption function and of the marginal propensity to consume out of cash-on-hand. Our approximate marginal propensity to consume is a linear transformation of the Fermi–Dirac distribution. The transformation is made to depend explicitly on the interest factor, on the unemployment probability, on the discount factor and on the constant

Acknowledgments

I would like to thank Chris Carroll, Alessandro De Martino, James Feigenbaum, Thomas Hintermaier, Tullio Jappelli, Luigi Pistaferri, the editor and two anonymous referees for comments and suggestions. The usual disclaimer applies.

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