## Neoclassical consumption models and Euler equations that do not fit reality

12 March, 2014 at 10:16 | Posted in Economics | 2 Comments

In the standard neoclassical consumption model — used in DSGE macroeconomic modeling — people are basically portrayed as treating time as a dichotomous phenomenon  today and the future – when contemplating making decisions and acting. How much should one consume today and how much in the future? Facing an intertemporal budget constraint of the form

ct + cf/(1+r) = ft + yt + yf/(1+r),

where ct is consumption today, cf is consumption in the future, ft is holdings of financial assets today, yt is labour incomes today, yf is labour incomes in the future, and r is the real interest rate, and having a lifetime utility function of the form

U = u(ct) + au(cf),

where a is the time discounting parameter, the representative agent (consumer) maximizes his utility when

u´(ct) = a(1+r)u´(cf).

This expression – the Euler equation – implies that the representative agent (consumer) is indifferent between consuming one more unit today or instead consuming it tomorrow. Typically using a logarithmic function form – u(c) = log c – which gives u´(c) = 1/c, the Euler equation can be rewritten as

1/ct = a(1+r)(1/cf),

or

cf/ct = a(1+r).

This importantly implies that according to the neoclassical consumption model that changes in the (real) interest rate and consumption move in the same direction.

So good, so far. But how about the real world? Is the neoclassical consumption as described in this kind of models in tune with the empirical facts? Hardly — the data and models are as a rule insconsistent!

In the Euler equation we only have one interest rate,  equated to the money market rate as set by the central bank. The crux is that — given almost any specification of the utility function  – the two rates are actually often found to be strongly negatively correlated in the empirical literature:

In this paper, we use U.S. data to calculate the interest rate implied by the Euler equation, and we compare this Euler equation rate with a money market rate. We find the behavior of the money market rate differs significantly from the implied Euler equation rate. This poses a fundamental challenge for models that equate the two rates.

The fact that the two interest rate series do not coincide – and that the spread between the Euler equation rate and the money market rate is generally positive – comes as no surprise; these anomalies have been well documented in the literature on the “equity premium puzzle” and the “risk free rate puzzle.” And the failure of consumption Euler equation models should come as no surprise; there is a sizable literature that tries to fit Euler equations, and generally finds that the data on returns and aggregate consumption are not consistent with the model.

If the spread between the two rates were simply a constant, or a constant plus a little statistical noise, then the problem might not be thought to be very serious. The purpose of this paper is to document something more fundamental – and more problematic – in the relationship between the Euler equation rate and observed money market rates … We compute the implied Euler equation rates for a number of specifications of preferences and find that they are strongly negatively correlated with money market rates …

Our results suggest that the problem is fundamental: alternative specifications of preferences can eliminate the excessive volatility, but they yield an Euler equation rate that is strongly negatively correlated with the money market rate.

Matthew Canzoneri, Robert Cumby and Behzad Diba