I’ve recently read some commenters talking about consumption behavior as if it’s a settled matter, particularly with respect to theory. Let me just put that idea to rest.

Suppose all identical agents behave by maximizing their utility derived from consumption (and only consumption) over all time into the infinite future, where income is derived in an endowment economy. There are no liquidity constraints, no asymmetric information. Then:

U'(C_t) = β(1+R)ε_tU'(C_t+1)

U(C_i) is flow utility of consumption in period t=i, U'(C_i) is the corresponding marginal utility, ∂U/∂C_i, β is the psychological discount factor (so if 0.95, something next year is 5% discounted relative to something this year), R is the real interest rate, and ε_t(.) is the expectations operator based on time t information.

Notice that one implication of this expression is that if the psychological rate of discount is 5% and the real interest rate is 5%, then the optimizing household will set the marginal utility in period t equal to that expected in period t+1.

Hard to say much more without imposing more functional form. A typical approach is to assume isoelastic utility.

U(C) = (C^1-η-1)/(1-η)

[Which seems undefined for η=1, but is U(C) = ln(C) — l’Hôpital’s rule strikes]

Marginal utility is then C^-η.

One can now see that changes in the real interest rate R changes the angle of expected consumption over time. Higher interest rates mean households should aim for lower marginal utility in the future, i.e., higher consumption, ignoring income effects (which depend on creditor or debtor status). In order to achieve that, given a constant endowment stream, consumption today must fall. How much? That depends on the magnitude of η among other things.

In general, for the model to mimic actual data, you need a really, really large η, much, much higher than the Arrow Conjecture of values between 2 to 4 — but since η is the inverse of the intertemporal elasticity of substitution in this framework, that means in practice there is very little response of consumption to interest rate changes.

That is why so much of modern macro entails use of more exotic utility functions, which in a way boost the “effective” coefficient of relative risk aversion — habit persistence, loss aversion. That breaks the direct link between the CRRA and the IES.

Bottom line:

1. Consumption behavior is complicated.
   
2. Microfoundations do not necessarily lead to better empirical models.
   

A formal, comprehensive, yet eminently clear, exposition by Pierre-Olivier Gourinchas, here.