A Simple Model With Consumer Durables

Nick Rowe on twitter:

How negative do real interest rates have to go before storing consumer goods (e.g. food) becomes profitable?

This question inspired me to try to come up with a model with durable consumer goods (i.e., goods that aren't immediately consumed when purchased, but instead last multiple periods) and sticky prices. I won't bother with adding sticky prices (or monopolistic firms) until I have a basic model sketched out, so here goes:

There is a representative agent which derives utility from their consumption of non-durable goods, their stock of durable goods, and leisure. The utility function is
$$(1)\: U = \sum^\infty_{t=0}\beta^t u(c^d_t,c^p_t,l_t)$$
where $\beta$ is the discount factor, $c^d_t$ is the current stock of durable goods, $c^p_t$ is the agent's current consumption of non-durable (perishable) goods, and $l_t$ is leisure. The agent uses income from working, holding government bonds, and profit from the representative firm to pay for perishable consumption, the increase in durable goods that it owns (which depreciates at rate $\delta$), new government bonds, and lump sum taxes. The budget constraint is:
$$(2)\: c^p_t + c^d_t - (1-\delta)c^d_{t-1} + B_t + T_t = R_{t-1}B_{t-1} + w_t (1-l_t) + \pi_t$$
The agent maximizes $1$ w.r.t. $2$, which yields the following first order conditions:
$$(3)\: \beta^t u_2(c^d_t,c^p_t,l_t) - \lambda_t = 0$$
$$(4)\: \beta^t u_3(c^d_t,c^p_t,l_t) + \lambda_t w_t = 0$$
$$(5)\: \beta^t u_1(c^d_t,c^p_t,l_t) -\lambda_t + E_t \lambda_{t+1} (1-\delta) = 0$$
$$(6)\: -\lambda_t + R_t E_t \lambda_{t+1} = 0$$
where $\lambda_t$ is the Lagrange multiplier on the maximization problem.

These can all be simplified to:
$$(7)\: u_2(c^d_t,c^p_t,l_t) = \beta E_t u_2(c^d_{t+1},c^p_{t+1},l_{t+1})R_t$$
$$(8)\: w_t = -\frac{u_3(c^d_t,c^p_t,l_t)}{u_2(c^d_t,c^p_t,l_t)}$$
$$(9)\: u_1(c^d_t,c^p_t,l_t) = u_2(c^d_t,c^p_t,l_t)\left(1 - \frac{1-\delta}{R_t}\right)$$

Given that the government budget constraint is
$$(10)\: B_t + T_t = R_{t-1} + B_{t-1}$$
it is possible to rewrite the agent's budget constraint as:
$$(11)\: c^p_t + c^d_t - (1-\delta)c^d_{t-1} = w_t (1 - l_t) + \pi_t$$

There is a representative firm that maximizes profit, $\pi_t = y_t - w_t(1-l_t)$, where $y_t$ is production and $w_t$ is the real wage, subject to the production function
$$(12)\: y_t = f(1-l_t)$$
which gives the following first order condition:
$$(13)\: w_t = f'(1-l_t)$$
Given that $\pi_t + w_t(1-l_t) = y_t$, it is possible to rewrite $11$ as
$$(14)\: y_t = c^p_t + c^d_t - (1-\delta)c^d_{t-1}$$

Now, we have the full equilibrium of the model:
$$(1)\: u_2(c^d_t,c^p_t,l_t) = \beta E_t u_2(c^d_{t+1},c^p_{t+1},l_{t+1})R_t$$
$$(2)\: w_t = -\frac{u_3(c^d_t,c^p_t,l_t)}{u_2(c^d_t,c^p_t,l_t)}$$
$$(3)\: u_1(c^d_t,c^p_t,l_t) = u_2(c^d_t,c^p_t,l_t)\left(1 - \frac{1-\delta}{R_t}\right)$$
$$(4)\: y_t = f(1-l_t)$$
$$(5)\: w_t = f'(1-l_t)$$
$$(6)\: y_t = c^p_t + c^d_t - (1-\delta)c^d_{t-1}$$

I'll probably do more analysis tomorrow, but what immediately strikes me as interesting (and relevant to Nick's question) is that consumer durables are basically the same as money in MIUF models, if the real return on government bonds falls below $-\delta$, then the demand for consumer durables will skyrocket -- basically representing a lower bound on the real interest rate in addition to the nominal one.

The only question should then be what the average depreciation rate of durable consumer goods is; if it's not too high, I may have just found an undeniable (real) constraint on monetary policy -- if the natural rate of interest should ever fall below $-\delta$, then the central bank genuinely can't do anything about it; only fiscal policy can (by raising the natural rate).