CLASS 2: THE PRICE OF DURABLE GOODS

• study how the price of a durable good (refrigerator) is determined
• emphasize how forward looking behavior (expectations) is -again- crucial in understanding the determination of the price of a durable good
• this is useful because money shares many features of durable goods

Price of Durable Goods in an Infinite Horizon Model

this model shares a number of assumptions with the previous one. In particular:

• the economy lasts an infinite number of periods, t=0,1,2,...
• there is a representative household in the economy
• in each period the households receives an amount of income y_t in terms of the consumption good
• the aggregate supply of refrigerators is normalized to 1, and is constant in all periods. However the household see itself as being able to choose how much to invest in refrigerators
• at the end of period t the refrigerator is worth q_t units of the consumption good
• in each period the household decides how much to consume, and how much to invest, that is, how many cows to own
  let us denote s_t the amount of refrigerators owned by the household at the end of period t (beginning of period t+1)

the main difference is that the durable good does not produce a dividend in each period, like a stock, but produces a service, in the sense that it raises the utility of the agent (in a sense, the ``dividend" from the durable good is in terms of ``utils", not in terms of consumption good). the household's problem is:

$$max_{{c_t}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(c_t,s_{t-1})$$

subject to the budget constraint (which holds in each period t):

$$q_t s_t+c_t \leq y_t + q_t s_{t-1}$$

the service the household receives in period t from holding the durable goods are proportional to the amount of durable good held at the beginning of period t, that is s_t-1 as before, in equilibrium all these constraints will hold with = sign, so we can substitute the constraint in the objective function:

$$max_{{s_t}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(y_t-q_t (s_t-s_{t-1}),s_{t-1})$$

Let us focus on the first order condition (FOC) with respect to s_t
Notice that the term s_t appears in the sum above only in the two terms:

$$\beta^t u(y_t-q_t (s_t-s_{t-1}),s_{t-1}) + \beta^{t+1} u(y_{t+1}-q_{t+1}(s_{t+1}-s_{t}),s_t)$$

so the first order condition is:

$$\beta^t q_t u_c(c_t,s_{t-1})=\beta^{t+1} [u_s(c_{t+1},s_t)+q_{t+1} u_c(c_{t+1},s_t)]$$

where $u_c \equiv \frac{du}{dc}$, and $u_s \equiv \frac{du}{ds}$ which can be rewritten as:

$$q_t u_c(c_t,s_{t-1}) =\beta [u_s(c_{t+1},s_t)+q_{t+1} u_c(c_{t+1},s_t)]$$

following what we did in the previous model, we can write the FOC for periods t+1,..,t+T, substitute again and again for q_t+1 u_c(c_t+1,s_t),...
and obtain the following expression:
$q_t u_c(c_t,s_{t-1})=\\ \sum_{i=1}^T \beta^i u_s(c_{t+i},s_{t+i-1}) +\beta^T q_{t+T} u_c(c_{t+T},s_{t+T-1})$ which can be rewritten as:

$$q_t= \sum_{i=1}^T \beta^i \frac{u_s(c_{t+i},s_{t+i-1})}{u_c(... ...\beta^T \frac{u_c(c_{t+T},s_{t+T-1})}{u_c(c_t,s_{t-1})} q_{t+T}$$

What happens to the right hand side of this expression as we let $T \rightarrow \infty$?
as in the previous model, assume for the time being that

$$lim_{T \rightarrow \infty}\beta^T \frac{u_c(c_{t+T},s_{t+T-1})}{u_c(c_t,s_{t-1})} q_{t+T}=0$$

, then we obtain:

$$q_t= \sum_{i=1}^{\infty} \beta^i \frac{u_s(c_{t+i},s_{t+i-1})}{u_c(c_t,s_{t-1})}$$

the above condition determines the price of a durable good (refrigerator) and can be interpreted as follows:

• the value of a durable good is a present discounted value of all future services provided by the good
• the price of the durable good depends on the marginal rate of substitution (MRS) between consumption of the non durable good (c) at time t and the services from the durable good at time t+i.
  This suggests that:
  • in periods when people consume a lot of the consumption good (booms) the price of the durable good is driven up as well: when c_t is high, the denominator u_c(c_t,s_t-1) is low
  • the value from durable goods depend crucially on the extent to which the services they provide and consumption of the non durable good are substitutes or complements

equilibrium

in equilibrium it must be that in each period c_t=y_t and s_t=1
the pricing formula then becomes:

$$q_t=\sum_{i=1}^{\infty}\beta^i \frac{u_s(y_{t+i},1)}{u_c(y_t,1)}$$

in particular, if income is constant over time (y_t=y all t), we obtain that the price of the non durable good is also constant:

$$q_t=\frac{\beta}{1-\beta} \frac{u_s(y,1)}{u_c(y,1)}$$

we can now discuss why the transversality condition must hold in this model:

$$lim_{T \rightarrow \infty}\beta^T \frac{u_c(c_{t+T},s_{t+T-1})}{u_c(c_t,s_{t-1})} q_{t+T}=0$$

• first, notice that is income y_t is bounded away from zero and infinity in all periods ( $0<a<y_t<A<\infty$ all t), the term $\frac{u_c(c_{t+T},s_{t+T-1})}{u_c(c_t,s_{t-1})}$ must also be bounded away from zero and infinity, since c_t=y_t and s_t=1 in equilibrium
• second, notice that the term $\beta^T$ goes to zero as T goes to infinity
• this implies that, as long as q_t+T does not explode towards infinity, the transversality condition must hold
• can we have an equilibrium in which q_t+T explodes towards infinity? No, because the agents' wealth in terms of the quantity of durable good they possess would also be shooting up to infinity, and it would be optimal for the agent to sell some of the durables they possess and try to increase their consumption of the non durable good

What did we learn so far?

• the value of an asset (durable) good is the price of the asset (durable) expressed in terms of the consumption good. we will think of the price level as reflecting the value of money in terms of the consumption good.
• people are forward looking, in the sense that the price of an asset and of a durable good depend on the expectations about, respectively, the future dividends generated by the asset and the services obtained from the durable good. in the remainder of the course we will show that this is true for money as well: the current price level depends on the future services generated by holding money balance.

references: Milton Friedman, The Optimal Quantity of Money and Other Essays, Chicago, Aldine, 1969