In Chapter 3, it was argued that f3 arises from a mixture of underlying parameters fl, a and p
3. The dynamic Dixit-Stiglitz model
Periodic utility is expressed by a Cobb-Douglas function (2)
where l[ is constant (O<l[ <1). Further, H and M are consumption of homogeneous goods and manufactured (differentiated) goods, respectively. M is aCES-aggregate
where N is the number of goods that are consumed in the particular period, b > 1 is a constant elasticity of substitution, and
Qi
is consumption of each good. The utility function is like a static one, but it expands the static definition as it applies in all periods throughout the history for a consumer who adjusts his perception of utility to the current level of productivity. This has been anticipated by the definition (1) that is embodied in (3). As the firm size is fixed, the relative size, which matters for utility, decreases along withQj.
Hence, the discount factor (which here might be called a productivity factor) establishes the updating mechanism. There are no subjective time preferences in the traditional sense, so equilibrium will be determined by maximizing utility each period conditional on an income constraint.By a standard argument, the consumers will spend an income share l[ on manufactures, and l-l[ on the homogeneous good. We get a set of demand functions for manufactured goods
where Pi is the price in terms of
Qi'
y is the total income rate, and G is a perfect price index:Since N is assumed to be large, each firm can act as a monopolist facing a constant price elasticity of demand equal to b. This is plotted in Figure 2.
~Eer~6~ ~P~r~oaduwc~t~D~i~~~er~e~n~ti~a~ti~on
Figure 2. Demand Functions in a Dynamic Dixit-Stiglitz Model.
Contrary to a static model, the firm size is fixed, so the firm has no opportunity to scale its production at a specific time. However, the patent fixes the size of the product, and growth (at rate p) is embodied in new patents. For this reason, a firm that does not activate immediately will find itself moving to the left in Figure 2, as its effective (relative) size shrinks. The effective size of a new firm is one, so the process will start at
Qi =
1, as indicated in the figure.As long as the properties of the goods do not change with time, we know from previous chapters how this development will be experienced by a small, waiting firm in equilibrium: It can act as if it faces a firm-specific price that increases at a constant rate, because this corresponds to a constant elasticity ofthediscount factor.
Let us show this formally. At entry, we have
Qi
=1, but the effective size shrinks by the process dQi=
-pQidt due to growth. The demand function in eqn. (4) can be written as- - b
Qi
=
(P / P;), where P is regarded as constant as N is large. This yields:dP;/dt=(dP;/dQi)·(dQi/dt)=(-P;/Qib)·(-QiP)=(p/b)p;. Hence, the firm will observe a price that increases at rate p / b==f.1det , which is smaller than the growth rate asb > 1.
Notice that J.ldet does not describe a "magic" preference for old products. It is simply a dynamic parameter that arises from fixed, but continuously updated, static preferences for variation. The underlying force is not like a process that makes homogeneous trees grow.
Chapter 6. Product Differentiation
To a large extent, the difference between the static and the dynamic model is a matter ofwhen the goods are consumed. In the static model this must take place instantaneously. Here, variation is obtained by time, as a number of goods are consumed each period.
This is a quite nice story about growing demand for old products. To put it simply, a veteran car may be popular not because it is old but because it is different. In such cases, price growth
(.udet > O) does not mean that preferences change; it is a consequence of a preference for variation. The larger Jidet (i.e., the smaller b),the more the consumer is willing to reduce the quantity of each good to obtain a larger number of categories. As there are no subjective time preferences, equilibrium follows by repeating this story each period, and by requiring optimal investment decisions, free entry, and clearing of the factor market.
The cost function of a representative firm becomes
where
w
is the wage rate, LA is the fixed number ofworkers required for a patent, and Le is the similar fixed number of workers that would be needed to activate a patent immediately. Due to growth, the labor requirement for activation decreases along with the discount factor for each good. Notice that the firm size is one, so we might have left out the tilda onQi
in eqn.(6). Nevertheless, we keep it just to emphasize the difference from the models with homogeneous goods in previous chapters.
Profits are maximized if the firm charges a fixed markup price as discussed several times in previous chapters. Leaving out firm subscripts and asterisks for optimal values, the optimal price at which to activate is
(7)
fl
P=-fl
-1wu.,
where
fl
=p/Jidet.4 Free entry is obtained by the zero profit requirement:4The replacement of bbyflis a matter of convention, as we assume that the firm acts according to the observed dynamic variables, Pde!and p.
Chanter 6. Product Differentiation
This free entry condition is ex ante, but with no uncertainty (as assumed for the moment), ex post profit will also be zero for all firms. By combining eqns. (7) and (8), we also find
(9)
Q = R(Ø -1),
where R ==LAlLe.Similar to the Chapter 3 model,
Q::;
1 is required, so there cannot be too much scale economies. Eqn. (9) shows that the age of each patent that is activated will be higher (the size of the good relative to a newone will be smaller) the stronger the preference for variation; i.e.,Q
will be smaller the smallerfl
(= b). This conforms with the intuitive arguments in the introduction.Further,
Q
in eqn. (9) does not depend on more thanfl
and the ratio of the two exogeneous cost elements. In a multi-region setting with "iceberg" trade costs (see below), the allocation of resources between the two types of investment (patenting and production) will then be independent of the distribution of firms, so the relative number of firms will be equal to the relative number ofworkers.Finally, if the total labor endowment in manufacturing is LM, the steady-state condition is N(LA +QLc)w
=
LMw. Using eqn. (9), this simplifies to:(lO)
If all patents are used (which is true in this deterministic case), Nis the constant rate of entry, activation and consumption in equilibrium. As eqn. (10) shows, the number of categories will be larger the stronger the preference for variation (i.e., the smaller /3). This also conforms with the introductory remarks.