that can be extracted from the dynamic framework, but for completeness we spell out the whole model first. Before doing so, it is convenient to summarize how the main assumptions and results relate to the static model:
1. The dynamic model is based on the same static preferences as Krugman' s model at any point in time (possibly in an expected sense). However, the model spans over an infinite time horizon, which includes exogenous growth of productivity. The representative consumer responds to this growth process by adjusting his perception of utility to the current level of productivity at any time.
2. Scale economies and market power are obtained by minimum requirements for patent investments, analogous to the fixed cost that applies in the static model.
3. All the rest of the main requirements in the static core-periphery model apply over the
entire time horizon. This includes assumptions on trade costs, factor endowments, factor
mobility etc.
Chapter 7. Agglomeration with Pecuniary Externalities
4. In broad terms, the dynamic framework brings forward a new interpretation of the core-periphery model. This can be used to study the two types of equilibria in the model: either a symmetric equilibrium, or one with agglomeration.
Finally, note that the static core-periphery model has been extended in a number of directions since the original publication; see e.g. Krugman and Venables (1995), Puga (1996), Baldwin and Forslid (1997), and Fujita, Krugman and Venables (1998). Hopefully, the analogy that is developed below also applies to several of these extensions.
2. The model
Non-technical description
The model has two regions (1 and 2), two goods (manufactures and agriculture), and two specific production factors (workers and farmers). The workers produce manufactured goods, the farmers produce agricultural goods, and the total endowments of workers and farmers are fixed. All agents are forward-looking, optimizing decision makers.
Agriculture is a homogenous, perfectly competitive good that can be traded at no cost. Half of the farmers live in each region, and they are not allowed to move. This implies equal wage rates in farming, and equal shares of total farmer income to each region. As in Chapter 5, the main role of the farmers is to make up a demand for manufactured goods.
Manufactures are differentiated goods, produced by firms that undertake irreversible investments. An infinite number of possible products exist, and a large number are produced each period. Hence, each firm is small compared to the size of the market. The workers are mobile, and will move - at no cost and with no delay - to the region that offers the higher real wage.
Two investments are necessary before a manufactured good can be consumed. First, there is an entry cost that gives the firm the exclusive right to produce a fixed quantity of the good.
This patent investment is irreversible in the sense that it locks on to a particular technology, and fixes the size of the firm. However, the firm is free to choose when to activate the patent.
Whenever activation takes place, the good must be consumed immediately.
ChaDIer 7. Agglomeration with Pecuniary Externalities
The productivity of the workers grows at a fixed, exogenous rate p in both activities. There are scale economies, as a fixed number of workers always must be hired for one period to acquire a patent. Thus, growth of productivity is embodied linearly in new patents, since the effective size of an old patent will be smaller than a new one, as in several previous models.
The formal setup
Preferences are fixed in the sense that the following Cobb-Douglas utility function applies in all periods:
(1) U
=
Hl-1r M" .Here H is consumption of the agricultural good, and M is a CES-aggregate in manufacturing:
(2) M -_ (N
J
Q-(b-I)/b.dlr
,whereQi
- _=Qiqi '
and Qi_=e-pT, .In eqn. (2), b >1 is a constant elasticity of substitution, and N is the number of manufactured goods that are consumed each period. The quantity measure for each one,
o:
takes intoaccount the age Ti of each good that is consumed, as the age is embodied in the discount factor Qj. Further, qj is the share of the good that each consumer gets, so we set qj = 1 when discussing the entire market. As in Krugman's model, the number of consumers can be scaled to one.
By letting this utility function apply in all periods, we make an important assumption about the consumer: He is "modem", as his perception of utility is related to the current state of productivity at any time. Utility is scaled so that the inner term in eqn. (2) equals one for a specific product ifit is basedon a new patent (Qj= 1), and the consumer gets all ofit (qj= 1).
No subjective time preferences in the traditional sense apply, so equilibrium follows by maximizing utility conditional on an income constraint each period, and by requiring optimal decisions, free entry, and full employment.
By a standard argument, the consumers will spend an income share reon manufactured goods, and l-re on agriculture each period. The demand function facing each patent holder is
Chapter 7. Agglomeration with Pecuniary Externalities
where G is a perfect price index
and Y is the rate of income, which is scaled to one each period. As in previous chapters, Pi is the price in terms of a good based on a new patent. Since each firm is small, and since trade costs will be of the iceberg type, it follows that a small firm can act as a monopolist with isoelastic demand in both markets.
The firm size is fixed, but as noted above, growth is embodied linearly in new patents as a fixed number of workers, LA' are needed for one period to acquire a patent at any time.
Similarly, Le workers would be needed for one period to activate such a patent immediately.
Hence, the fixed quantity embodied in the patent i that LA workers were able to make Ti periods ago, is only Qi (::;;1) times as large as the quantity embodied in a new patent. On the other hand, due to growing productivity, only Q;Le (::;;Le) workers will be needed to activate the old patent. Thus, if
w
is the wage rate in manufacturing, and we define A = wLA and C=
wLo the total cost function for a firm contemplating entry in equilibrium becomes:As in Chapter 6, we could have left out the tilda on
Qi
in this equation, but we keep it to emphasize that this is a model with true product differentiation.As also discussed in the previous chapter, a firm that does not activate a patent immediately will observe the demand function (3) as a firm-specific price that increases at a fixed rate
p/b == f-ldet
«
p). Hence, instead ofbeing able to choose the relative quantity that it can supplyat a specific point in time, the firm can use the option established by the patent, to slide along the demand curve. Then it is optimal to activate when time has brought the firm a fixed
Chamer 7. Agglomeration with Pecuniary Externalities
markup price over the cost of activation. The unit cost of activation for a small firm in equilibrium is C from eqn. (5), so the markup price becomes
(6)