New technologies are developed using final output. As we will see shortly, due to a market size effect in the creation of new technologies, countries in the South will perform no R&D. All technological progress will therefore originate in the North. But the South can adopt these technologies. All consumers have linear preferences given by f Ce~ridt, where С is consumption and r is the discount rate, which will also be the interest rate. We suppress time indexes when this causes no confusion.

Technology

We first describe the production technology which is common across countries, and the R&D technology in the North. To simplify notation, we omit the country indexes for

where kz(i, v) is the quantity of machines of type v used in sector i together with workers of skill level г (i.e. this is sector and skill-specific capital). There is a continuum of machines, denoted by j [0,A^], that can be used with unskilled workers, and a continuum of machines (different) j € [0, N# ] used with skilled workers. Technical progress in this economy will take the form of increases in NL and JV#, that is, technical change expands the range of machines that can be used with unskilled and skilled workers.

This is similar to the expanding variety model of Romer (1990) (see also Grossman and Helpman, 1991), but allows for technical change to be skill-or labor-complementary as in Acemoglu (1998). Equation (2) also implies that each good can be produced by skilled or unskilled workers, using the technologies suited to their needs. The terms (1 — г) and i imply, however, that unskilled labor is relatively more productive in producing goods with low indexes. The parameter Z (where Z > 1) enables a positive skill premium. Feasibility requires that Jq l(i)di < L and /0г h(i)di < H.
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Producers of good г € [0,1] take the prices of their products, р(г), wages, wl and wH, and the rental prices of all machines, Xl(v) and Xh{v), given, and maximize profits. This gives the following sectoral demands for machines:

A (technology) monopolist owns the patent for each type of machine. We assume that it also owns machines and rents them out to users at the rental rates Xz(v)- Machines depreciate at the rate 6 and investments in machines are reversible. Consider the monopolist owning the patent to a machine v for skill class z, invented at time 0. Define the total demand for machine v for skill type z as Kz(v) = Jq1 k(i,v)di. The monopolist chooses an investment plan and a sequence of capital stocks so as to maximize the present discounted value of profits, as given by Vz{v) — Jo°° e~rt [Xz{y)Kz(v) — 9Iz{v)\ dt—0K®(v), subject to Kz{y) = h{v) — 6Kz(u) and to the set of demand constraints given by (3), where we have suppressed time indexes. 0 denotes the marginal cost of machine production, assumed to be constant; is the quantity of machines produced by the monopolist at the time when the variety v is invented (in this case, at time 0); and Iz{v) denotes gross investment.

Since (3) defines isoelastic demands, the solution to this program involves Xz{v) — 0(r -f <5)/(l — /3), that is, all monopolists charge a constant rental rate, equal to a mark-up over the marginal cost times the interest rate plus the depreciation rate. We assume that the marginal cost of machine production in the North is 0 = (1 — P)2/(r + <5), so that \ — (1 ~ £0- Profit-maximization also implies Kz(v) = Kz{v) = Kz and Iz{v) = 5Kz{v) = 6KZ, that is, each monopolist rents out the same quantity of machines in every period. Notice also that Vz(i/) = Vz for all v, that is all machines produced for skill type г are equally profitable (though this profitability can change over time). Substituting (3) and the machine prices into (2), we obtain