Equilibrium: Transitional Dynamics

n_N,tdt= (1−n_N,t)ιtdt−n_N,tµdt, (1.30) and dividing by dtgives

˙

n_N,t= (1−n_N,t)ι_t−n_N,tµ. (1.31) Trade between North and South is balanced at each point of time, so there is no interna-tional debt.²⁶ In the North, the firm’s profits are given to households via dividends, and the government’s tariff revenue is distributed via lump-sum transfers to households. So, household expenditure equals firm revenue plus government revenue,

cN,tLN,t=nN,t(cN,tLN,t+cS,tLS,t) + (1−nN,t)cN,tLN,t

1 +τ_N τN. (1.32) Equation (1.32) can be rearranged to

cN,tLN,t= (1 +τN)cS,tLS,t

nN,t

1−n_N,t, (1.33)

and I refer to this as the balanced-trade condition.

1.2.4 Equilibrium: Transitional Dynamics

We are now in a position to solve the model. The Southern labour market condition, (1.12), reduces, after replacing cN,t using the balanced trade condition, (1.33), to

c_S,tL_S,t wS

=L_S,t, (1.34)

so

cS,t=cS=wS = 1, (1.35)

25It can also be interpreted as the average share of time in which the quality leader is located in the North.

26This is standard in this literature, see e. g. Grossman and Helpman (1991b, p. 149), Arnold (2002) or Grieben and S¸ener (2009a).

which means that Southern per-capita consumption equals the Southern wage,w_S, which is the numeraire.

I use the firm value, (1.27), in the free-entry-in-innovation condition, (1.16), to replacev^I_N,t, which yields:

which I use in the Keynes-Ramsey rule (1.9) for Northern per-capita consumption:

˙

To replace πN,t, I use the balanced-trade condition, (1.33), in the profit equation, (1.29), and withc_S = 1, I get

where Λ is the ratio of profit to revenue. Replacing the wage of rent-protection workers w_N,t^RP by equation (1.28) and using the definition of ∆ in (1.26), we can rearrange the above equation (1.38) to Using the labour market equation for specialised workers (1.11), we can derive that

D˙t

and using the free-entry-in-innovation condition, (1.16), we can show that

˙ v_N,t^I v_N,t^I = D˙_t

Dt

, (1.42)

sinceaR and ¯w are constant over time. Hence, we can rewrite equation (1.40) as

˙

N,t, I take logs in the balanced-trade condition (1.33) and differentiate it with

respect to time. That yields Using equation (1.17) and the equilibrium equation (1.11) for specialised workers to replaceDt, and using the equation for balanced industry flows, (1.31), in equation (1.46) to replace ι_t, we obtain after solving for ˙nN,t

˙

v_N,t^I is deterministically given by the dynamic version of the free entry in innovation condition, (1.42), and the growth rate of R&D difficulty, (1.41). We now have a nonlinear, non-autonomous differential equation of first order and first degree forn_N,t.

Using again (1.31) to replaceι_tin equation (1.28) yields the wage of rent-protection workers in terms ofn_N,t,

To finish the model’s solution, we need to state the Northern unemployment rate, u_N,t, in terms of nN,t and the model’s parameters. Therefore, we use the balanced trade condition, equation (1.33), in the Northern labour market equation (1.10). To replaceR_t, we use first the definition ofι_tin equation (1.5) and then the equation for balanced industry flows (1.31). This yields

The first component, 1−s, is the share of general purpose workers. The second component is

the share of production workers, and the third component is the share of R&D workers.

We now consider an equilibrium of the model in which the wage of rent-protection workers is above the minimum wage,w^RP_N,t >w, and in which there is positive unemployment. The way¯ unemployment is generated in this model may not be obvious at first sight, as the optimisation problem of challenging firms is linear in R&D activity R_m,t. If the free-entry-in-innovation condition holds, a single firm could employ more and more R&D staff without violating this condition. First, in equilibrium, there is no incentive to deviate in any direction. Second, this would prop up the innovation rate, and a higher innovation rate would result in a lower interest rate. The decline in the interest rate and the resulting decline in ^c_c^˙N,t

N,t would not match the increase in ⁿ_n^˙N,t

N,t

1

1−n_N,t. But that means that the expected marginal benefit from R&D would be lower than the marginal cost. If this were the case, people would stop doing R&D, which would generate unemployment.

Let us now determine the short-run effects of globalisation on unemployment and, in partic-ular, which employment share reacts immediately to changes in globalisation parameters. The short-run effect is what happens immediately, that is if we keep time constant. Keeping time constant also means to have a constant share of Northern quality leaders, nN,t, which is the state variable. It does not change in the short run because it only depends on the history of the model’s parameters.

Starting from any state of nN,t, a decrease in the Northern import tariff, τN, translates only into an immediate fall of ˙nN,t and hence lowers the share of R&D workers, but there is no immediate change in the share of production workers. A decrease in τ_N decreases the price markup in the North, and hence the incumbents’ profit and the incentives to innovate. It has no immediate effects on production because the lower tariff revenue from higher import tariffs is returned to Northern households, such that the price decrease for Northern consumers is neutralised. For Southern demand, there is also no change. Hence, production is still the same.

A decrease in the imitation rate, µ, yields also no immediate change in the share of produc-tion workers. But what is surprising is that the share of R&D workers decreases, caused by a decrease in the innovation rate ι_t. The intuition for this effect is easy to see: Suppose that the share of quality leaders is lower than in steady state whenµdecreases. As the danger of being imitated is now lower, the North has to put less effort into R&D to gain additional market shares. More formally, the decrease in µ leads to both an increase in r_N,t and hence ^c_c^˙N,t

N,t (see equation (1.38)). The decrease inµalso leads to an increase in ⁿ_n^˙N,t

N,t

1

1−n_N,t (see equation (1.31)), but this increase is stronger for a constant innovation rate.²⁷ Hence, to reestablish equality in equation (1.45), innovation is reduced: It raises the interest rate and hence ^c_c^˙N,t

N,t, but it lowers

By contrast, a change in the Southern market size, LS,t, affects both production and R&D worker shares. Demand from the South increases relative to demand in the North and produc-tion capacities are immediately increased. But this also increases profits and hence innovaproduc-tion

27Differentiating equation (1.38) gives

∂^cN,t˙ cN,t

∂µ =−1, while using equation (1.31) yields

∂^nN,t˙ nN,t 1

1−nN,t

∂µ =−_1−n¹

N,t.

activities increase. Let us summarise these findings in the first main result:

Proposition 1 (Short run effects)

Starting at any state, a decrease of the Northern import tariff, τ_N, or of the Southern imita-tion rate, µ, increase unemployment u_N,t. An increase of Southern market size, L_S,t, yields a decrease of unemployment.

The proofs are provided in Appendix 1.A.1. To distinguish these results from the paper by Dutt et al. (2009), we need to take into account that they consider frictional instead of structural unemployment. In their model, the short-run increase in unemployment comes from job destruction in import-competing sectors, which is faster than job creation in exporting sectors. So, the short-run effect and the transition towards the new steady state is a result from time-consuming frictions in the labour market.

In this model, the short-run effects and the transition to the new steady state result from time-consuming R&D processes and slow adjustments of the share of Northern quality leaders.

As I want to single out this effect, I consider structural unemployment. Trade liberalisation does not lead to firm exit and job destruction, as it has no effects on product demand. Con-sumers spend their income equally across all industries, and whether they buy a good from an industrialised country or from a developing country does not result from relative prices, but from quality leadership. This is not affected in the short run by import tariffs, but by innova-tive activities. However, if I had a matching framework, there would not be much difference.

It would only take longer to employ additional research workers. Nevertheless, job creation in research would start earlier than job creation in production.

Compared to models with full employment, this result also reveals that a model with un-employment yields different predictions on changes in un-employment shares. If the labour market were perfectly competitive, any immediate change in the share of R&D workers would require the opposite change in the share of production workers. In our case, the share of unemployed gives us an additional degree of freedom, as one of the three shares of workers can be kept constant.