2.4 The Large Country Case
2.4.3 Production Side
The producers in the home country employ capital and labor to produce manufac-tured and agricultural goods. We assume that capital is sector specific in the short run but perfectly mobile in the long run, whereas labor is always perfectly mobile.
Producers thus will have to decide how much capital to employ in which sector, before the state of the world gets revealed. Labor can subsequently be shifted from one sector to the other, in order to react to the realization of the economic outcome.22 The production functions for the two sectors are
ym1 =ϕ(KmγL1−γm )δ
E[ya1] =π[s(KaγL1−γa )δ] + (1−π)[(KaγL1−γa )δ]
(2.18)
ϕ is a productivity parameter which is used later in order to model comparative advantage. The larger ϕ, the bigger the relative productivity of the manufacturing
21See the Appendix for a discussion of the need for such a normalization and a derivation of the normalization we use.
22The assumption that labor is more mobile in the short-run than capital is common in the economic literature, see e.g. Neary (1978).
Labor rigidity can also have influences on the production structure, as shown in Trentinaglia De Daverio (2013). She finds that firms will install over-capacities if labor is rigid and there is demand uncertainty.
sector, compared to the agricultural sector for which productivity is normalized to one in both countries.23
π describes the probability that a negative production shock s will occur. We assume that 0< δ < 1 which implies decreasing returns to scale. In this way, we assure that perfect specialization in one sector is not a viable option. γis the capital share of output.
For the sake of simplicity, we set π = 12 and normalize the shock such that it becomes zero in expectation.
E[ya1] = 1
2(1 +σ1)(KaγL1−γa )δ+1
2(1−σ1)(KaγL1−γa )δ (2.19) σ1∈[0,1] is the shock parameter which measures volatility in the agricultural sec-tor for the home country. We model it such that it takes the value−σ1 orσ1 with probability 12. Therefore,E[σ1] = 0.
The subscript 1 identifies variables that belong to thehome country. The produc-tion funcproduc-tions of the foreign country are assumed to be
ym2= (KmγL1−γm )δ E[ya2] = 1
2(1 +σ2)(KaγL1−γa )δ+1
2(1−σ2)(KaγL1−γa )δ
(2.20)
The correlation between the shocks in the two countries is r, where a positive r makes states more probable, where the output shock has the same sign in both countries. For a negative r, asymmetric cases with high agricultural output in one country and low output in the other become more likely to occur.
Note that, for large shocks in both countries, situations are possible, in which the effective comparative advantage can be reversed. This is the case if one country has a favorable shock in the sector where it does not have its comparative advantage, and vice versa. If the following equations are fulfilled, this will not occur.
ϕ 1 +σ1
> 1 1−σ2
for comparative advantage in manufacturing ϕ
1−σ1
< 1 1 +σ2
for comparative advantage in agriculture
(2.21)
In most cases, such an ex-post reversal of comparative advantage would not imply a trade reversal, because capital is allocated to the different sectors before the shock occurs. Thus, with sufficient specialization, a country will still produce more of the good for which it has an expected comparative advantage, even if, ex post, it has a
23We chose to use similar production functions for both sectors for mathematical convenience, and since differences in the return to capital and labor in the two sectors will not influence the intuition of our results. Note that for our purposesϕis sufficient to model productivity differences between the sectors.
disadvantage.24
We assume that the domestic country imposes an ad valorem tariff ton all im-ported goods. The social planner announces this trade policy before the producers allocate capital to the different sectors. We assume that the legislation is such that trade policy cannot be changed in the short run. Therefore the social planner has to commit to an ad-valorem tariff rate that cannot be state dependent. Since the policy is irrevocable, producers will take the announced policy as a credible threat and maximize their expected profits accordingly. The domestic producers maximize their expected profit, given the normalized world market prices and the announced trade policyt. The domestic country is assumed to have a comparative advantage in the manufacturing sector in all possible states of the world. This implies that there is no trade-reversal, and the home country will always be a net exporter of the manufactured good. The trade policy t that the social planner can set is then a tariff on the net imported agricultural good. The tariff will change the relative domestic price. With our uncertainty robust price normalization, the tariff changes the relative price for the manufactured good to (pw/(1 +t))1−α.
In the aggregate, the profit maximizing behavior of the producers leads to a maxi-mization of the expected value of the total output of the domestic country, evaluated at the expected relative domestic prices. This is given by
E[I(t, p)] =E[( pw
1 +t)1−αym+ (1 +t
pw )αya] (2.22) wherepw indicates the world market relative price. Setting in the production tech-nology and resource constraints for the home country, this becomes
E(I) =E[( pw
1 +t)1−αϕ(Km1γ L1−γm1 )δ +(1 +t
pw )α((K1−Km1)γ(L1−Lm1)1−γ)δ] i, j∈(h, l)
(2.23)
whereϕagain measures the comparative advantage in producing the manufactured good.
We limit ourselves to a discrete version of uncertainty, where each country has either a high or low productivity in the agricultural sector. Shocks to the foreign agricultural output will also have an influence on the domestic market, since these shocks influence the world market price. A positive shock to the foreign agricultural productivity will drive the relative price of agricultural products down, thus leading to a more favorable situation for the domestic country, since we assume that it is a net exporter of the manufactured good. With two productivity outcomes in each country, this gives us four different states of the world. We can thus rewrite (2.23) as
24See Appendix for an illustration.
E[I1] = 1
4(1 +r)Ihh(Km1, Lm1, phh)|σ1>0+1
4(1−r)Ihl(Km1, Lm1, phl)|σ1>0
+1
4(1−r)Ilh(Km1, Lm1, plh)|σ1<0+1
4(1 +r)Ill(Km1, Lm1, pll)|σ1<0
(2.24)
Producers maximize their expected profit by choosing capital and labor optimally.
Since labor is mobile in the short run, producers can re-optimize their production point once the state of the world gets revealed. Labor is a state dependent function of the relative price of the state, the previously determined capital allocation and the trade policy. The optimal labor allocation follows in each state by maximizing within state profit. It can be written as
Lt∗m1ij(Km1, pwij, t) = L1
(p
w ij
1+t)1−αϕKm1δγ1−δ(1−γ)1
(p
w ij
1+t)1−αϕKm1δγ
1−δ(1−γ)1 +
(1±σ1) 1+t
pwij
α
(K1−Km1)δγ
1−δ(1−γ)1 i, j∈(h, l)
(2.25)
The producers anticipate that they will choose labor optimally in each state of the world and can thus determine the optimal capital allocation by backwards induction.
They will therefore plug the above expression for optimal labor into the long run maximization problem and choose capital accordingly.
Equation (2.24) can thus be expressed as a function that depends only on prices, capital and the trade policy.
E[I1(Km1, L∗m1ij(Km1, pwij, t), pwij, t)] =E[I1(Km1, pwij, t)] i, j∈(h, l) (2.26) Taking the derivative with respect to Km1 and setting it equal to zero gives us the first order condition for the domestic producers’ problem. This is the first equation that determines the equilibrium of the economy.
∂E[I1]
∂Km1 =f1(Km1∗ , pwij, t)= 0! i, j∈(h, l) (2.27) The maximization problem for the foreign producers is similar to that of the do-mestic producers. What changes is that ϕ2 = 1, where the subscript 2 denotes country 2, and that the endowments of the foreign country might differ from those of the home country. Furthermore, we assume that there is no trade policy abroad, so that the foreign producers trade at the world market prices. The foreign pro-duction decision is indirectly influenced by the domestic trade policy, since a tariff
influences the domestic allocations and thus change the equilibrium prices.
The first order condition of the foreign country is the second equation that deter-mines the equilibrium of the economy.
∂E[I2]
∂Km2 =f2(Km2, pwij, t)= 0! i, j∈(h, l) (2.28)