The Cournot-Ricardo Solution under Domestic Free Movement of Labour

Abstract

This paper analyses the properties of the Cournot-Ricardo solution if workers have incentives to move from low to high profit industries. By construction, these movements contribute to the improvement of the competitiveness of countries with free movement of labour, due to the increase in size of industries with a comparative advantage. However, in general, it is not sufficient to guarantee convergence to the Ricardo-Mill solution. In conclusion, under the inter-industry movement of labour, economies are more competitive, but specialisation might not coincide with that predicted by the Ricardo-Mill model.

Appendix: Proofs

Proof

(Proposition 1) Under conditions 1 and 2, the reaction function R ^c(z, t; x ^c′)can be found by solving the maximisation problem described in Eq. 2:

$$ R^{c}(z,t;x^{c\prime}) =\left( M(t)\alpha(z)x^{c\prime}(z,t)\right)^{1/2}\left( \omega^{c}(t)a^{c}(z)\right)^{-1/2}-x^{c\prime}(z,t). $$

(5)

The property P.1.proof is obtained from the algebraic manipulation of Eq. 5 for R ^c(z, t; x ^c′) > 0. Solving Eq. 5 for x ^c′ = R ^c′, it satisfies P.2. Solvingequations ∂ R ^c/∂ x ^c′ = 0, and ∂ ² R ^c/∂(x ^c′)² < 0, the reaction functionattains a maximum at x ^c′(z, t) = M(t)α(z)/(4a ^c(z)ω ^c(t))and P.3.holds for all z, since thismaximum is strict and global. □

Proof

(Proposition 2) Given the Definition 2 of the reactionfunction, the production level satisfies simultaneously x ^c(z, t) = R ^c(z, t; R ^c′),and x ^c′(z, t) = R ^c′(z, t; R ^c)at the Cournot equilibrium. Then, by Proposition1.- P.2it holds that:

$$ R^{c\prime}(z,t;R^{c})\geq \frac{M(t)\alpha(z)\omega^{c}(t)a^{c}(z)}{(2\omega^{c}(t)a^{c}(z))^{2}} =\frac{M(t)\alpha(z)}{4\omega^{c}(t)a^{c}(z)}, $$

(6)

since ω ^c(t)a ^c(z) + ω ^c′(t)a ^c′(z) ≤ 2ω ^c(t)a ^c(z)if c ^′has acomparative advantage.

The last term of the inequality (6) coincides with the property P.3.given in Proposition1. Then, if country c ^′has a comparative advantage, according to Definition 1, the reaction function R ^c(z, t; R ^c′)is non-increasing inthe Cournot solution. □

Proof

(Proposition 3) The total profit of the industry z and country c is π^c(z, t) = p(z, t)x ^c(z, t) − ω(t)ℓ ^c(z, t).Under Condition 1, x ^c(z, t) = ℓ ^c(z, t)/a ^c(z),and profits are π^c(z, t) = p(z, t)ℓ ^c(z, t)/a ^c(z) − ω(t)ℓ ^c(z, t).Under Condition 2 c),each worker supplies exactly one unit of labour, then π ^c(z, t) = π(z, t)/ ℓ ^c(z, t)is the per-worker participation in profits. Plugging π ^c(z, t)and p(z, t) = ω ^c(t)a ^c(z) + ω ^c′(t)a ^c′(z)in Eq. 3we find:

$$ m^{c}(z,t)=\frac{p(z,t)}{a^{c}(z)} =\frac{\omega^{c}(t)a^{c}(z)+\omega^{c\prime}(t)a^{c\prime}(z)}{a^{c}(z)} =\omega^{c\prime}(t)\left( \frac{\omega^{c}(t)}{\omega^{c\prime}(t)} +\frac{a^{c\prime}(z)}{a^{c}(z)}\right). $$

(7)

The result of Eq. 7proofs that m ^c(z, t) − m ^c(z + δ ^c, t)satisfies the first part of Eq. 4. The inequality can be proved observing that a(z)is strictly increasing.Then, for all z + δ such that z + δ ∈ (0, 1),and δ > 0, then 1/a(z) > 1/a(z + δ), and theinequality holds. □

Proof

(Proposition 4) Under Assumption 1, workers move from/to industries z ∈ I _R (δ).Since the strategies of industries and workers are independent, the z ∈ I _R (δ)industries cannot decide the amount of labour to be hired(or fired). Therefore, the labour demand of these industries is ℓ ^c(z, t) = ℓ ^c(z, 0) + d ℓ ^c(z, t),since d ℓ ^c(z, t)–the amount of labour hired (or taken)– is a scalar exogenously determined. The z ∈ I _S (δ)industries are under the Cournot rules. Thus the labour demand ℓ ^c(z, t)for z ∈ I _S (δ),depends on ω(t).Given the total supply of labour, L ^c,the equilibrium in the labour market satisfies:

$$ L^{c}={\int}_{I_{R}(\delta)}\left( \ell^{c}(z,0)+d\ell^{c}(z,t)\right)dz+{\int}_{I_{S}(\delta)} \ell^{c} (z,t)dz, $$

(8)

since all workers find a new job in some of the z ∈ I _R (δ)industries,${\int }_{I_{R}(\delta )}d\ell ^{c}(z,t)dz=0$. Hence, Eq. 8 does not depend on d ℓ ^c(z, t):

$$ L^{c}={\int}_{I_{R}(\delta)}\ell^{c}(z,0)dz+{\int}_{I_{S}(\delta)} \ell^{c} (z,t)dz $$

(9)

Since the reaction functions hold simultaneously at t = 0, total productionsatisfies x(z, 0) = R ^c(z, 0; x ^c′) + x ^c′(z, 0). Plugging x ^c(z, 0) = ℓ ^c(z, 0)/a ^c(z)in Eq. 5, andmanipulating terms ℓ ^c′(z, 0) = ω(t)ℓ ^c(z, 0)and z ∈ I _S (δ)holdsfor all t > 0.Also, ω(0) = L ^c′/L ^cat t = 0.Plugging these results in Eq. 9, the following relation holds for all t ≥ 0:

$$ \frac{\omega(0)L^{c}-\omega(0){\int}_{I_{R}(\delta)}\ell^{c}(z,0)dz}{L^{c}-{\int}_{I_{R}(\delta)}\ell^{c}(z,0)dz} =\omega(t)\frac{{\int}_{I_{S}(\delta)} \ell^{c} (z,t)dz}{{\int}_{I_{S}(\delta)} \ell^{c} (z,t)dz} $$

(10)

and, the equality ω(0) = ω(t)holds for all t ≥ 0. □

Proof

(Proposition 5) The labour demand at t > 0can be expressed as a function of those at the initial time t = 0,i.e. ℓ ^c(z, t) = ℓ ^c(z, 0) + d ℓ ^c(z, t).Since ω(0)ℓ ^c(z, 0) + d ℓ ^c′(z, t) = ω(t)(ℓ ^c(z, 0) + d ℓ ^c(z, t))for all t ≥ 0(see proof of Proposition 4), arranging terms, a necessarycondition for the symmetric equilibrium for any c and period t ≥ 0is

$$ \ell^{c}(z,0)\left( \omega(0)-\omega(t)\right)= \omega(t) d\ell^{c}(z,t)-d\ell^{c\prime}(z,t) $$

(11)

Let us assume that d ℓ ^c(z, t) > 0 workers decide to move between industries z ₀and z ₁. Proposition 3 shows that z ₀ < z ₁;and, d ℓ ^c(z ₀, t) > 0, d ℓ ^c′(z ₀, t) < 0, d ℓ ^c(z ₁, t) < 0, d ℓ ^c′(z ₁, t) > 0. From Eq. 11,

$$\begin{array}{@{}rcl@{}} \ell^{c}(z_{0},0)\left( \omega(0)-\omega(t)\right)= \omega(t) d\ell^{c}(z_{0},t)-d\ell^{c\prime}(z_{0},t)\geq 0 \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} \ell^{c}(z_{1},0)\left( \omega(0)-\omega(t)\right)= \omega(t) d\ell^{c}(z_{1},t)-d\ell^{c\prime}(z_{1},t)\leq 0 \end{array} $$

(13)

Equations 12 and 13 hold simultaneously if and only if $\left (\omega (0)-\omega (t)\right )=0$, since ℓ ^c(z, 0) > 0for z = z ₀, z ₁. Then, ω(t)d ℓ ^c(z, t) = d ℓ ^c′(z, t) and, for any ω(t) > 0 it satisfies that d ℓ ^c(z, t) = 0, since d ℓ ^c(z, t) and d ℓ ^c′(z, t) have the opposite sign for any non zero movement of labour. □

Proof

(Proposition 6) Since the labour market is in equilibrium –see Eq. 8– for all z ∈ I _S (δ) the labour demands satisfy ℓ ^c′(z, t) = ω(t)ℓ ^c(z, t) (see proof of Proposition 4), then the following relation holds:

$$ \frac{L^{c\prime}-{\int}_{I_{R}(\delta)}R^{c\prime}(z,t;x^{c})a^{c\prime}(z)dz}{L^{c}-{\int}_{I_{R}(\delta)}\ell^{c}(z,0)dz} =\omega(t)\frac{{\int}_{I_{S}(\delta)} \ell^{c} (z,t)dz}{{\int}_{I_{S}(\delta)} \ell^{c} (z,t)dz}=\omega(t) $$

(14)

Since all workers find a new job or do not move, then${\int }_{I_{R}(\delta )}d\ell ^{c}(z,t)dz=0$. Thus, the denominator on the left-hand side of Eq. 14 is constant or, equivalently, independent of the inter-industry movements.

Under Assumption 1, workers leave industries z ∈ [1 − δ, 1] (without a comparative advantage) and seek a new job in industries z ∈ [0, δ](with a comparative advantages). Since a(z) is an increasing function, x ^c(z, t) increases if z ∈ [0, δ], and it decreases if z ∈ [1 − δ, 1]. Proposition 2 shows that R ^c′(z, t; x ^c)is non-increasing in z ∈ [0, δ]and non-decreasing in z ∈ [1 − δ, 1]. Then, ℓ ²(z, t) ≤ ℓ ²(z, 0) for all z ∈ I _R (δ), and ω(t) ≥ ω(0). □