Growth in an optimizing model of migration (based on Braun, 1993) Consider Braun's model of migration, which we presented in section 9.1.3. Toward the end of that section, we mentioned an extension to allow for the dynamics of the world economy. Assume that the framework of section 9.1.3 applies, including the production function in equation (9.27), except that the world now consists of only two economies, country 1 and country 2. The natural resources in each country, R1 and R2, are fixed.
The populations of each country are denoted by L1 and L2, where L = L1 + L2 is world population. Natural population growth rates are 0 in each economy, and the initial conditions are such that the flow of migrants is from country 2 to country 1.
The cost of moving from country 2 to country 1 is still given by equation (9.34), except that w world is replaced by w2.
The moving cost for each migrant again approaches 0 as the number of migrants goes to 0. Capital is perfectly mobile across the economies. The total capital stock, K = K1 + K2, is allocated across the economies to equalize the net marginal products of capital at each point in time. The world rate of return, r-which can now vary over time-equals the net marginal product of capital.
Assume for simplicity that technological progress and depreciation are absent. Consumers in each country have Ramsey preferences with infinite horizons, as assumed in chapter 2.
a. Work out the dynamic system in terms of the variables k, L2, B, and c, where B is the present value of the benefit from a permanent move from country 2 to country 1 (an analogue to equation [9.31]) and c ≡ C/L is the world's average consumption per person. Note that the state variables for the system are k and L2; for given L, the variable L2 determines the allocation of population between the two countries. (Hint: people who start in country 1 never move, and the path of consumption, c1, is the same for each person. For people who start in country 2, the path of consumption, c2, must be the same regardless of when they move to country 1 or whether they ever move. These considerations, along with the standard Ramsey formula for consumption growth, determine the behavior of c in relation to the rate of return, r.)
b. What are the steady-state values of k, L2, and B?
c. Consider a log-linear approximation of the dynamic system in the neighborhood of the steady state.
(i) Observe that, close to the steady state, a small change in L2 has a negligible effect on wage rates in the two countries, world output, and the rate of return. Use these facts to break the four-dimensional system into two separate parts: one that applies to the world variables, k and c, and another that applies to the migration variables, L2and B.
(ii) Find the speed of convergence, β, for the world variables and relate the answer to the solution of the Ramsey model from chapter 2.
(iii) Find the speed of convergence, µ, for L2. Show how the convergence speed for per capita output in one country, y1, depends on β and µ (see equation [9.45]).