Assignment

1. In this question, you will consider what would happen if a society has access to both the Malthus and neoclassical technologies at the same time. To keep things simple, we will abstract from capital accumulation.

Suppose that there are two production processes available to produce the same good. The first uses land and labor, and the second uses only labor. In particular, suppose Y_1,t  = S_t^α(A_t L_1,t)^1-α and Y_2,t = A_tL_2,t , where L_i,t is the amount of labor employed in process i, S is the stock of land, and A is technological progress. Notice that both of these production functions display constant returns to scale. Assume that the stock of land is constant over time (equal to 1) and that A_t+1 = (1+ g) A_t.

Assume, initially, that the population, L, is constant over time. Let θ be the fraction of the population employed in the land-intensive sector, where 0 ≤ θ ≤ 1 . That is, L_1,t = θ_tL and L_2,t = (1-θ_t)L. Total output is given by Y_t = (θ_tA_tL)^1-α + (1-θ_t )A_tL.

A. Suppose that a planner wanted to maximize total output by choosing θ. Derive an expression for the optimal choice of θ as a function of A. [Hint: Remember that θ must between 0 and 1.] How large must A be, in terms of L and Y, for the less land-intensive technology to be adopted (for θ to be less than 1)?

B. Set L = 1, A₀ = .001, α = .4 , and g = .5 . Use Excel to simulate this economy for 30 periods. Produce graphs showing the time series for θ and y (per capita income). [Hint: Use the min function in Excel to simulate θ , i.e. θ = min(1,some expression).]

2. Suppose that there are two production processes available to produce the same good. The first uses land and labor, and the second uses only labor. In particular, suppose Y_1,t  = S_t^α(A_tL_1,t)^1-α and Y_2,t = At L_2,t , where L_i,t is the amount of labor employed in process i, S is the stock of land, and A is technological progress. Notice that both of these production functions display constant returns to scale. Assume that the stock of land is constant over time (equal to 1) and that A_t+1 = (1+ g) A_t. In addition, suppose that a period is 35 years and that the population growth rate is a linear function of per capita income. Set α = .4, L₀ = 1, and A₀ = 0.1.

A. Show that in period 0,θ_t = 1 . That is, the economy starts off using only the Malthusian technology.

B. Suppose that the economy is initially in the Malthusian steady state and that the population doubles every 230 years. What is the per period population growth rate in this steady state? What value for g is consistent with maintaining this population growth rate in a Malthusian steady state? What is per capita income in this steady state?

C. Suppose that the linear population growth function is of the form,η_t = a₀ + a₁y_t , where η_t is the growth rate of population from period t to t +1 and yt is per capita income. Find values for a0 and a1 consistent with the following two facts: (i) (η0, y0 ) are the Malthusian steady state values computed in (b); and (ii) the population doubles every 35 years (every period) when per capita income is twice the Malthusian level.

D. Using the parameter values found (or given), simulate this economy for 30 periods. Produce graphs showing the time series for θ and y (per capita income). About how many periods does it take for the industrial revolution to be complete once it begins (for 99 percent of the labor to be employed in the less land intensive sector)?