Question

a. Show, stating all required assumptions, that the growth rate of efficiency in any

economy can be calculated as (where z = Z/L) (In malthusian economy)

gA = gy - agk - cgz

b. Show that the result in part (a) above implies that the growth rate of efficiency in

Malthusian economies over the long run is given by

gA = cgN

where c is the share of land rents in all incomes, and gN is the population growth rate.

Answer 1

Solution: a) We will denote the quantity of output, capital, land and labor in any economy by Y, K, Z, and L and the prices paid for output, capital, land and labor per unit as p, r, s, and w. Also assume that the efficiency of the economy can be measured by a single index number A.

Thus we can write Y = AF(K, L, Z) ---------- 1

Equation 1 says output is the product of the efficiency level of the economy times some function F(..)of the amounts of capital, labor and land.

Let a ΔX indicate the change in the amount of any quantity or price X in a year. Thus ΔY is the change in output in any year, ΔK the change in the capital stock in a year, and ΔA the change in the level of efficiency of the economy. In this case the annual growth rate of output g_y will be

g_{Y = Y/Y}

Similarly annual growth rate of the capital stock will be,

g_{K= K/K}

Also the growth rate of the efficiency of the economy will be,

g_{A= A/A}

Thus change in output from a change of ΔK, ΔL, ΔZ, in inputs and ΔA in the level of efficiency is given by,

ΔY = mpK ΔK + mpLΔL + mpZΔZ + F(K,L,Z)⋅ΔA

In a competitive economy with constant returns to scale, all factors get paid the value of their marginal products. Therefore the value of the marginal product of labor, mpL, is just the wage w. Similarly the value of the marginal product of capital will be r, and the value of the marginal product of land the rent s.

ΔY = rΔK + wΔL + sΔZ + F(K, L,Z)⋅ΔA

Dividing both sides by Y we get

rK + wL + sZ = pY = Y (since p=1)

Thus rK/Y = a is the share of capital in national output, wL/Y = b is the share of labor in national output, and sZ/Y = c is the share of natural resources in national output (a + b + c = 1)

Hence we can rewrite

gY = a.gK + b.gL + c.gZ + gA

For small changes in Y and in L year by year this is approximately equivalent to,

Thus gy = gY/L ≈ g Y – gL

Similarly the rate of growth of capital per worker (k = K/L) is g_K– g_L, and the rate of growth (or more often of decline) of resources per worker (z=Z/L) is g_Z – g_L.

If we subtract g_L, the rate of growth of the labor supply from each side of equation we get, g_Y – g_L = ag_K + bg_L + cg_Z + g_A – g_L

= a(g_K – g_L) + c(g_Z– g_L) + g_A

gy = agk + cgz + g_A

gA = gy - agk - cgz

Solution b.

Before 1800 we have a special case of equation gy = agk + cgz + g_A

where in the long run gy = gk = 0. Also gz = -g_N , where N is the level of population. Thus if population was growing at 1 percent per year, then land per person was falling at this rate.

Substituting these values gives, for the long run,

Hence gA = cgN

Since income per person does not change over the long run in the Malthusian economy, and since to a first approximation wages and the return on capital should be constant.