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The Black Death - bubonic plague - wiped out between a third and a half of the population of medieval Western Europe. Many historians report that real wages rose and the real rents on land and capital fell. Why?
In England, the plague struck in 1348-1349, 1360-1361, 1369, and 1375. According to one historian, the population fell from 3.76 million in 1348, 3.13 million in 1348-1350, 2.75 million in 1360, 2.45 million in 1369, 2.25 million in 1374, and recovered to 3.1 million in 1430.
English nominal wages rose in the second half of the fourteenth century compared to the first half: Thatchers earned 1.35 times as much, thatcher's helpers 2.05, carpenters 1.40, masons 1.48, mowers 1.24, oat thresher 1.73, and oat reaper 1.61. In Pistoia, Italy, rents in kind on land fell by about 40% and the rate of return on capital fell by about the same proportion.
English relative prices changed substantially across products, with non-farm product prices generally rising relative to those of farm products. Again comparing the second half to the first half of the fourteenth century, grain prices were about the same (0.97 times) on average, livestock was up 1.17 times, farm products 0.98, wool 1.01, textiles 1.59, wood and metal 1.73, building materials 1.78, agricultural implements 2.14, and foreign products 1.54. According to one source, the real wage rose by about 25%.
What effect does the Black Death have on the marginal product of labor? From Chapter 15, we know that the marginal product of labor for our Cobb-Douglas production function is MPL = Q/L. When labor falls from L before the Black Death to L* = qL after, output falls from Q = ALaK1-a to Q* = A(qL)aK1-a = qaALaK1-a = qaQ. Thus, the output to labor ratio changes from Q/L to qaQ/(qL) = qa-1Q/L > Q/L (because a-1 < 0, raising a fraction, q, to that power results in a number greater than one). Consequently, the marginal product of labor rises from MPL = aQ/L to MPL* = qa-1Q/L = qa-1MPL.
The factor demand equations are determined by setting the factor price equal to its marginal revenue product (Chapter 15). The competitive labor demand equation is w = pMPL. Rearranging this expression, we find that the real wage equals the marginal product, w/p = MPL. (We refer to w/p as the real wage because there is only one price, p.) Thus, the real price of labor rises because the MPL rises.
Because output falls and capital remains the same, the marginal product of capital falls from MPK = (1-a)Q/K to MPK* =qa(1-a)Q/K = qaMPK. Consequently, the real price of capital, r/p = MPK, drops.
Because of our assumption that the production function has constant returns to scale, the sum of the labor and non-labor earnings exactly equal the amount spent on food. Wage earnings are wL = ap(Q/L)L = apQ, non-labor earnings are rK = (1-a)p(Q/K)K = (1-a)pQ, so their sum is pQ. If we normalize the price so that p = 1, then w = a and r = (1-a), so labor and capital split total output, Q, in proportions determined by the production function.
We can use a numerical example to illustrate the basic idea. Suppose that a = ½, A = 1, K = 100, and L = 100. Then output is Q = LaK1-a = 100½100½ = 100. The marginal product of labor is MPL = aQ/L = ½(100/100) = ½, so the real wage is w/p = ½. Similarly, the marginal product of capital and the real price of capital are ½.
Suppose that the labor force plummets to 25 (a larger drop than occurred, but an assumption that leads to relatively simple calculations). Output falls to Q = 25½100½ = 50, MPL = ½ (50/25) = 1 = w/p, and the MPK = ½ (50/100) = ¼ = r/p. Thus, the real wage doubles and the real rental rate on capital is sliced in half.
Answer: If a worker's utility function is Cobb-Douglas, U = YbN1-b, where Y is the amount of a good purchased and N is the number of hours of leisure, then the labor supplied per day is 24b, regardless of the wage.
Answer: Given a vertical supply curve—one that is not sensitive to the wage—the only effect of the wage control is to create excess demand for labor and to transfer wealth from workers (peasants and craftsmen) to their employers (the nobility and merchants).