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Black Death Raises Wages

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?

Background
The plague is characterized by large dark lumps in the groin or armpits followed by livid black spots on the arms, thighs, and other parts of the body. The Black Death led to a horrible demise in virtually all of its victims within one to three days.

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%.

Task
Your task is to use microeconomic theory to explain these shifts in real wages and rents. For simplicity, we'll assume that a single good, food, is produced using two factors, labor, L, and capital (though, it is often land or some other non-labor input), K according to a constant-returns-to-scale Cobb-Douglas production function: Q = ALaK1-a. The amount of land or capital is fixed. To ensure that the supply curve of labor is vertical, we'll assume that workers make their labor-leisure choice using a Cobb-Douglas utility function or the number of hours they work ("sun to sun") is set by social convention or by legal requirements. The Black Death causes the amount of labor available to fall from L to qL, where q is a number between a half and two-thirds. Because food is the only product workers and owners of land and capital can buy, they spend all their money on that good. Labor, capital, and output markets are competitive.
Analysis
Our basic intuition is that the capital-labor ratio rose, so the marginal product of labor increased, causing an increase in the wage. A full analysis requires that we consider what happens in output markets as well (see Chapter 10 on general equilibrium models). We've simplified this analysis by assuming that there is only one output market and that the supply curves of capital and labor are vertical.

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 = alphaQ/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.

Questions
  1. Why does assuming that workers have a Cobb-Douglas utility function over their labor-leisure choice guarantee that the number of hours they work is independent of the wage rate?

    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.

  2. To prevent wage rates from rising, English authorities imposed wage controls (maximum wage rates). According to the Statute of Labourers in 1351, no peasant could be paid more than the wage paid in 1346. In our model, what effect does a wage control have?

    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).


SOURCES: Wellington (1990); www.history-magazine.com/black.html; ww.historylearningsite.co.uk/black_death_of_1348_to_1350.htm; ww.bric.postech.ac.kr/science/97now/00_11now/001127a.html.




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