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The problem is asking you to analyze the rise in wages after the Black Death using a production model and the Cobb-Douglas production function, which is given as:

\[
Y = \bar{A}K^{1/3}L^{2/3}
\]

Where:
- \( Y \) is the total output (GDP),
- \( \bar{A} \) is total factor productivity (a measure of technological efficiency),
- \( K \) is the capital stock,
- \( L \) is labor,
- The exponents \( 1/3 \) and \( 2/3 \) represent the output elasticities of capital and labor, respectively.

### (a) Explanation of why wages may have increased using the production model:

In the Cobb-Douglas production function, labor (\( L \)) and capital (\( K \)) contribute to output. The marginal product of labor (MPL) determines the wage rate in a competitive market. The marginal product of labor is given by:

\[
MPL = \frac{\partial Y}{\partial L} = \frac{2}{3} \cdot \frac{Y}{L}
\]

When the Black Death occurred, it significantly reduced the labor force (\( L \)) by about one-third. Holding capital (\( K \)) and productivity (\( \bar{A} \)) constant, this reduction in labor increased the marginal product of labor, which leads to an increase in wages.

This increase occurs because with fewer workers, the remaining laborers become more valuable to the production process, and their productivity (and thus wages) rises, assuming the demand for goods remains stable or increases. This also assumes that the capital stock remains relatively unchanged, which makes each worker more productive due to the higher capital-to-labor ratio.

### (b) Attaching a number to the explanation:

Given that the Cobb-Douglas function is \( Y = \bar{A}K^{1/3}L^{2/3} \), we can calculate the percent change in wages by examining the change in \( L \).

Let the population (labor) drop by 1/3, so \( L \) becomes \( \frac{2}{3}L \). We want to see how this affects the marginal product of labor (MPL), which is proportional to:

\[
MPL \propto \frac{Y}{L} \propto \frac{K^{1/3}L^{2/3}}{L} = K^{1/3}L^{-1/3}
\]

So, after the population decreases to \( \frac{2}{3}L \), the new marginal product of labor is:

\[
MPL' \propto K^{1/3}\left(\frac{2}{3}L\right)^{-1/3} = K^{1/3}L^{-1/3} \left(\frac{2}{3}\right)^{-1/3}
\]

This simplifies to:

\[
MPL' \propto MPL \times \left(\frac{3}{2}\right)^{1/3}
\]

Therefore, wages rise by a factor of \( \left(\frac{3}{2}\right)^{1/3} \), which is approximately \( 1.1447 \). This corresponds to an **increase in wages by about 14.47%** after a third of the population dies.

Would you like further details or clarification on this calculation?

### Follow-up Questions:
1. What assumptions are made in the Cobb-Douglas model when analyzing changes in wages?
2. How does the change in capital affect wages if the labor force decreases?
3. How would a decrease in total factor productivity (\( \bar{A} \)) during the Black Death affect wages?
4. What other factors, aside from the labor force, could influence wages after the Black Death?
5. How would wages change if labor were to rebound in the years following the Black Death?

#### Tip:
When analyzing economic models, always keep track of how each variable (like labor or capital) changes in response to external shocks. This helps predict outcomes more accurately.

In the middle of the fourteenth century, an epidemic known as the Black Death killed about a third of Europe’s population, about 34 million people. While this was an enormous tragedy, the macroeconomic consequences might surprise you: Over the next century, wages are estimated to have been higher than before the Black Death.

(a) Use the production model to explain why wages might have been higher.
(b) Can you attach a number to your explanation? In the model, by what percent would wages rise if a third of the population died from the disease assuming the standard Cobb-Douglas production function (Y = ĀK^(1/3)L^(2/3))?

Explore how the Black Death led to a rise in wages using the Cobb-Douglas production model. Understand the marginal product of labor and how a one-third population decrease impacts wages.

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