Suppose that a significant fraction of the civilian working-age population was decimated in a short period of time due to a war, while capital and land remained virtually unaffected. According to the production model with production function Y=AKαL1−α and α∈(0,1), which of the following statements is true?

Question

BlackTom AI · Accepted Answer

First, let's restate the problem to make sure it's understood: we have a production function Y = A K^α L^{1−α} with α ∈ (0,1). A war decimates a significant fraction of the civilian working-age population, so the labor input L falls sharply, while capital K and land stay largely unaffected. We are asked which statement is true given this setup, and the available answer is: 'Total output dropped, but output per capita increased.'

Now, analyze the impact on total output Y. Since Y = A K^α L^{1−α}, a fall in L lowers L^{1−α} (because 1−α > 0). With K and A unchanged, this reduced L translates directly into a smaller total output Y. In other words, total output falls when labor input collapses, assuming capital stock does not instantly adjust upward to offset the loss.

Next, analyze output per capita, y = Y/L. Using the production function, y = (A K^α L^{1−α}) / L = A K^α L^{−α}. Here, the exponent on L is −α, which is negative since α > 0. This means that as L falls, L^{−α} rises, so y rises if K remains unchanged. Intuitively, with fewer workers but the same amount of capital, each worker can, on average, produce more, pushing income per person up even though total output declines.

With this understanding, evaluate the given statement: 'Total output dropped, but output per capita increased.' The first part is correct: total output declines due to the drop in labor input L. The second part is also correct: output per capita increases because y = A K^α L^{−α} increases as L decreases, assuming K does not immediately collapse. Therefore, the statement is internally consistent with the production function and the implied shock to labor.

Since there are no other answer options provided to compare, we cannot evaluate alternatives. However, based on the standard neoclassical Cobb‑Douglas-like production function, the qualitative result is clear: a large, persistent decline in labor lowers Y but raises Y per worker, given capital does not fall in the same period.

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