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Say a natural disaster strikes. People were killed but the losses were small relative to the size of the work force. However, many buildings and infrastructures have been severely damaged. I'm thinking how would such an event impact the shape of the production function. Assume the production function is of Cobb-Douglas form.

On one hand, I'm compelled to say that the production function shifted down. The labor force is still there but there are now perhaps too many people to work the reduced amount of capital, so productivity of labor decreases, shifting the function downwards.

However, I also feel like it's just a movement. The technological level remains the same. We would just move further down the function as capital decreases.

Those are my thoughts. I'm having trouble thinking this through.

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Suppose the Cobb-Douglas production function is $Y = AK^{\alpha}L^{\beta}$, and the effect of the disaster is to reduce $K$ from $K_0$ to $K_1$. If any change in $A$ and $L$ is insignificant, then you can look at this in two ways:

  1. Focusing on a partial production function in terms of $L$ only (that is, treating $A$ and $K_i$ as fixed parameters in the short term), there has been a change from $Y=AK_0^{\alpha}L^{\beta}$ to $Y=AK_1^{\alpha}L^{\beta}$. With $K_1<K_0$, that implies a shift downwards, with labour productivity having decreased.

  2. However, the full production function in terms of $K$ and $L$ is unchanged, and there has been a movement on the surface defined by the function. If the function were plotted in three dimensions, with $K$ and $L$ on horizontal axes and $Y$ vertical, then this would be a movement downward (and along the $K$ dimension).

These are both valid ways of describing the situation: which is more useful would depend on what you are trying to do.

One caveat. Although production functions often suggest otherwise, labour is not homogeneous. Even if only a few people are killed, the effect on production could be significant if those people happen to have key skills that are in short supply, or knowledge the loss of which would affect the technological level.

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