Suppose that a hurricane damages some physical capital. You happen to remember that this loss of capital affects the marginal product of capital in the Solow model. Taking this effect into account and assuming that the real interest rate
𝑅
𝑡
remains constant, the IS curve predicts
[ Select ]
a drop
no change
a rise
in investment and
[ Select ]
a drop
no change
a rise
in short-run output.

Note: First, you need to determine whether the destruction of physical capital increases or decreases the marginal product of capital. If you are unsure, you can use the neoclassical Cobb-Douglas production function
𝑌
=
𝐴
𝐾
𝛼
𝐿
1
−
𝛼
as a example and compute the MPK by taking the derivative with respect to
𝐾
. You can then determine the direction of the shift of the IS curve.

Question

Suppose that a hurricane damages some physical capital. You happen to remember that this loss of capital affects the marginal product of capital in the Solow model. Taking this effect into account and assuming that the real interest rate 
𝑅
𝑡
 remains constant, the IS curve predicts 
[ Select ]
a drop
no change
a rise
 in investment and 
[ Select ]
a drop
no change
a rise
 in short-run output.

Note: First, you need to determine whether the destruction of physical capital increases or decreases the marginal product of capital. If you are unsure, you can use the neoclassical Cobb-Douglas production function 
𝑌
=
𝐴
𝐾
𝛼
𝐿
1
−
𝛼
 as a example and compute the MPK by taking the derivative with respect to 
𝐾
. You can then determine the direction of the shift of the IS curve.

BlackTom AI · Accepted Answer

Question restatement: We’re told a hurricane damages physical capital, and this damage affects the marginal product of capital (MPK) in the Solow model. With the real interest rate R_t held constant, the IS curve’s predictions are asked for two choices: first, the change in investment, and second, the change in short-run output. The given answer pattern is that both selections are 'a rise'.

Analyzing the MPK effect: In the neoclassical Solow framework, MPK is the derivative of output with respect to capital, MPK = ∂Y/∂K. Using a typical Cobb-Douglas production function Y = A K^α L^{1−α} with 0 < α < 1, we get MPK = α A K^{α−1} L^{1−α}. Since α−1 is negative, MPK decreases when K increases and increases when K decreases. Therefore, the hurricane reducing physical capital raises MPK, not lowers it.

Implication for investment (first blank): If MPK rises, the return to capital increases at the existing real interest rate. With the interest rate held constant, firms observe a higher marginal product of capital and thus have an incentive to invest more today to exploit the now more productive capital. In standard IS analysis, an increase in MPK translates into higher desired investment, all else equal. Consequences: the first blank should be 'a rise'.

Implication for short-run output (second blank): Higher investment goods, driven by the elevated MPK, require more spending on capital goods, boosting aggregate demand in the short run. With investment up and a constant interest rate path, output expands in the near term as the economy moves toward the new, higher capital stock path. Thus the second blank also should be 'a rise'.

Why not the other options for the first blank? A drop would imply investment falls despite a higher MPK, which contradicts the intuition that a more productive capital stock should encourage more investment when the cost of borrowing (the real rate) is unchanged. No change would ignore the explicit shift in MPK and the consequent change in investment incentives. Why not for the second blank? If investment rises, and prices/R_t are constant in the short run, the demand-side effect raises output, so 'a rise' is the coherent outcome rather than 'a drop' or 'no change'.

Put together, the consistent sequence is: the hurricane-induced capital loss raises MPK, which, with R_t fixed, stimulates investment, and the higher investment raises short-run output. Therefore, the two selections being 'a rise' and 'a rise' align with the standard neoclassical/Solow intuition about capital-destructive shocks and their short-run effects on investment and output.

FA25 ECON 302 002 Homework #8 (IS Curve)

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In the Solow model with no population growth (i.e., L is fixed), a rise in the saving rate leads to a higher steady-state capital stock, and a rise in the depreciation rate leads to a lower steady-state capital stock.

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