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

For a Cobb–Douglas production function Y = A K^α L^{1−α} the marginal product of capital (MPK) is
MPK = ∂Y/∂K = α A K^{α−1} L^{1−α}.
Because 0 < α < 1, the exponent α − 1 is negative, so a fall in K raises K^{α−1} and therefore raises MPK. Concretely, if K falls to 0.9K, then Y becomes Y_new = A (0.9K)^α L^{1−α} = 0.9^α Y, and
MPK_new = α A (0.9K)^{α−1} L^{1−α} = 0.9^{α−1} MPK_old,
and since 0.9^{α−1} > 1 (because α−1 < 0), MPK increases.

Step-by-step reasoning and channels at work
1) Supply-side effect: the destruction of K raises MPK (shown above). Holding the real interest rate r constant, a higher MPK tends to make investment more attractive from a profitability perspective and would, other things equal, push planned investment up.
2) Demand-side / short-run income effect: the direct effect of the hurricane is a lower capital stock and therefore a lower current output Y (Y_new = 0.9^α Y, so Y falls). In standard IS analysis investment is a function of output and the interest rate (I = I(Y,r)); lower current output reduces demand for investment goods (the accelerator effect and lower retained earnings/cash flow), generating a reduction in planned investment at the given r.
3) In the short run the negative output/income (demand) effect dominates the MPK profitability effect: firms face disrupted demand, financing and adjustment frictions, and immediate capacity loss that reduces sales. Thus aggregate planned investment falls and the IS curve shifts left, lowering short-run output further.

Why the correct answer is a drop / a drop
- Investment falls because the immediate decline in output and demand after the hurricane outweighs the mechanical rise in MPK (firms do not instantly and fully offset destroyed capital by new investment at the same r). 
- Short-run output falls directly because K is lower and because the IS shift left (lower planned spending) reduces equilibrium output.

Why other choices are incorrect
- “No change” is wrong because capital destruction changes both Y and investment incentives. 
- “A rise” is wrong because although MPK mechanically rises, the dominant short-run effect on planned investment is negative through reduced output and demand, so investment does not rise in equilibrium at fixed r.

Conclusion
Combining these channels, the IS prediction is a drop in investment and a drop in short-run output — matching the provided answer (a drop, a drop).

类似问题

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

In this same Solow economy, you observe that GDP grows at a rate of 2.7 percent per year. Based on this information you know that the economy is _______.

An increase in leads to a higher steady-state capital stock, and a decline in leads to a lower steady-state capital stock.

Let 𝑠 ¯ be the investment rate, 𝛿 the depreciation rate, 𝐴 ¯ aggregate productivity (or TFP), 𝐿 ¯ the population, and 𝛼 the exponent on capital in the production function of a Solow economy. What is the steady-state level of investment per person 𝑖 ∗ ?

In the Solow model with population growth rate 𝑛 ¯ = 3 % , you know that total GDP grows at a rate of ________ percent in steady state.

Using the Solow model, if, in time 𝑡 = 0 , the initial capital stock is 𝐾 0 = 100 , investment is 𝐼 0 = 25 , and 𝑑 ¯ = 0.1 is the depreciation rate, capital accumulation from period 0 to period 1, i.e., Δ 𝐾 1 is:

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