Consider an economy described by the textbook Solow model with a Cobb-Douglas production function with constant returns to scale with respect to K and L (jointly). Moreover, you know that the economy is producing 80 units of total output and the productivity parameter is equal to 1. If the depreciation rate is 10%, the investment rate is 10%, and there are 75 workers, the growth rate of GDP per person ____________.

Question

BlackTom AI · Accepted Answer

In the Solow model the key steady-state condition for capital per worker is that investment equals depreciation: sY = δK (aggregate) or equivalently s y = δ k (per worker), where y = Y/L and k = K/L.

Step 1 — compute output per worker and investment per worker
- Total output Y = 80, workers L = 75 → y = Y/L = 80/75 = 1.0666667.
- Investment rate s = 0.10 → investment per worker s y = 0.10 × 1.0666667 = 0.10666667.

Step 2 — use the capital-accumulation/steady-state identity
- At steady state aggregate investment equals aggregate depreciation: sY = δK.
- Rearranging gives K = (s/δ) · Y. Here s = δ = 0.10, so K = (0.10/0.10) · Y = 1 · Y = Y. Thus K = 80.
- Therefore capital per worker k = K/L = 80/75 = 1.0666667, which equals y.

Step 3 — verify change in capital per worker is zero
- The law of motion for capital per worker is Δk = s y − δ k.
- Substituting s y = 0.1066667 and δ k = 0.10 × 1.0666667 = 0.1066667 gives Δk = 0.1066667 − 0.1066667 = 0.
- With Δk = 0, capital per worker is not changing and so output per worker y is constant over time.

Why the correct option is correct
- In the Solow model, a zero change in capital per worker implies no growth in GDP per person. Because the given parameters imply sY = δK and hence s y = δ k, the economy is at its steady state and GDP per person does not grow.

Why the other choices are incorrect
- “Negative because above its steady state” and “positive because below its steady state” both imply s y ≠ δ k (investment ≠ depreciation); here they are equal so neither holds.
- “Cannot be determined” is wrong because the provided numbers allow checking the steady-state condition and concluding Δk = 0.

Conclusion
- The growth rate of GDP per person is equal to zero because the economy is at its steady state, which matches the provided answer.

类似问题

24. In countries with well-developed economies, economic growth tends to:

24. In countries with well-developed economies, economic growth tends to:

24. In countries with well-developed economies, economic growth tends to:

Based on the information you have so far, you know that the economy is [ Select ] at above below its steady state and GDP per capita is growing at [ Select ] a positive rate a negative rate a rate of zero at time t.

In the long run, the change in the trajectory above features [ Select ] level effect and [ Select ] growth effect.

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