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. 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 ____________.

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BlackTom AI · Accepted Answer

We begin by identifying key components of the Solow model in this problem. The production function is Cobb-Douglas with CRS, so output per worker is y = f(k) = k^α, where k = K/L is capital per worker and α is the capital share. The per-worker investment is s y, where s = 0.1, and depreciation per worker is δ k with δ = 0.1.

First, determine the steady-state capital per worker k*. In the Solow model, at the steady state s y = δ k. Substituting y = k^α gives s k^α = δ k, which can be rearranged as s k^{α-1} = δ. Solving for k* yields k* = (s/δ)^{1/(1-α)}. Here s/δ = 0.1 / 0.1 = 1, so k* = 1^{1/(1-α)} = 1 regardless of α. This means the steady-state capital per worker is k* = 1.

Next, compute current per-capita output and infer current capital per worker. We are told total output Y = 80 and workers L = 75, so output per worker is y = Y/L = 80/75 ≈ 1.0667. Since y = k^α, we can solve for k: k = y^{1/α} = (1.0667)^{1/α}.

Because 1.0667 > 1 and α is between 0 and 1 for a typical Cobb-Douglas production function, the exponent 1/α > 1, so k = (1.0667)^{1/α} > 1. In other words, the current capital per worker k is greater than the steady-state level k* = 1.

Now look at the dynamics of capital per worker: Δk = s y − δ k. We have s y = 0.1 × 1.0667 ≈ 0.10667 and δ k = 0.1 k. Since k > 1, δ k > 0.1, and because k is large enough that 0.1 k > 0.10667 (given k > 1.0667 as established earlier), we get Δk < 0. Therefore, capital per worker is falling over time, and with it, output per worker y = k^α is also falling (at least initially).

Putting this together, the growth rate of GDP per person is negative in this situation because the economy is operating above its steady state (k > k*). The statement that the growth rate is negative because the economy is above its steady state is consistent with the mechanics of the Solow model under the given parameters. Note: since the problem statement provides a single choice that you labeled as the correct option, the reasoning above supports why that option is correct; if there were other options, we would evaluate them similarly by checking whether Δk would be positive or negative given the current k relative to k* and how that affects y growth.

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Similar Questions

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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