Suppose you are given the data for Brazil and Portugal. In Brazil, the saving rate is 10 percent while in Portugal the saving rate is 20 percent. Portugal and Brazil have a common capital share
𝛼
∈
(
0
,
1
)
, a common depreciation rate
𝑑
¯
∈
(
0
,
1
)
, and the same positive population growth rate
𝑛
. Unfortunately, you don't know which country is more productive and you're not sure how much bigger Brazil's population is. Despite this missing information you know that when both countries are in their respective steady states, __________

Question

Suppose you are given the data for Brazil and Portugal. In Brazil, the saving rate is 10 percent while in Portugal the saving rate is 20 percent. Portugal and Brazil have a common capital share 
𝛼
∈
(
0
,
1
)
, a common depreciation rate 
𝑑
¯
∈
(
0
,
1
)
, and the same positive population growth rate 
𝑛
. Unfortunately, you don't know which country is more productive and you're not sure how much bigger Brazil's population is. Despite this missing information you know that when both countries are in their respective steady states, __________

BlackTom AI · Accepted Answer

We start by restating the setup and the options so we can compare them directly. - Context: Brazil has saving rate s_B = 0.10, Portugal has s_P = 0.20. They share the same depreciation rate d̄ and population growth n, with a common capital share α ∈ (0,1). The question asks about the steady-state relationships between the two countries, given these shared parameters but unknown level of productivity and unknown population sizes. Option 1: "Brazil has a lower level of aggregate (total) output (Y) than Portugal." - What this implies: Y_B < Y_P in steady state. - Analysis: In the Solow framework with identical n and d̄ but different saving rates, the steady-state output per worker y and the steady-state capital per worker k both depend on the saving rate s. Specifically, the steady-state capital per worker k* scales with s/(n+d̄), and output per worker y* = k*^α. Since s_P > s_B, Portugal will attain a higher k*_P and hence a higher y*_P. However, total output Y = y * L where L (population) is unknown and could differ between Brazil and Portugal. If Portugal’s population L_P is much smaller than Brazil’s, it is not guaranteed that Y_P > Y_B without knowing L. Therefore, we cannot deduce Y_B < Y_P for certain; the statement is not guaranteed given only the information provided. Option 2: "Brazil has a higher capital-output (K/Y) ratio than Portugal." - What this implies: K/Y_B > K/Y_P. - Analysis: From the Solow model with Y = K^α L^(1-α) and the steady-state condition s Y = (n+d̄) K, we can derive K/Y in per-country terms. In per-capita terms, K/Y equals (K/L) / (Y/L) = k / y. With y = k^α, we get K/Y = k / k^α = k^(1-α). The steady-state k* is proportional to [s/(n+d̄)]^(1/(1-α)). Thus K/Y = k*^(1-α) ∝ s/(n+d̄). Since n+d̄ is common to both countries, K/Y is proportional to the saving rate s. Because s_P = 0.20 is greater than s_B = 0.10, Portugal has the higher K/Y ratio. Therefore the claim that Brazil has a higher K/Y is incorrect. Option 3: "Portugal has a lower level of output per person (y) than Brazil." - What this implies: y_P < y_B. - Analysis: As shown above, y* = k*^α and k* increases with s. Since s_P > s_B, Portugal’s steady-state per-worker capital k*_P is higher than Brazil’s k*_B, leading to a higher y*_P = (k*_P)^α. Therefore, Portugal does not have a lower y; it has a higher output per worker. The statement is incorrect. Option 4: "Portugal and Brazil have the same capital-labor ratio." - What this implies: k*_P = k*_B. - Analysis: The steady-state per-worker capital k* depends on s/(n+d̄) and α, not on population size directly. Since s_P ≠ s_B, we have k*_P ≠ k*_B; higher s leads to higher k*. Hence the capital-labor ratio cannot be the same across the two countries under these conditions. This statement is false. Option 5: "Portugal has a higher capital-output (K/Y) ratio than Brazil." (This is the option listed as the answer in the data.) - What this implies: K/Y_P > K/Y_B. - Analysis: As established in Option 2, K/Y scales with s/(n+d̄) and since s_P > s_B, Portugal indeed achieves a larger K/Y ratio. Therefore this statement correctly describes the comparative statics of K/Y between the two countries under the given assumptions. The reasoning hinges on the linear relationship between K/Y and the saving rate when n and d̄ are equal across countries. Option 6: "Portugal has a higher capital-labor ratio than Brazil." - What this implies: k*_P > k*_B. - Analysis: Since k* ≈ [s/(n+d̄)]^(1/(1-α)) and α ∈ (0,1), a higher saving rate s directly raises k*. With s_P = 0.20 > s_B = 0.10 and identical n+d̄, Portugal achieves a higher steady-state capital per worker than Brazil. Therefore this statement is correct as well. Summary of reasoning highlights: - The saving rate difference drives both K/Y and k* in a predictable way: higher s leads to higher K/Y and higher k*, holding n and d̄ constant. - Output per worker y* also rises with s, since y* = k*^α and α ∈ (0,1). - Total output Y depends on per-worker output and population, so without knowing relative population sizes, statements about aggregate Y cannot be asserted with certainty. - Consequently, the correct qualitative conclusions given the data are that Portugal has a higher K/Y ratio than Brazil and also a higher capital-labor ratio (k*), while statements asserting Brazil has higher Y or higher K/Y or higher y for Brazil do not hold under the given setup.

FA25 ECON 302 002 Midterm 2

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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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