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HW4_sol

# HW4_sol - Answers to Textbook Questions and Problems 4 The...

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4. The higher the population growth rate is, the lower the steady-state level of capital per worker, and therefore there is a lower level of steady-state income per worker. For example, Figure 7–1 shows the steady state for two levels of population growth, a low level n 1 and a higher level n 2 . The higher population growth n 2 means that the line rep- resenting population growth and depreciation is higher, so the steady-state level of cap- ital per worker is lower. In a model with no technological change, the steady-state growth rate of total income is n : the higher the population growth rate n is, the higher the growth rate of total income. Income per worker, however, grows at rate zero in steady state and, thus, is not affected by population growth. Problems and Applications 1. a. A production function has constant returns to scale if increasing all factors of pro- duction by an equal percentage causes output to increase by the same percentage. Mathematically, a production function has constant returns to scale if zY = F ( zK , zL ) for any positive number z . That is, if we multiply both the amount of capital and the amount of labor by some amount z , then the amount of output is multi- plied by z . For example, if we double the amounts of capital and labor we use (set- ting z = 2), then output also doubles. To see if the production function Y = F ( K , L ) = K 1/2 L 1/2 has constant returns to scale, we write: F ( zK , zL ) = ( zK ) 1/2 ( zL ) 1/2 = zK 1/2 L 1/2 = zY . Therefore, the production function Y = K 1/2 L 1/2 has constant returns to scale. b. To find the per-worker production function, divide the production function Y = K 1/2 L 1/2 by L : If we define y = Y/L , we can rewrite the above expression as: y = K 1/2 / L 1/2 . Defining k = K/L , we can rewrite the above expression as: y = k 1/2 . 58 Answers to Textbook Questions and Problems ( δ + n 2 ) k ( δ + n 1 ) k sf ( k ) Investment, break-even investment Capital per worker k k 2 * k 1 * Figure 7–1 Y L K L L = 1 2 1 2 .

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c. We know the following facts about countries A and B: δ = depreciation rate = 0.05, s a = saving rate of country A = 0.1, s b = saving rate of country B = 0.2, and y = k 1/2 is the per-worker production function derived in part (b) for countries A and B. The growth of the capital stock Δ k equals the amount of investment sf ( k ), less the amount of depreciation δ k . That is, Δ k = sf ( k ) – δ k . In steady state, the capital stock does not grow, so we can write this as sf ( k ) = δ k . To find the steady-state level of capital per worker, plug the per-worker pro- duction function into the steady-state investment condition, and solve for k * : sk 1/2 = δ k . Rewriting this: k 1/2 = s / δ k = ( s / δ ) 2 . To find the steady-state level of capital per worker k * , plug the saving rate for each country into the above formula: Country A: k = ( s a / δ ) 2 = (0.1/0.05) 2 = 4. Country B: k = ( s b / δ ) 2 = (0.2/0.05) 2 = 16.
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HW4_sol - Answers to Textbook Questions and Problems 4 The...

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