Homework 3 Solutions

Homework 3 Solutions - Economics 3213 Answers to Problem...

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Economics 3213 Answers to Problem Set 3: The Lion King Prof. Xavier Sala-i-Martin 1. The Circle of Life a. Assuming the Cobb-Douglas production function Y = AK α L 1- , the growth rate of output is given by the growth rate of productivity plus the sum of the growth rates of capital and labor weighted by their respective shares: ( Δ Y / Y ) = ( Δ A / A ) + ( Δ K / K ) + (1- )( Δ L / L ). Hence, the growth rate of productivity is given by: ( Δ A / A ) = ( Δ Y / Y ) - ( Δ K / K ) - (1- )( Δ L / L ). Labor share is the aggregate wage bill divided by the aggregate GDP: 1- = wL / Y = 60/100 = 0.6. Capital share is = 1 - 0.6 = 0.4. Plugging in numerical values for growth rates, we get: ( Δ A / A ) = 0.1 - 0.4 0.1 - 0.6 0.05 = 0.03. Hence, the growth rate of productivity is 3 percent per year. b. The labor and capital shares are now 1- = 80/100 = 0.8 and = 1 - 0.8 = 0.2. The growth rate of productivity is: ( Δ A / A ) = 0.1 - 0.2 0.1 - 0.8 0.05 = 0.04, or 4 percent per year. Productivity growth is the part of GDP growth that cannot be attributed to the increases in capital or labor. Here capital grows at the same rate as GDP and at twice the rate of labor. Giving a higher share to a slower growing factor, i.e. labor, means that the residual, i.e. productivity growth, must be higher. c. With ( Δ K / K ) = ( Δ L / L ) = 0.10, productivity growth for part (a) is ( Δ A / A ) = 0.1 - 0.4 0.1 - 0.6 0.1 = 0, and for part (b) is
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( Δ A / A ) = 0.1 - 0.2 0.1 - 0.8 0.1 = 0. In both cases capital and labor are growing at the same rate. Since the Cobb-Douglas function exhibits constant returns to scale, output must be growing at the same rate. Hence, no growth is attributed to changes in productivity. 2. I Can't Wait to be King i. If we double both capital and labor, output also doubles: A (2 K ) + B (2 K ) 1/2 (2 L ) 1/2 = 2 AK + B 2 1/2 K 1/2 2 1/2 L 1/2 = 2( AK + BK 1/2 L 1/2 ) = 2 Y . Hence, this production function exhibits constant returns to scale (CRS). ii. If we double only capital without doubling labor, output will less than double: A (2 K ) + B (2 K ) 1/2 L 1/2 = 2 AK + 2 1/2 BK 1/2 L 1/2 < 2( AK + BK 1/2 L 1/2 ) = 2 Y , because 2 1/2 = 1.41. .. . iii. Substitute an expression for total output Y into the defintion of output per person y = Y / L : iv. Divide the expression for y from the previous section by k : When k becomes bigger and bigger, i.e. when k goes to infinity, the square root of k also becomes bigger and bigger. The constant B divided by a bigger and bigger number approaches 0. Hence, the expression for y / k approaches A . We say that the limit of y / k as k goes to infinity is A :
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v. Substitute an expression for y / k into the Solow-Swan fundamental equation, i.e. into the expression for the growth rate of capital per person: vi. As k goes to 0, the savings curve sy / k approaches infinity as usual because the constant B is divided by a smaller and smaller number: As k goes to infinity, sy
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Homework 3 Solutions - Economics 3213 Answers to Problem...

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