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Suppose the economy has TFP = 10, there are 400 hours worked, and 60 unit of capital and 210 units of land the Cobb-Douglas production function is:
Output = TFP x Hours 0.3 x Capital 0.3 x Land 0.2 .
-In this economy,approximately how fast would the capital stock need to grow to produce growth of 1% per annum?
A) 1% per annum
B) 2.5% per annum
C) 3.3% per annum
D) 4.8% per annum
E) 5.2% per annum

Question

Quizplus · Accepted Answer

Given the Cobb-Douglas production function, the output is proportional to the product of labor, capital, and land each raised to a power that represents their respective shares in production. The function provided is:$$ \text{Output} = 10 \times \text{Hours}^{0.3} \times \text{Capital}^{0.3} \times \text{Land}^{0.2} $$To achieve a 1% growth in output, we need to understand how changes in capital (given its exponent of 0.3) contribute to overall growth. Since labor and land quantities are held constant, the growth in output must come from the growth in capital. The exponent on capital (0.3) means that a 1% increase in output requires a larger percentage increase in capital because capital's contribution to output is less than proportional (since 0.3 is less than 1). To calculate the required growth in capital to achieve a 1% increase in output, we use the fact that the elasticity of output with respect to capital is 0.3. This means that a 1% increase in output can be achieved by a $ \frac{1\%}{0.3} = 3.33\% $ increase in capital, rounding to the nearest tenth gives us approximately 3.3% per annum.

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