Consider the “Cobb-Douglas” production function given by where A is a positive constant. Suppose that the capital stock grows at 2% per year and the labour force grows at 1% per year. According to the production function, what will output growth be approximately?

Question

Consider the “Cobb-Douglas” production function given by  where A is a positive constant. Suppose that the capital stock grows at 2% per year and the labour force grows at 1% per year. According to the production function, what will output growth be approximately?

BlackTom AI · Accepted Answer

Restating the question: We have a Cobb-Douglas production function of the form Y = A K^α L^(1-α) with A > 0, where α is the capital share. Capitals grows at 2% per year (g_K = 2%), and the labor force grows at 1% per year (g_L = 1%). The task is to determine the approximate output growth rate.

Key idea: For a Cobb-Douglas production function with constant returns to scale (α + (1-α) = 1), the growth rate of output is the weighted sum of the growth rates of inputs, with weights given by their income shares. Specifically:
- g_Y = α g_K + (1-α) g_L.

Plugging in g_K = 2% and g_L = 1% gives:
- g_Y = α(2%) + (1-α)(1%) = 2α% + (1 - α)%. 
- This simplifies to g_Y = 1% + α(1%), since 2α% + (1 - α)% = 1% + α%.

Interpretation: The output growth rate depends on α, the capital share, which is between 0 and 1 for a standard Cobb-Douglas. Therefore, the possible range for g_Y is:
- If α = 0 (all labor), g_Y = 1%.
- If α = 1 (all capital), g_Y = 2%.
- For typical α values (e.g., α in [0.3, 0.5]), g_Y would be between 1.3% and 1.5%.

Because the problem statement did not specify α (the exponent on K), we cannot pinpoint a single numeric value beyond this range. The most you can say is that output growth lies between 1% and 2% per year, with the exact figure increasing in α. If an answer option were to be chosen, it would correspond to the α used in the model; without that value, any specific single percentage (like exactly 1%) would be an incomplete determination.

Notes on potential misconceptions:
- Assuming g_Y equals the average of input growth rates (i.e., (g_K + g_L)/2 = 1.5%) would be incorrect for a Cobb-Douglas without specifying α, because the correct formula weights by α, not by a simple average.
- Treating α as 0.5 (a typical but not universal midpoint) would yield g_Y = 1% + 0.5% = 1.5%, illustrating how the exact α shifts the result.
- If one misremembers the production function form and uses a mis-specified exponent sum, results could be outside the 1%–2% range or misattribute the growth to the wrong input.

In summary, the approximate output growth rate is contextually tied to α and will fall between 1% and 2% per year, increasing with larger capital share α.

ECON10003_2025_SM1 Pre-Tutorial Quiz 7

Consider the “Cobb-Douglas” production function given by where A is a positive constant. Suppose that the capital stock grows at 2% per year and the labour force grows at 1% per year. According to the production function, what will output growth be approximately?

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