An economy's output is determined by its inputs—capital and labor—and how efficiently these inputs are used, known as Total Factor Productivity (TFP). Consider an economy with the following Cobb-Douglas production function:

Where:

Y = Total output
A = Total Factor Productivity
K = Quantity of capital input
L = Quantity of labor input
α = Capital's share of output (a constant between 0 and 1)

Given:

Capital's share of output (α) = 0.3
In Year 1:
Total output = 2,000 units
Capital input = 1,000 units
Labor input = 1,500 units
In Year 2:
Total output = 2,400 units
Capital input = 1,100 units
Labor input = 1,550 units

Which of the following factors contributed to the increase in output per worker between Year 1 and Year 2?

Question

An economy's output is determined by its inputs—capital and labor—and how efficiently these inputs are used, known as Total Factor Productivity (TFP). Consider an economy with the following Cobb-Douglas production function:

Where:

Y = Total output
A = Total Factor Productivity
K = Quantity of capital input
L = Quantity of labor input
α = Capital's share of output (a constant between 0 and 1)

Given:

Capital's share of output (α) = 0.3
In Year 1:
Total output = 2,000 units
Capital input = 1,000 units
Labor input = 1,500 units
In Year 2:
Total output = 2,400 units
Capital input = 1,100 units
Labor input = 1,550 units

Which of the following factors contributed to the increase in output per worker between Year 1 and Year 2?

BlackTom AI · Accepted Answer

To unpack the question, we start by noting the Cobb-Douglas production function: Y = A * K^α * L^(1−α), with α = 0.3. This allows us to separate growth in output into two components: (1) a change in total factor productivity A, and (2) a change in the input mix given K and L.

First, compute Year 1 (Y1) inputs: K1 = 1000, L1 = 1500, Y1 = 2000. With α = 0.3, K1^α = 1000^0.3 and L1^(1−α) = 1500^0.7. Numerically, 1000^0.3 ≈ 7.95 and 1500^0.7 ≈ 167.0. Their product is ≈ 1327.0. Therefore A1 ≈ Y1 / (K1^α L1^(1−α)) ≈ 2000 / 1327 ≈ 1.51.

Year 2 (Y2) inputs: K2 = 1100, L2 = 1550, Y2 = 2400. Compute K2^α = 1100^0.3 ≈ 8.18 and L2^(1−α) = 1550^0.7 ≈ 170.2. Product ≈ 8.18 * 170.2 ≈ 1392.0. Thus A2 ≈ Y2 / (K2^α L2^(1−α)) ≈ 2400 / 1392 ≈ 1.72.

Now compare A2 to A1 to assess TFP change: A2 / A1 ≈ 1.72 / 1.51 ≈ 1.14, i.e., roughly a 14% increase in total factor productivity from Year 1 to Year 2.

Next, evaluate capital deepening per worker and output per worker:
- Output per worker in Year 1: y1 = Y1 / L1 = 2000 / 1500 ≈ 1.333.
- Output per worker in Year 2: y2 = Y2 / L2 = 2400 / 1550 ≈ 1.548.
The per-worker output rose by about 0.215, which is a substantial increase (≈16%).
- Capital per worker in Year 1: k1 = K1 / L1 = 1000 / 1500 ≈ 0.667.
- Capital per worker in Year 2: k2 = K2 / L2 = 1100 / 1550 ≈ 0.709.
Capital per worker thus increased, indicating capital deepening contributed to higher output per worker.

Since output per worker rose and we have evidence of both an increase in capital per worker (k2 > k1) and an increase in total factor productivity (A2 > A1), the drivers of the higher output per worker between Year 1 and Year 2 are both: an increase in capital per worker and an increase in Total Factor Productivity.

Option-by-option evaluation:
- Increase in Total Factor Productivity only: This ignores the observed rise in capital per worker, so it is incomplete.
- Increase in capital per worker only: This ignores the observed rise in A, so it is incomplete.
- Increase in labor input per worker: The data show θ per-worker metrics (Y/L) rising due to both capital deepening and TFP, not solely a shift in labor input per worker. This option misstates the mechanism.
- Both an increase in capital per worker and Total Factor Productivity: This matches the combined contributions identified above (k/L rising and A rising), aligning with the computed increases in y, k, and A.

ECON 2002.01 SP2025 (25230) Midterm Exam 2

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