Q: Using the Cobb-Douglas function $$Y = A K$$^{0.3}$$ N$$^{0.7}$$, what is the marginal product of labor (MPN) when K$ is fixed?

$$MPN = 0.7 \times A K$$^{0.3}$$ N$$^{-0.3}$$

Q: Suppose total factor productivity ($$A) $$increases by 20%, capital ($$K) $$increases by 10%, and labor ($$N) $$increases by 5%. Using the Cobb-Douglas function with $$\alpha = 0.3$$, what is the approximate growth rate of output ($$Y$$)?

$$\Delta Y / Y \approx 0.20 + 0.3 \times 0.10 + 0.7 \times 0.05 = 0.20 + 0.03 + 0.035 = 0.265$$

Question 1

Which equation best represents the aggregate production function as described in the study materials?

Accepted Answer

$$Y = A F(K, N)$$

Question 2

In the Cobb-Douglas production function $$Y = A K$$^{\alpha}$$ N$$^{1-\alpha}$$, what does the parameter $$\alpha$$ represent?

Accepted Answer

The share of capital in production

Question 3

Refer to the first image. What does the increase from $$N_1 to$$ $$N_2$$ illustrate about the production function when capital is fixed?

Accepted Answer

Output increases, but at a decreasing rate due to diminishing marginal returns to labor

Question 4

Refer to the second image. Why is $$Y_3$$ - $$Y_2$$ \u003c $$Y_2$$ - $$Y_1$$ when increasing labor input?

Accepted Answer

Because of diminishing marginal productivity of labor

Question 5

What does the third image illustrate about the marginal product of labor (MPN) as the number of workers increases?

Accepted Answer

MPN decreases as the number of workers increases, showing diminishing marginal returns

Question 6

What is the effect of an increase in total factor productivity (TFP) on the marginal product of labor (MPN), as shown in the fourth image?

Accepted Answer

The MPN curve shifts upward, increasing MPN for any given level of labor

Question 7

If the production function exhibits constant returns to scale, what happens to output if both capital and labor are doubled?

Accepted Answer

Output is doubled

Question 8

Using the Cobb-Douglas function $$Y = A K$$^{0.3}$$ N$$^{0.7}$$, what is the marginal product of labor (MPN) when K$ is fixed?

Accepted Answer

$$MPN = 0.7 	imes A K$$^{0.3}$$ N$$^{-0.3}$$

Question 9

Suppose total factor productivity ($$A) $$increases by 20%, capital ($$K) $$increases by 10%, and labor ($$N) $$increases by 5%. Using the Cobb-Douglas function with $$\alpha = 0.3$$, what is the approximate growth rate of output ($$Y$$)?

Accepted Answer

$$\Delta Y / Y \approx 0.20 + 0.3 	imes 0.10 + 0.7 	imes 0.05 = 0.20 + 0.03 + 0.035 = 0.265$$

Question 10

How can total factor productivity (TFP) growth be estimated using growth accounting?

Accepted Answer

$$\Delta A / A \approx \Delta Y / Y - \alpha \Delta K / K - (1 - \alpha) \Delta N / N$$

Question 11

If GDP per capita is given by $$Y/L = (N/L) 	imes (Y/N)$$, what does $$Y/N$$ represent?

Accepted Answer

Labor productivity (output per worker)

Question 12

If the fraction of the population in the labor force ($$N/L) $$does not change over time, what is the main source of growth in GDP per capita?

Accepted Answer

Growth in labor productivity ($$Y/N)$$

Question 13

In the Cobb-Douglas production function, how is labor productivity ($$Y/N) $$expressed in terms of capital-labor ratio?

Accepted Answer

$$Y/N = A (K/N$$)^{\alpha}$$

Question 14

Which of the following best describes a positive supply (productivity) shock?

Accepted Answer

An increase in total factor productivity ($$A) $$that allows more output to be produced for any combination of capital and labor

Question 15

If a country experiences a sustained increase in TFP, what is the likely long-term effect on GDP per capita?

Accepted Answer

GDP per capita will increase due to higher labor productivity

Question 16

Why are materials and other inputs often ignored in the aggregate production function?

Accepted Answer

Because capital and labor are considered the primary factors of production in macroeconomic models

Question 17

If the marginal product of capital (MPK) declines as capital increases, what economic principle does this illustrate?

Accepted Answer

Diminishing marginal productivity of capital

Question 18

Suppose a firm doubles its labor input while keeping capital fixed. According to the Cobb-Douglas production function, what happens to output?

Accepted Answer

Output increases by less than double due to diminishing returns to labor

Question 19

What is the main reason for diminishing marginal product of labor in the context of a fixed capital stock?

Accepted Answer

As more workers are added, each additional worker has less capital to work with, reducing their contribution to output

Production, Productivity, and Growth Accounting in Macroeconomics

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