Quiz: Solow Model and Economic Growth — 24 questions

1. Why cannot endless capital accumulation produce ever-increasing growth rates?

A. Technology growth makes the production function become linear over time

B. Depreciation makes total capital fall whenever output increases

C. Population growth makes the saving rate decline as capital rises

D. Concavity makes the marginal product of capital decline as capital rises

Answer: (D)

Explanation

A concave production function implies a decreasing marginal product of capital, so additional capital contributes progressively less to output growth. The other mechanisms do not explain the limiting effect identified here.

2. Which treatment of technology distinguishes the Solow model from endogenous growth models?

A. Technology changes unpredictably in response to short-run market prices

B. Technology is created within the model through firms’ research decisions

C. Technology grows exogenously at a constant rate outside the model

D. Technology is excluded because production uses capital and labor alone

Answer: (C)

Explanation

The Solow model treats technological progress as exogenous, with its growth specified outside the model. Endogenous growth models instead explain technological change through mechanisms within the model.

3. Which equation describes the evolution of capital per effective worker in the Solow model?

A. $\dot k=(s+n+g)f(k)-\delta k$

B. $\dot k=sY-\delta K$

C. $\dot k=sf(k)+(n+g+\delta)k$

D. $\dot k=sf(k)-(n+g+\delta)k$

Answer: (D)

Explanation

Capital per effective worker rises with saving per effective worker and falls through dilution from population and technology growth plus depreciation. The last expression describes total capital rather than capital per effective worker.

4. What is the primary purpose of the Solow model?

A. To predict monthly fluctuations in employment and consumer spending

B. To explain the proximate causes of long-run growth and income differences across countries

C. To explain how firms choose prices in markets with limited competition

D. To describe how governments set taxes and regulate financial markets

Answer: (B)

Explanation

The Solow model focuses on the mechanics of long-run economic growth and cross-country differences in income. Monthly economic movements concern short-run analysis rather than the model’s central purpose.

5. A Solow analysis divides output and capital by effective labor $AL$. Which definitions result?

A. $y=\frac{Y}{K}$ and $k=\frac{AL}{K}$

B. $y=\frac{Y}{L}$ and $k=\frac{K}{A}$

C. $y=\frac{Y}{A}$ and $k=\frac{K}{L}$

D. $y=\frac{Y}{AL}$ and $k=\frac{K}{AL}$

Answer: (D)

Explanation

Output per effective worker is defined as $y=\frac{Y}{AL}$, and capital per effective worker is $k=\frac{K}{AL}$. These definitions lead to the reduced production relationship $y=f(k)=F(k,1)$.

6. What does it mean for a firm to be a price taker in a competitive market?

A. The firm accepts the market price because its own decisions cannot influence it

B. The firm selects the market price by controlling the quantity supplied

C. The firm raises its price whenever its production costs increase

D. The firm negotiates a separate price with every household in the market

Answer: (A)

Explanation

A price-taking firm is one among many firms and lacks the market power to affect the prevailing price. A price-setting firm, in contrast, can influence the price through its market position.

7. How does the treatment of technology differ between the Solow model and endogenous growth models?

A. Both approaches take technology as given, but they assign different roles to capital accumulation.

B. The Solow model makes technology endogenous, whereas endogenous models exclude technological change.

C. The Solow model explains technology through household choices, whereas endogenous models treat it as fixed.

D. The Solow model takes technology and its growth rate as given, whereas endogenous models explain them within the model.

Answer: (D)

Explanation

In the Solow model, $A$ and its growth rate $g$ are exogenous, meaning they are taken as given. Endogenous growth models instead explain technological change through mechanisms within the model, so the second option reverses the distinction.

8. What does break-even investment represent?

A. Investment needed to make total capital equal to total output

B. Investment needed to keep capital per effective worker constant

C. Investment generated by saving after depreciation is added

D. Investment required to raise population and technology growth

Answer: (B)

Explanation

Break-even investment offsets population growth, knowledge growth, and depreciation so that capital per effective worker does not change. It is therefore distinct from total investment generated by saving.

9. What happens when capital per effective worker is below its steady-state level?

A. Actual investment equals break-even investment, so $k$ remains fixed

B. Break-even investment exceeds actual investment, so $k$ falls

C. Actual investment exceeds break-even investment, so $k$ rises

D. Depreciation disappears, so $k$ rises at the technology growth rate

Answer: (C)

Explanation

Below the steady state, saving-funded investment is greater than the investment needed to maintain $k$, causing capital per effective worker to increase. The opposite comparison applies when $k$ is above its steady-state level.

10. Which developments are motivated by the limitations of the Solow model?

A. Endogenizing technology, modeling intertemporal household choices, and incorporating broader capital externalities

B. Treating institutions as irrelevant, simplifying investment decisions, and reducing the role of knowledge

C. Holding technology fixed, removing household decisions, and narrowing capital to machines

D. Replacing capital accumulation with population growth and excluding productivity differences

Answer: (A)

Explanation

The model's limitations motivate theories that endogenize $A$, include households' intertemporal choices, and recognize broader forms of capital with important externalities. The other combinations move away from, rather than address, these extensions.

11. What did Robert Solow's growth-accounting calculations identify as the main driver of long-run economic growth?

A. Technological progress or productivity growth

B. Higher population growth combined with unchanged productivity

C. Greater saving that permanently raises the capital growth rate

D. Accumulation of physical capital and expansion of labor

Answer: (A)

Explanation

Solow's calculations attributed the main source of economic growth to technological progress or productivity growth. Capital accumulation can raise the level of output, but it does not by itself sustain the long-run growth rate.

12. What is the long-run effect of an increase in the saving rate?

A. It raises output per worker permanently, while capital per effective worker returns to its former level

B. It raises steady-state capital per effective worker, while the output-per-worker effect is temporary

C. It leaves both steady-state capital and output per worker unchanged after adjustment

D. It lowers steady-state capital per effective worker, while output per worker rises permanently

Answer: (B)

Explanation

A higher saving rate shifts actual investment upward and leads to a permanently higher steady-state level of capital per effective worker. Output per worker rises during the transition, but its growth effect is temporary.

13. If all inputs in a production function are doubled under constant returns to scale, what happens to output?

A. Output rises by less than twice its original level

B. Output doubles in the same proportion as the inputs

C. Output remains unchanged because productivity is fixed

D. Output rises by more than twice its original level

Answer: (B)

Explanation

Constant returns to scale require $F(cK,cAL)=cF(K,AL)$, so setting $c=2$ makes output twice as large. More-than-proportional and less-than-proportional changes correspond to increasing and decreasing returns, respectively.

14. What is the steady-state growth rate of output per worker when output per worker is given by $Y/L=Af(k)$?

A. It is $n+g$ because labor and technology both expand

B. It is zero because output per worker is constant in the steady state

C. It is $g+n+\delta$ because break-even investment determines growth

D. It is $g$ because $k$ is constant in the steady state

Answer: (D)

Explanation

In the steady state, $k$ is constant, so changes in $f(k)$ contribute no growth and output per worker grows at the technology rate $g$. The rate $n+g$ can arise during convergence, not in the steady state.

15. What condition defines the steady state for capital per effective worker?

A. $\dot k=g$ and $sf(k^*)=nk^*$

B. $\dot K=0$ and $sY=\delta K$

C. $\dot Y=0$ and $f'(k^*)=n+g+\delta$

D. $\dot k=0$ and $sf(k^*)=(n+g+\delta)k^*$

Answer: (D)

Explanation

At the steady state, capital per effective worker is constant, so actual investment equals break-even investment. This does not require total capital to stop growing.

16. Which production function represents labor-augmenting technological progress?

A. $Y(t)=A(t)F(K(t),L(t))$

B. $Y(t)=F(A(t)K(t),L(t))$

C. $Y(t)=F(A(t)K(t),A(t)L(t))$

D. $Y(t)=F(K(t),A(t)L(t))$

Answer: (D)

Explanation

Labor-augmenting technology enters by multiplying labor, so effective labor is $A(t)L(t)$. Capital-augmenting technology multiplies capital, while neutral technology multiplies the production function itself.

17. On a balanced-growth path, at what rates do total output and output per worker grow in the steady state?

A. Total output grows at $g$, while output per worker grows at $n+g$.

B. Both total output and output per worker grow at $n+g$.

C. Both total output and output per worker grow at $g$.

D. Total output grows at $n+g$, while output per worker grows at $g$.

Answer: (D)

Explanation

Population growth contributes to the expansion of total output, giving it a rate of $n+g$, while output per worker reflects technological progress at rate $g$. The reversed pattern incorrectly assigns population growth to output per worker.

18. Which combination correctly describes the reduced production function and its Inada conditions?

A. $f'(k)>0$, $f''(k)<0$, with marginal product approaching infinity near zero capital and zero at very high capital

B. $f'(k)<0$, $f''(k)>0$, with marginal product approaching zero near zero capital and infinity at very high capital

C. $f'(k)<0$, $f''(k)<0$, with marginal product increasing near zero capital and declining near high capital

D. $f'(k)>0$, $f''(k)>0$, with marginal product remaining constant as capital increases

Answer: (A)

Explanation

The reduced function satisfies $f(0)=0$, has a positive but diminishing marginal product, and obeys the Inada limits $\lim_{k\to0}f'(k)=\infty$ and $\lim_{k\to\infty}f'(k)=0$. Thus, additional capital raises output while its marginal contribution declines.

19. How does the capital stock change when output generates saving and existing capital depreciates?

A. It changes according to $\dot K=nK+gK$

B. It changes according to $\dot K=sY+\delta K$

C. It changes according to $\dot K=(n+g+\delta)K$

D. It changes according to $\dot K=sY-\delta K$

Answer: (D)

Explanation

Capital increases through saving, which equals investment, and decreases through depreciation, giving $\dot K=sY-\delta K$. Depreciation is subtracted rather than added because it removes capital from the stock.

20. Which condition identifies the Golden Rule steady state that maximizes consumption?

A. $f(k^*)=n+g+\delta$

B. $f'(k^*)=s+n+g$

C. $sf(k^*)=(n+g+\delta)k^*$

D. $f'(k^*)=n+g+\delta$

Answer: (D)

Explanation

Golden-rule consumption is maximized where the marginal product of capital equals the slope of break-even investment, yielding $f'(k^*)=n+g+\delta$. The equality involving $sf(k^*)$ defines a general steady state rather than the consumption-maximizing one.

21. In the competitive labor market described by the model, what determines the wage?

A. The depreciation rate, meaning the fraction of capital lost during production

B. The total output of the economy, regardless of how much labor is employed

C. The marginal product of labor, meaning the extra output from one more unit of labor

D. The average product of capital, meaning output divided by the capital stock

Answer: (C)

Explanation

With households supplying labor inelastically, the competitive wage equals the marginal product of labor. The marginal product measures the additional output generated by one more unit of labor, not the return on capital.

22. Why does the Solow model predict that poorer countries can catch up with richer countries?

A. They can access technology as a public good and raise their productivity.

B. They begin with higher productivity because their capital stock is smaller.

C. They grow faster because population growth is lower in every poor country.

D. They receive automatic capital transfers from richer economies.

Answer: (A)

Explanation

The model predicts convergence because technology is treated as a public good available across countries, enabling poorer economies to improve productivity and catch up. Automatic capital transfers and universally lower population growth are not assumptions of the model.

23. Which growth process is represented by $\dot L(t)=nL(t)$ and $\dot A(t)=gA(t)$?

A. Labor grows at rate $g$ and technology grows at rate $n$

B. Labor growth depends on saving while technology grows at rate $g$

C. Both labor and technology grow at rate $n+g$

D. Labor grows at rate $n$ and technology grows at rate $g$

Answer: (D)

Explanation

The equations assign $n$ to labor growth and $g$ to technology growth. Reversing these parameters confuses the demographic and technological growth processes.

24. Which condition describes capital-market clearing and the associated net return on capital?

A. $K^{supply}(t)<K^{demand}(t)$ and $r(t)=r_K(t)-\delta$

B. $K^{supply}(t)>K^{demand}(t)$ and $r(t)=r_K(t)+\delta$

C. $K^{supply}(t)=K^{demand}(t)$ and $r(t)=r_K(t)+\delta$

D. $K^{supply}(t)=K^{demand}(t)$ and $r(t)=r_K(t)-\delta$

Answer: (D)

Explanation

The capital market clears when supplied capital equals demanded capital, expressed as $K^{supply}(t)=K^{demand}(t)$. The net return subtracts depreciation from the gross return, giving $r(t)=r_K(t)-\delta$.