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**1. ** Suppose an economy described by the Solow model has the following production function:

*Y* = *K*^{1/2}(*LE*)^{1/2}.

For this economy, what is

*f*(*k*)?Use your answer to part (a) to solve for the steady-state value of

*y*as a function of*s*,*n*,*g*, and*δ*.Two neighboring economies have the above production function, but they have different parameter values. Atlantis has a saving rate of 28 percent and a population growth rate of 1 percent per year. Xanadu has a saving rate of 10 percent and a population growth rate of 4 percent per year. In both countries,

*g*= 0.02 and*δ*= 0.04. Find the steady-state value of*y*for each country.

**2. **
• An economy has a Cobb–Douglas production function:

*Y* = *K*^{α}(*LE*)^{1−α}.

(For a review of the Cobb–Douglas production function, see **Chapter 3**.) The economy has a capital share of a third, a saving rate of 24 percent, a depreciation rate of 3 percent, a rate of population growth of 2 percent, and a rate of labor-augmenting technological change of 1 percent. It is in steady state.

At what rates do total output, output per worker, and output per effective worker grow?

Solve for capital per effective worker, output per effective worker, and the marginal product of capital.

Does the economy have more or less capital than at the Golden Rule steady state? How do you know? To achieve the Golden Rule steady state, does the saving rate need to increase or decrease?

Suppose the change in the saving rate you described in part (c) occurs. During the transition to the Golden Rule steady state, will the growth rate of output per worker be higher or lower than the rate you derived in part (a)? After the economy reaches its new steady state, will the growth rate of output per worker be higher or lower than the rate you derived in part (a)? Explain your answers.

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**3. **
• In the United States, the capital share of GDP is about 30 percent, the average growth in output is about 3 percent per year, the depreciation rate is about 4 percent per year, and the capital–output ratio is about 2.5. Suppose that the production function is Cobb–Douglas and that the United States has been in a steady state. (For a discussion of the Cobb–Douglas production function, see **Chapter 3**.)

What must the saving rate be in the initial steady state? [

*Hint*: Use the steady-state relationship,*sy*= (*δ*+*n*+*g*)*k*.]What is the marginal product of capital in the initial steady state?

Suppose that public policy alters the saving rate so that the economy reaches the Golden Rule level of capital. What will the marginal product of capital be at the Golden Rule steady state? Compare the marginal product at the Golden Rule steady state to the marginal product in the initial steady state. Explain.

What will the capital–output ratio be at the Golden Rule steady state? (

*Hint*: For the Cobb–Douglas production function, the capital–output ratio is related to the marginal product of capital.)What must the saving rate be to reach the Golden Rule steady state?

**4. ** Prove each of the following statements about the steady state of the Solow model with population growth and technological progress.

The capital–output ratio is constant.

Capital and labor each earn a constant share of an economy’s income. [

*Hint:*Recall the definition*MPK*=*f*(*k*+ 1) −*f*(*k*).]Total capital income and total labor income both grow at the rate of population growth plus the rate of technological progress,

*n*+*g*.The real rental price of capital is constant, and the real wage grows at the rate of technological progress

*g*. (*Hint:*The real rental price of capital equals total capital income divided by the capital stock, and the real wage equals total labor income divided by the labor force.)

**5. ** Two countries, Richland and Poorland, are described by the Solow growth model. They have the same Cobb–Douglas production function, *F*(*K, L*) = *A K ^{α}L^{1−α}*, but with different quantities of capital and labor. Richland saves 32 percent of its income, while Poorland saves 10 percent. Richland has population growth of 1 percent per year, while Poorland has population growth of 3 percent. (The numbers in this problem are chosen to be approximately realistic descriptions of rich and poor nations.) Both nations have technological progress at a rate of 2 percent per year and depreciation at a rate of 5 percent per year.

What is the per-worker production function

*f*(*k*)?Solve for the ratio of Richland’s steady-state income per worker to Poorland’s. (

*Hint*: The parameter*α*will play a role in your answer.)If the Cobb–Douglas parameter

*α*takes the conventional value of about 1/3, how much higher should income per worker be in Richland compared to Poorland?Income per worker in Richland is actually 16 times income per worker in Poorland. Can you explain this fact by changing the value of the parameter

*α*? What must it be? Can you think of any way of justifying such a value for this parameter? How else might you explain the large difference in income between Richland and Poorland?

**6. ** The amount of education the typical person receives varies substantially among countries. Suppose you were to compare a country with a highly educated labor force and a country with a less educated labor force. Assume that education affects only the level of the efficiency of labor. Also assume that the countries are otherwise the same: they have the same saving rate, the same depreciation rate, the same population growth rate, and the same rate of technological progress. Both countries are described by the Solow model and are in their steady states. What would you predict for the following variables?

The rate of growth of total income

The level of income per worker

The real rental price of capital

The real wage

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**7. ** This question asks you to analyze in more detail the two-sector endogenous growth model presented in the text.

Rewrite the production function for manufactured goods in terms of output per effective worker and capital per effective worker.

In this economy, what is break-even investment (the amount of investment needed to keep capital per effective worker constant)?

Write down the equation of motion for

*k*, which shows Δ*k*as saving minus break-even investment. Use this equation to draw a graph showing the determination of steady-state*k*. (*Hint*: This graph will look much like those we used to analyze the Solow model.)In this economy, what is the steady-state growth rate of output per worker

*Y*/*L*? How do the saving rate*s*and the fraction of the labor force in universities*u*affect this steady-state growth rate?Using your graph, show the impact of an increase in

*u*. (*Hint*: This change affects both curves.) Describe both the immediate and the steady-state effects.Based on your analysis, is an increase in

*u*an unambiguously good thing for the economy? Explain.

**8. ** Choose two countries that interest you—one rich and one poor. What is the income per person in each country? Find some data on country characteristics that might help explain the difference in income: investment rates, population growth rates, educational attainment, and so on. (*Hint*: The Web site of the World Bank, http://www.worldbank.org, is one place to find such data.) How might you figure out which of these factors is most responsible for the observed income difference? In your judgment, how useful is the Solow model as an analytic tool for understanding the difference between the two countries you chose?

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