Practice problems Solow model and economic growth

Economics DDIM 2013

Instructor: Nicola Borri

Problem 1

Assume the aggregate production function in the economy is:

Y = K1/44√L3.

1. Does this production function have constant returns to scale? If so, why?

2. Capital has diminishing returns:prove it.

3. Does labor have diminishing returns? Prove your answer.

4. Transform this production function in a relationship between output per worker and capital

per worker. Plot this relationship and explain the economic intuition of your graph.

5. For a given saving rate s, and depreciation rate δ, write the expression for capital per worker

in steady state.

6. Compute the values of the expressions from point 4) and 5) when the saving rate and depre-

ciation rate are both equal to 10%.

7. Assume now that the saving rate doubles (everything else stay the same). What happens to

output per worker in steady state? And to consumption per worker?

8. Write down the general expression for consumption per worker in this model.

1

Problem 2

Assume the population growth rate to be equal to zero. Assume further that the rate of capital

depreciation suddenly and unexpectedly increases. The production function does not change.

1. Use a graphical analysis to show what is the effect of this change to the level of steady state

per capita capital if the saving rate does not change.

2. What happens to the level of per capita capital that maximises per capita consumption?

Explain with the help of a graph and/or analytical expressions.

2

Problem 3

Consider a closed economy. The production function is the standard neoclassical function

Y = AF (K,L) = K1/3L2/3. The saving rate is s = 0.3 and the depreciation rate δ = 0.1. The

total factor productivity A is assumed to be equal to 1. Both the rate of growth of L and A are

equal to zero.

1. Compute and plot the per worker production function. Show in the graph what happens if

the level of technology A increases for any capital per worker level.

2. Describe the relationship between investment, depreciation rate and changes in the stock of

capital per worker.

3. Suppose the present level of capital per worker is equal to 3. What do you expect is going to

happen in the long run to the levels of capital, consumption and investment per worker with

respect to current levels?

4. Compute the golden rule level of capital per worker. Suppose further that a benevolent

dictator can control the saving rate. What should the dictator do to align the current level

of capital per worker to its golden rule level?

3

Problem 4

Consider a standard Solow model with exogenous labor augmenting technical progress. In

particular, let A be the efficiency of labor and define the production function as follows:

Y = F (K,AL) = Kα(AL)1−α.

Define the output per worker as y = Y/L and the capital per worker as k = K/L. Consider the

following table:

Country y i/yUS k i/kUS Ai/AUS

USA 1 1 1

Canada 0.81 0.95 –

France 0.67 0.92 –

South Korea 0.48 0.61 –

Argentina 0.33 0.32 –

Mexico 0.26 0.23 –

Indonesia 0.11 0.08 –

1. Assume that α = 1/3 in all countries, compute the empty entries in the table.

2. Assume that the US has converged to a steady state, as predicted by the Solow model with

labor-augmenting technical progress. By looking at the table, and assuming that all countries

converge to the same capital-labor ratio in efficiency units (k̃∗), which countries are likely to

have a capital labor ratio k below the one they would be able to reach at the steady state?

4

Problem 5

Consider two countries. In both of them there is investment of 20% of output, in both of them

depreciation is 7%. In country A population grows at 1% and in country B it grows at 2%. What

does the Solow model predict about their long run output differences in the absence of technical

change? What about long run differences in capital?

5

Problem 6

Given the law of motion for per capita capital associated with the Solow model with both

population growth at rate n and exogenous growth of labor productivity gA:

K̇ = sKα(AL)1−α − δK,

derive an expression for how capital per capita per level of technology evolves over time. Find

an expression for this variable such that if the economy is ever in this position, it will stay there.

6

Short questions

Answer briefly the following questions:

1. Why convergence is the expected outcome within the Solow model? What is the crucial

assumption that generates this result?

2. Why some economists argue that education and human capital, rather than technology, may

be a better explanation for the lack of convergence in per capita income.

3. Can you think of any other explanation for the lack of convergence?

4. Why are the Penn World Tables a useful tool to compare outputs of different countries?

5. What was the productivity slow down?

6. Describe what happens in the growth model without population growth if an epidemic of

yellow fever instantaneously wiped out one half of the population.

7