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ECON 5011, Macroeconomics (MA), Fall 2023

Assignment #1

Due: Monday, October 26, 2023 (before class)

1 Growth and Income Inequality

Thomas Piketty’s book (Piketty and Goldhammer (2014)), Capital in the Twenty-First Century, has attracted widespread attention of the academics and the popular media.  The book makes two main contributions to the discussion of income inequality and growth.  First, Piketty and his co- authors (such as Emanuel Saez, Anthony Atkinson, and Gabriel Zucman) collected historical data on income and wealth of very rich households; i.e., those at the top 5%, top 1%, top 0.1% of the income distribution in major developed countries dating back to 19th century.  They document that the share of income that goes to top earners has increased significantly since the early 1980s and today’s wealth accumulation (as a ratio of national income) has reached its highest level since World War I.

Figure 1: The transformation of the top 1% in the United States

For example, Figure 1 plots the share of total income claimed by top 1% since early 20th cen- tury. A striking fact stands out from this chart is that both labor and capital income inequality has followed a U-shaped pattern in the US during the last century. Prior to the 1980s, income inequal- ity greatly diminished: the top 1% share of total income decreased to around 9% from around 20% in late 1920s. Yet, after the 1980s income inequality dramatically rose in the US. Today the income share of the top 1% of the population has reached the very high levels of America’s Gilded Age.

Furthermore, he also documented that today’s wealth accumulation (as a ratio of national in- come) has reached its highest level since World War I (WWI). Figure 2 plots the wealth to income ratios in major developed economies since 1970s (source: Piketty and Zucman (2014)). All major economies have seen their wealth increase at a faster pace than their national income.

Figure 2: Figure 1: Private wealth / national income ratios 1970-2010

Source: Piketty and Zucman (2014)

The second contribution of the book is a theoretical prediction: Piketty predicts that there will be a further increase in the wealth to income ratio in the future as income growth slows down due to slower productivity and population growth.  Furthermore, he also argues that the marginal product of capital (rate of return on capital) will diminish only slowly that a rise in the ratio of wealth (capital) to income will result in an increase in capital’s share of income.1   Since capital income is highly concentrated, this will lead to an increase in income inequality in the economy.

In the first part of this assignment you will analyze Piketty’s predictions about the evolution of the wealth to income ratio, capital’s share of income, and income inequality theoretically through the lenses of Solow and neo-classical growth models. For this purpose please discuss the following questions:

1.  Show that the share of capital income, a, equals to the rate of return on capital,r times the wealth to income ratio, β (β =Y/K , where K is capital (wealth) stock in the economy and Y is the output.): a = rβ .2  (1 point)

2.  Piketty argues that in the long run capital-net income (income net of depreciation) ratio

converges to the ratio of net saving rate˜(s)(savings net of depreciation) to technology’s growth rate, g; i.e., β(˜) = s/g .3  For example, if a country saves 10% of its output net of depreciation each year, and the technology grows at at a rate of 2%, then wealth to income ratio in this country would be 500%.

(a)  Suppose that instead of gross saving rate s, net saving rate, ˜(s) is constant in the Solow  growth model (for now let’s assume no population growth, n = 0).  Furthermore, we  need to assume that marginal product of capital is bounded below by δ ; i.e., FK(K;AL) - δK ! 0 as K ! 0.

i. Define the following “net” variables (variables net of depreciation) in terms of

gross variables: net saving rate (˜(s)), net output (Y(˜)), net investment (I(˜)). (1 point)

ii.  Show that there is a balanced growth path in which capital K and net output Y(˜) are

growing at the rate of 1 + g. (2 points)

iii. Show that β(˜) =

iv.  Derive the gross saving rate (s = in terms of the model parameters. (3 points)

v.  Piketty argues that as the economy’s growth rate declines to zero (g ! 0), capital- net income ratio may increase indefinitely (β(˜) ! ¥).  What happens to the gross

saving rate, s in this case?  Discuss Piketty’s claim in light of your answer to the above question. (2 points)

vi. Does this model satisfy basic growth (Kaldor) facts? Explain. (2 points)

vii.  How would your answers to above questions change if the population grows at the rate of 1 + n? (2 points)

(b)  Now suppose that gross saving rate s is constant as in the version of Solow model we saw in the class.

i.  Derive β in the steady state in terms of the model parameters. (2 points)

ii.  Derive the net saving rate in the steady state in terms of the model parameters. (2 points)

iii. Deriveβ(˜) capital to net income ratio. (2 points)

References

PIKETTY, T. and GOLDHAMMER, A. (2014). Capital in the twenty-first century. Belknap Press.

— and ZUCMAN, G. (2014). Capital is back: Wealth-income ratios in rich countries, 1700-2010. The Quarterly Journal of Economics.