Intermediate Macroeconomics Questions, due on April 29, 3:00pm (UTC+8)

09WealthInequality.docx

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Wealth and Intergenerational Inequality

1. Introduction.

a) Overview

In this lecture, we will explore the relationship between income inequality and wealth inequality. More concretely, we will explore how wealth inequality is both a cause and a consequence of income inequality.

To this end, we will develop an occupational choice labour market model. In this model, we will show that the wealth distribution plays a critical role for the degree of income inequality. In turn, the degree of income inequality influences the wealth inequality in future periods.

More concretely, we will attempt to establish the following causal link:

Asset inequality in the current period => Income inequality in the current period => Asset inequality in the following period

In the next lecture, we will augment this labour market model with intergenerational transfers of assets, i.e., inheritance. This extra characteristic will allow us to discuss the implications of asset accumulation (e.g., pensions, housing, etc.) on intergenerational income inequality.

b) Wealth inequality vs. income inequality

· Assets vs. income

The relationship between assets (also referred to as wealth) and income could be demonstrated through the following equation:

The assets in 2021, , equal the assets in 2020, , plus the interest earned on those assets, , plus the share of income one saves from their income in 2020, .

· Wealth inequality vs. income inequality

Income inequality uses income as the unit of analysis, while wealth inequality uses assets (or wealth) as the unit of analysis.

c) Key facts

In this section, we will analyze some key evidence about the evolution of wealth inequality in the United States due to the absence of available comparable data for Canada.

First, we need to demonstrate that Canada and the United States have followed a similar pattern with respect to the income distribution in the past few decades.

Figure 1

Source: IRPP(2016)

Figure 1 presents evidence that only the top 10 %, i.e., the high-income segment of the population in both Canada and the US has benefited from economic growth. This is manifested in the positive average growth rates of their incomes. In contrast, the bottom 90 % have seen no gains.

Figure 1 also indicates that the largest income gains are concentrated among the top 1 % and the top 0.1 % of the population.

Figure 2

Source: Washington Post (2019)

Figures 1 and 2 suggest that the sustained increases in income inequality are mapped to sustained increase in the gap in asset holdings between the richest 1 % and the bottom 80/90 % of the population. In addition, the gains in asset holdings from the last few decades are almost exclusively concentrated among the richest 0.1 % of the population.

What we can infer from the two figures is that the stagnant incomes of the bottom 90 % translate into a declining share of wealth. In the rest of the lecture, we will investigate if there is a causal relationship between income inequality and wealth inequality.

2. Occupational choice labour market model.

a) Overview

In this section, we will go over a model of a dual labour market developed by Ghatak and Jiang (2002), in which individuals make an occupational choice decision that in part depends on one’s asset holdings.

The model also assumes credit market imperfections in the sense that individuals must present collateral (e.g., mortgage a property) to obtain credit.

The model develops the idea of a poverty trap at two levels:

· poor individuals (with low asset holdings) end up earning low income because they are born poor,

· if a large enough share of the population that is poor (with low asset holdings), the economy has:

· a lower standard of living,

· more unequal income distribution.

b) Economic environment

· Agents

· Demographics: Infinitely lived dynasties of individuals who live for one period.

· In this lecture, we will analyze the model for one period. In the next lecture, we will develop the idea of dynasties, where children inherit their parents’ assets.

· Production technologies

This is a dual labour market model as there are two sectors of the economy that use different technology.

· Entrepreneurial (industrial) technology:

· it uses one unit of capital and one unit of labour, worker, to produce output ;

· workers are paid a wage that is endogenously determined.

· Self-employment technology:

· it uses no capital and one unit of labour.

· workers are paid an exogenous wage .

The entrepreneurial technology is superior in the sense that the net output of using this technology is greater than were two units of labor using the self-employment technology. That is,

· Occupations

· Self-employed

The income of a self-employed, , is generated from a payment for completing a project, , as well as a return on their assets, . is the amount of assets, which could be positive, zero or negative, and is the interest rate.

· Worker

The income of a worker, , is generated from a wage payment, , as well as a return on their assets, . is the amount of assets, which could be positive, zero or negative, and is the interest rate.

· Entrepreneur

The income of a worker, , is generated from the profit an entrepreneur earns, , as well as a return on their assets, .

The profit is the difference between the revenue the entrepreneur generates and the wage to the single worker hired by the entrepreneur.

To earn the profit , the entrepreneur must invest the amount .

is the amount of assets, which could be positive, zero or negative, and is the interest rate.

c) Credit market imperfections

· Constraints

For an individual to find in their self-interest to become an entrepreneur, we must demonstrate that:

· being an entrepreneur pays the highest income of all three occupations (participation constraint)

· repaying the debt pays off relative to defaulting (incentive compatibility constraint).

· Participation constraint:

The idea behind a participation constraint is closely related to that behind an opportunity cost.

For instance, an individual has an incentive to become an entrepreneur if their income as an entrepreneur is at least as large as their highest valued alternative. Their highest valued alternative is being a worker if the income of a worker exceeds the income of a self-employed.

Numerical example:

Suppose that . Suppose that 70 % of the population have asset holdings and the remaining 30 % of the population .

For a rich individual with

A rich individual with will become an entrepreneur if

The highest value alternative is being a worker if or equivalently .

· Incentive compatibility constraint (ICC)

An ICC checks whether an individual who intends to become an entrepreneur would have an incentive to default on their debt.

– amount of punishment (through legal system or social sanctions)

– probability of being caught

This condition implies that the individuals who will receive a loan are those whose assets are at least as large as .

Therefore, only the relatively wealthy individuals could become entrepreneurs.

Numerical example:

Suppose that . Suppose that 70 % of the population have asset holdings and the remaining 30 % of the population .

The non-credit-constrained individuals are the 70 % who have assets . These individuals have the option of becoming either an entrepreneur, a worker or self-employed.

The credit-constrained individuals are the 30 % who have assets . These individuals have the option of becoming either a worker or self-employed but not an entrepreneur.

In the subsequent section, we will demonstrate that the poor, i.e., those with insufficient asset holdings, could end up earning less because they have insufficient asset holdings they can use as collateral.

The threat of punishment from lenders is that the collateral will be seized if loans are not repaid back. This threat of punishment works less well against the poor since lenders have nothing or little to confiscate in the event of default. As a result, the poor cannot become entrepreneurs.

· Occupational distribution

The ICC,, provides us with the share of individuals who can become entrepreneurs, . These individuals can select into any of the three occupations.

The remaining share of the population cannot become entrepreneurs. These individuals can become eithers workers or self-employed.

d) Labour market equilibrium

There are two possible equilibria: one for and another one for .

We will show that there is a high-wage labour market equilibrium if the share of individuals are not credit constrained, , is above a critical share.

This critical share equals 50 % if the technological requirement of the entrepreneurial technology is one entrepreneur hiring one worker.

We will also show that there is a low-wage labour market equilibrium if the share of individuals who are not credit constrained is below the same critical share, i.e., .

Numerical example:

Suppose that . Suppose that 70 % of the population have asset holdings and the remaining 30 % of the population .

To determine the labour market equilibrium, we must analyze each of the relevant wage ranges. These wage ranges are determined by the participation constraints of the credit constrained and the non-credit constrained individuals.

We have found those threshold level wages to be and We need to determine labour demand at these two wages as well wages that fall into the following intervals:, and .

If , everyone wants to be a worker () but no one wants to be an entrepreneur () or self-employed. A wage cannot be sustained as an equilibrium since .

If , all credit-constrained individuals want to become workers and all non-credit-constrained workers are indifferent between becoming entrepreneurs and becoming workers. No individual has incentive to become self-employed. This implies that and . That is, a wage can be sustained as an equilibrium since there is a potential intersection between and .

For this to be an equilibrium, each entrepreneur must be matched with one worker, that is , due the production requirements of the entrepreneurial technology. In addition, no individual would have an incentive to switch their occupation.

If , all non-credit constrained individuals want to become entrepreneurs () and all credit constrained individuals want to become workers (). No one has an incentive to become self-employed. A wage cannot be sustained as an equilibrium since .

If , all non-credit constrained individuals want to become entrepreneurs () but the credit constrained individuals are indifferent between becoming workers and becoming self-employed (). All credit constrained individuals want to become self-employed. A wage cannot be sustained as an equilibrium since .

If , all non-credit constrained individuals want to become entrepreneurs () but no one wants to become a worker (). All credit constrained individuals want to become self-employed. A wage cannot be sustained as an equilibrium since .

Figure 3

e) Labour market outcomes

The labour market equilibrium results in:

· occupational distribution (shares individuals who become entrepreneurs, workers and self-employed), i.e., and .

· labour market outcomes in the entrepreneurial sector, i.e., employment and wage .

· Income distribution: there is perfect income equality for individuals with the same asset holdings since but there is some inequality for individuals with different asset holdings due the differential return on assets .

For the individuals with :

f) Modified wealth distribution

· Wealth distribution

In this section, we will demonstrate that a for a different distribution of asset holding, we may obtain a very different equilibrium characterized by a lower wage, a lower average standard of living, a more unequal income distribution.

Suppose that we have the same per capita assets, , but they are distributed differently across the population.

Suppose that 40 % of the population have assets and the remaining 60 % of the population have assets .

For this distribution, we have the same per capita assets, .

Except for this modified distribution of asset holdings, we will assume that all else in the economy is the same. That is, .

· Labour market equilibrium

What changes in the solution of the model?

First, the share of non-credit-constrained individuals is 40 % since . Only this group of individuals can become entrepreneurs if they choose to.

Secondly, the occupational shares and the labour markets outcome change.

We will show that the equilibrium outcome is and and the occupational distribution is and .

If , all non-credit-constrained workers want to become entrepreneurs and all credit-constrained workers are indifferent between becoming self-employed and becoming workers. This implies that . That is, a wage can be sustained as an equilibrium since there is a potential intersection between and .

· Labour market outcomes

The labour market equilibrium results in:

· occupational distribution (shares individuals who become entrepreneurs, workers and self-employed), i.e., and .2,

· labour market outcomes in the entrepreneurial sector, i.e., employment and wage ,

· Income distribution: there is income inequality since ,

· Average income is lower since for any pair of individuals operating the entrepreneurial technology as entrepreneur and worker generates more income than these same two individuals becoming self-employed.

g) Key insights

Figure 4

Source: Statistics Canada (2015)

The model demonstrates for a different distribution of asset holding we may obtain a very different economy that exhibits structural changes with respect to the formation of a self-employed sector accompanied with lower wages and rising income inequality.

In recent decades, we have observed each of the three phenomena in the data, including rising share of self-employed. Figure 4 illustrates the increase rate of self-employment in recent decades.

The only problematic prediction of the economic model is about the economic growth rate. However, we must be careful in its interpretation. The relevant question is by how much more would the economic growth have been in the absence of an increasing self-employment rate.