Unit 9 Lenders and borrowers and differences in wealth

9.10 Inequality: Lenders, borrowers, and those excluded from credit markets

Gini coefficient

A measure of inequality of a quantity such as income or wealth, varying from a value of zero (if there is no inequality) to one (if a single individual receives all of it). It is the average difference in, say, income between every pair of individuals in the population relative to the mean income, multiplied by one-half. Other than for small populations, a close approximation to the Gini coefficient can be calculated from a Lorenz curve diagram. See also: Lorenz curve.

We can assess inequalities between borrowers and lenders (and among borrowers) using the Gini coefficient.

Imagine an economy with one lender and five self-employed people who each borrow an amount, L, to finance their small business. The revenue from selling the goods they produce is greater than the loan; each borrower receives revenue (1 + R)L, where R is the profit rate.

At the end of the year, the borrowers repay the loans with interest, at rate r. To simplify the analysis, we assume that all of the loans are repaid. The main message would not be altered if we included the probability that loans were not repaid (as in the Chambar case study), but the mathematics would be more complicated. After the loan is repaid:

• The lender’s net income from each business is rL.
• The borrower’s net income is (R – r)L.
• The total net income from each business is I = RL.
• The lender’s share of the income I is s = \(\frac{r}{R}\).
• The borrower’s share is 1 – s.

For example, if r = 0.10 and R = 0.15, then the lender’s share of the net income from each business is \(s=\frac{r}{R}=\frac{0.10}{0.15}=\frac{2}{3}\), and the borrower’s is \(\text{1 } – \text{ s} = \frac{1}{3}\).

In all the calculations in this section, we will assume that the interest rate is sufficiently high that the lender always has higher total income than the borrowers.

We can use the Gini coefficient to measure the level of inequality in this economy. The Gini measures the average of the differences between all individuals in the population, relative to the average income. (The differences are always written as positive values.) It is calculated as:

\[g = \frac{1}{2} \times \frac{\text {average income difference between every pair of individuals}}{\text {average income}}\]

The Gini coefficient is always a number between 0 and 1. It is 0 if everyone’s income is the same (perfect equality); the more inequality there is, the higher the average difference is and the closer g is to 1. If one person obtains all the income in the economy, \(g=1\).

In our example there are six individuals. We could draw a diagram showing the people as circles and the differences using lines joining the circles, as in the example with four individuals in Figure 5.29. But with more individuals, the number of lines increases rapidly. So instead we show the population in a table, in Figure 9.19a.

The top part of the table shows the income of each person, measured in units of I, the net income of each business. The lender receives a proportion, s, from each of the five businesses, and hence a total income of 5s; each of the borrowers obtains 1 − s. Total income is 5, so average income in the economy is \(\frac{5}{6}\).

The lower part of the table shows a way of calculating the average difference. The first row shows the differences between the income of the lender and the incomes of each of the five borrowers—(\(6s − 1\)) in each case.

The second row gives the differences between the income of borrower 1 and those of the other four borrowers (all of which are zero); we have already included the difference with the lender on the first row. As we go down the table there is one fewer difference to count each time, because the others have already been counted. With a population of 6, there are a total of 15 differences to count.

Adding up all of the differences and dividing by 15 gives us an average difference of \((6s-1)/3\).

We can now use the average difference and average income to calculate the Gini. Using the formula above:

\[g = \frac{1}{2} \times \frac{6s - 1}{3} \div \frac{5}{6}\]

and simplifying this expression, we get:

\[g = \frac{6s - 1}{5}\]

For example, if r = 0.10 and R = 0.15 as above, the lender’s share is \(s=\frac{2}{3}\), and the Gini coefficient is 0.6. If the interest rate were increased, giving the borrower a higher share of income, the level of inequality would rise.

Excluding prospective borrowers raises the Gini coefficient

In previous sections, we showed why some would-be borrowers (those unable to provide collateral or lacking their own funds to finance a project) might be excluded entirely from borrowing, even if they would be willing to pay the interest rate. How does this affect the Gini coefficient?

Figure 9.19b shows the same table for the case when borrowers 4 and 5 are excluded, and hence receive no income (assuming that the lender still has the highest income, so \(s\geq\frac{1}{4}\)). Compared with the previous case:

• Total and average incomes in the economy are lower.
• The lender’s income is lower, reducing income differences between the lender and the borrowers who can borrow.
• But there is additional inequality, between the borrowers who can borrow, and those who are excluded.

Calculating the Gini as before from the expressions in the table, we find:

\[g = \frac{4s + 1}{5}\]

For the example in which \(s=\frac{2}{3}\), the Gini rises from 0.6 to 0.73. Exclusion has increased the level of inequality in the economy. You can verify, by comparing the expressions for g, that exclusion raises the Gini coefficient for any value of \(s<1\).