Leibniz

3.7.1 Mathematics of income and substitution effects

We have seen that when you are deciding how many hours per day you would like to work, the effect on your decision of a change in your wage rate can be decomposed graphically into an income effect and a substitution effect. This Leibniz shows how to do this decomposition mathematically.

We have modelled the working hours decision by supposing that you choose your consumption $c$ and hours of free time $t$ to maximize your utility, given that your consumption depends on how much you earn. We can write this mathematically as a constrained optimization problem:

$$\begin{align*}\text{maximize}\ U(t,\ c) \text{ subject to}\ c=w(24-t)+I \end{align*}$$

where $w$ is your wage rate, and $I$ is the income that you would receive irrespective of your hours of work (for example, from the mysterious benefactor).

We will solve this problem for a particular utility function

$$U(t,\ c) = tc$$

to find the optimal choice of free time. Then, we can work out how the solution changes as the wage, $w$, changes, and decompose the change into an income effect and a substitution effect.

The first-order condition for optimization equates the marginal rate of transformation (MRT) to the marginal rate of substitution (MRS). As we saw in the text, your MRT is $w$. To see this directly, remember that the budget constraint $c=w(24-t)+I$ is the equation of the feasible frontier. The MRT is the absolute value of the slope of the feasible frontier:

$$\text{MRT} = \left|\frac{dc}{dt}\right|=w$$

The MRS can be calculated using the formula familiar from earlier Leibnizes:

$$\text{MRS} = \left| \frac{\partial U}{\partial t} \left/ \frac{\partial U}{\partial c} \right.\right| = \frac{c}{t}$$

So in this case the first-order condition MRS = MRT gives us $c/t=w$, or equivalently $wt-c=0$. The optimal values of $t$ and $c$ are found by solving a pair of simultaneous equations, the first-order condition and the constraint:

$$wt - c = 0,\ \ c = w\left( 24 - t \right) + I$$

The solution is:

$$t = 12 + \frac{I}{2w}, \quad c = 12w + \frac{I}{2}$$

Therefore, the optimal number of hours of free time is a function of the wage rate and additional income, with the following partial derivatives:

$$\frac{\partial t}{\partial w} = - \frac{I}{2w^2} \lt0 , \quad \frac{\partial t}{\partial I} = \frac{1}{2w} \gt0$$

These two partial derivatives tell us how the hours of free time chosen will change if the wage changes, or if income changes. The inequality $\partial t/\partial I \gt0$ tells us that with this utility function, free time $t$ increases if income $I$ increases and there is no change in the wage. From $\partial t/\partial w\lt0$, we see that $t$ decreases if $w$ increases and there is no change in income.

In other words, the overall effect of a wage increase with no change in income $I$, given by $\partial t/\partial w$, is negative. This overall effect can be decomposed into income and substitution effects. We will look at a numerical example to see how.

A numerical example

Suppose that $w = 16$ and $I= 160$. Putting these values into the equations for the solution above, we find that the optimal choice of hours of free time and dollars of consumption is:

$$t_0 = 12 + \frac{160}{2 \times 16} =17, \quad c_0 = 12 \times 16 + \frac{160}{2} = 272$$

The level of utility is $U_0=t_0 \times c_0 =17 \times 272 = 4,624$.

Now suppose the wage rises to 25, while income remains at 160. We will first find the overall effect of the wage rise on the choice of free time, and then decompose it.

1. What is the overall effect of the wage rise?

With $w=25$ and $I=160$, the optimal choice for $t$ and $c$ is:

$$t_1 = 12 + \frac{160}{2 \times 25} =15.2, \quad c_1 = 12 \times 25 + \frac{160}{2} = 380$$

The level of utility rises to $U_1=15.2 \times 380 = 5,776$.

We can see that the overall effect of increasing the wage from 16 to 25, while income remains constant at 160, is to reduce free time:

$$t_1-t_0=15.2-17=-1.8$$

2. What change in income would have the same effect on utility as the wage increase?

Suppose that the wage had remained at $w=16$, but income had changed from 160 to $J$. To give a utility level of 5,776, $J$ must satisfy the equation:

$$\left(12 + \frac{J}{32}\right)\left(192 + \frac{J}{2}\right) = 5,776$$

We may write this equation as $(384 + J)^2 = 64 \times 5,776$. Taking the square root of both sides (only the positive square root makes economic sense), we obtain $384 + J = 608$ and hence $J=224$.

So increasing income from 160 to 224, while keeping the wage at 16, would have the same effect on utility as the wage rise from 16 to 25.

3. Find the income effect

We do this by working out how the income change that is equivalent to the wage change would affect the choice of free time.

With income increased to 224 and a wage of 16, the optimal choice of hours of free time is:

$$t_2 = 12 + \frac{224}{2 \times 16} =19$$

This tells us the income effect of the wage rise. The wage rise increases utility to 5,776. If this had been achieved by increasing income instead, hours of free time would have changed from $t_0=17$ to $t_2=19$. The income effect is $t_2-t_0=19-17=+2$.

4. Find the substitution effect

The overall effect of the wage rise is to change free time from $t_0$ to $t_1$. It is the sum of the income effect (the change from $t_0$ to $t_2$) and the substitution effect, which is therefore the change from $t_2$ to $t_1$.

Income effect	$t_2-t_0=+2$
Substitution effect	$t_1-t_2=-3.8$
Overall effect	$t_1-t_0=-1.8$

The substitution effect is the effect on the choice of free time of changing the wage from 16 to 25, but also adjusting income to keep utility constant at 4,624.

Read more: Sections 14.1, 17.1 and 17.3 of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press.