Combined Income & Substitution Effects

Income Effect (IE) Substitution Effect (SE)

Combined Income/Substitution Effects §

In cases where price of one good changes, income and substitution effects happen together. Imagine when gasoline prices rise:

• Graph (a) shows the change in the budget line due to the increase in price of gasoline.
• Graph (b) shows the compensated budget line to maintain the level of utility.
• Graph (c) shows the ultimate, uncompensated bundle C.

The process for both effect is as follows

1. Move along the same utility curve from bundle A → B; substitution effect
2. Move to the new budget line from bundle B → C; income effect → this determines if good is normal (homo-thetic or not), quasi-linear, inferior; see below)

Income and Substitution Effects on Types of Goods §

The above case was when gasoline was a normal good. When gasoline is inferior the move due to income effect (B → C) is to the right. The degree to which this occurs coincidentally determines if the good is regular inferior, or (the strange case of) a Giffen good.

Observe that for both graphs B → C is a rightward move, because lowering income (blue → red) increased consumption, meaning this is an inferior good. If C has more of gasoline than A, it is the strange Giffen good.

Bringing it Together §

The final is combining all of these movements together. In the following graph, the consumer consumes only shirts and pants. Then they’re given a coupon that discounts the price of shirts (blue → red). Using the compensated budget we determine the substitution effect (b, A → B). Then, depending on whether shirts/pants are normal, inferior (or quasilinear) we can determine where the final income effect (B → C) will lie.

Mathematically… §

Assume:

• Utility curve $u_A=u(x_1,x_2)$
• Budget line $I=p_1{x}_1+p_2{x}_2$
• Your bundle $A(x_1^A,x_2^A)$,

Then occurs a change in price $p_1\to p_1^{\prime }$, causing a change in the budget line from $I\to I^{\prime }$. Then, consider how much you should compensate your income, to stay in the same indifference curve with utility $u_A$. That is, what is the minimum income that I can spend to reach a certain indifference curve?

$\text{min }{p}_1^{\prime }{x}_1+p_2{x}_2\text{ s.t.}{u}_A=u(x_1,x_2)$

Thus you find the bundle $B(x_1^B,x_2^B)$, with the size of the substitution effect $x_1^A\to x_1^B$

Since this compensated budget is a hypothetical scenario, we will need to move back to the original budget.

$\text{max }u(x_1,x_2)\text{ s.t.}{I}^{\prime }=p_1^{\prime }{x}_1+p_2{x}_2$

to find bundle $C(x_1^C,x_2^C)$.