<p>Consider a setting where someone consumes <span class="math-container">$c$</span> at price <span class="math-container">$p$</span> and provides labour <span class="math-container">$L$</span> with wage <span class="math-container">$w$</span>. The budget constraint is then:
<span class="math-container">$$
pc = wL.
$$</span>
Now instead of labor <span class="math-container">$L$</span> let us express things in terms of leisure <span class="math-container">$\ell$</span>. If <span class="math-container">$T$</span> is total available time then <span class="math-container">$T = L + \ell$</span>. So we can substitute <span class="math-container">$L = T - \ell$</span> into the budget constraint to get:
<span class="math-container">$$
pc = w(T - \ell) \to pc + w \ell = w T.
$$</span>
This budget constraint states that total potential income <span class="math-container">$wT$</span> must be split  between consumption <span class="math-container">$pc$</span> and leisure <span class="math-container">$w\ell$</span>. It is as if you have two goods <span class="math-container">$c$</span> and <span class="math-container">$\ell$</span> and you divide your total income <span class="math-container">$w T$</span> between <span class="math-container">$c$</span> at price <span class="math-container">$p$</span> and leisure <span class="math-container">$\ell$</span> at price <span class="math-container">$w$</span>.</p>
<p>Now, an increase in <span class="math-container">$w$</span> will make <span class="math-container">$\ell$</span> more costly compared to <span class="math-container">$c$</span>. If <span class="math-container">$c$</span> and <span class="math-container">$\ell$</span> are substitutes, this will lead to a decrease in <span class="math-container">$\ell$</span>, which is what you call the &quot;substitution efffect&quot;. On the other hand, an increase in <span class="math-container">$w$</span> also makes total potential income <span class="math-container">$wT$</span> higher. This is the &quot;income effect&quot;. If <span class="math-container">$\ell$</span> is a &quot;normal good&quot;, this might increase <span class="math-container">$\ell$</span>. So the total effect is ambiguous.</p>

CC BY-SA 4.0
2024-12-22T13:24:17.703Z
Your explanation was fantastic! I followed completely and fully understood the substitution effect. However, I'm still a little confused about the income effect. What does a higher wT have to do with pc? If you could break down the steps further then that would be great.
BeginnerCode776
CC BY-SA 4.0
2024-12-21T11:03:41.083Z
2024-12-21T11:03:41.083Z
tdm
<p>Leisure is assumed to be a normal good,  i.e. an increase in income (from any source) increases demand for leisure.  With an increase in hourly wage, you could reduce your number of hours working and increase your amount of leisure while having the same total income.  So there is an &quot;income effect&quot;.</p>
<p>The &quot;substitution effect&quot; is that the marginal cost of leisure time is the marginal income that could have been obtained by working instead, so this increases with an increase in hourly wage.</p>
<p>So an increase in hourly wage tends to have both effects, pulling in opposite directions, and either could dominate the other.  This is one of reasons that overtime pay tends to be higher than standard pay especially when newly introduced, as the employer tries to minimise the income effect from existing working time while maximising the substitution effect with additional hours worked.</p>

CC BY-SA 4.0
2024-12-21T21:02:55.697Z
2024-12-21T21:02:55.697Z
Henry
<p>The important distinction to make is that a rise in wage is simply one's hourly wage, whereas income refers to unearned income that is not tied to his working hours. An example could be a lottery win of some small but relatively significant amount, perhaps 30% of his monthly salary.</p>
<p>In this case, if there is only a rise in hourly wage, why would there be an income effect? How can both an income effect and a substitution effect both happen? I understand that the opp cost of free time increases so there is a substitution effect, but why is there also an income effect? To have an income effect, he would first have to work more, thereby allowing his income to rise! So it is assumed that he will work more hours instead of increasing free time which is what the income effect says he would do.</p>

CC BY-SA 4.0
2024-12-21T16:27:42.087Z
Why would they have to work more? If the wage goes up, they receive more for the same time worked. Or they could work less to keep their total income the same and increase free time.
Barmar
<optimization>
Does a rise in hourly wage (not unearned income) have an income effect, or just a substitution effect?
BeginnerCode776
# Does a rise in hourly wage (not unearned income) have an income effect, or just a substitution effect?

(Asked by: username_0 on 2024-12-21)

The important distinction to make is that a rise in wage is simply one's hourly wage, whereas income refers to unearned income that is not tied to his working hours. An example could be a lottery win of some small but relatively significant amount, perhaps 30% of his monthly salary.

In this case, if there is only a rise in hourly wage, why would there be an income effect? How can both an income effect and a substitution effect both happen? I understand that the opp cost of free time increases so there is a substitution effect, but why is there also an income effect? To have an income effect, he would first have to work more, thereby allowing his income to rise! So it is assumed that he will work more hours instead of increasing free time which is what the income effect says he would do.

**Tags:** optimization
> **Comments:**
> **username_1**: Why would they have to work more? If the wage goes up, they receive more for the same time worked. Or they could work less to keep their total income the same and increase free time.  

---

## 2 answers

### Answer by username_2 on 2024-12-21. Score: 6

Consider a setting where someone consumes $c$ at price $p$ and provides labour $L$ with wage $w$. The budget constraint is then:
$$
pc = wL.
$$
Now instead of labor $L$ let us express things in terms of leisure $\ell$. If $T$ is total available time then $T = L + \ell$. So we can substitute $L = T - \ell$ into the budget constraint to get:
$$
pc = w(T - \ell) \to pc + w \ell = w T.
$$
This budget constraint states that total potential income $wT$ must be split between consumption $pc$ and leisure $w\ell$. It is as if you have two goods $c$ and $\ell$ and you divide your total income $w T$ between $c$ at price $p$ and leisure $\ell$ at price $w$.

Now, an increase in $w$ will make $\ell$ more costly compared to $c$. If $c$ and $\ell$ are substitutes, this will lead to a decrease in $\ell$, which is what you call the "substitution efffect". On the other hand, an increase in $w$ also makes total potential income $wT$ higher. This is the "income effect". If $\ell$ is a "normal good", this might increase $\ell$. So the total effect is ambiguous.

> **Comments:**
> **username_0**: Your explanation was fantastic! I followed completely and fully understood the substitution effect. However, I'm still a little confused about the income effect. What does a higher wT have to do with pc? If you could break down the steps further then that would be great.  

---

### Answer by username_3 on 2024-12-21. Score: 3

Leisure is assumed to be a normal good, i.e. an increase in income (from any source) increases demand for leisure. With an increase in hourly wage, you could reduce your number of hours working and increase your amount of leisure while having the same total income. So there is an "income effect".

The "substitution effect" is that the marginal cost of leisure time is the marginal income that could have been obtained by working instead, so this increases with an increase in hourly wage.

So an increase in hourly wage tends to have both effects, pulling in opposite directions, and either could dominate the other. This is one of reasons that overtime pay tends to be higher than standard pay especially when newly introduced, as the employer tries to minimise the income effect from existing working time while maximising the substitution effect with additional hours worked.


---

