<p>I'm surprised nobody suggested a quadratic (in consumption) and linear (in leisure) utility function! Suppose we have a single price-taking (wrt both the wage and price of the good) consumer with preferences over a consumption good <span class="math-container">$c$</span> and leisure.</p>
<p>Let <span class="math-container">$c$</span> be a numeraire and let <span class="math-container">$w$</span> be the wage, <span class="math-container">$T$</span> the consumer's time endowment. This utility function should work whether the second decision variable is the amount of leisure <span class="math-container">$l$</span> or the amount of labor <span class="math-container">$L$</span>.</p>
<p>Let's first consider the case where leisure is the decision variable, and so the budget constraint is <span class="math-container">$c \le w(T-l)$</span>. The utility function we will be considering is
<span class="math-container">$$U(c,l)= \alpha c - c^2 +l$$</span>
for some <span class="math-container">$\alpha&gt;0$</span></p>
<p>Since utility is monotone increasing in leisure we know the constraint will hold with equality at an optimal bundle, so <span class="math-container">$c=w(T-l)$</span>. Substituting back into our utility function we obtain</p>
<p><span class="math-container">$$U(l)=\alpha w (T-l) - w^2 (T-l)^2 + l$$</span>
Taking the FOC we have
<span class="math-container">$$U'=-\alpha w + 2w^2(T-l) + 1=0$$</span></p>
<p><span class="math-container">$$T-l^*=L(w)=\frac{\alpha w -1}{2w^2}$$</span></p>
<p>Plotting <span class="math-container">$L(w)$</span> for <span class="math-container">$\alpha=10$</span> we get</p>
<p><img src="https://github.com/user-attachments/assets/cea5a696-bb7a-4cf5-a38f-5eef104ad108" alt="plot" /></p>
<p>In the case where labor <span class="math-container">$L$</span> is the decision variable we have that the consumer's problem is</p>
<p><span class="math-container">$$\max_{c,L} \{\alpha c - c^2 + (T-L)\}$$</span> s.t <span class="math-container">$$c \le wL $$</span></p>
<p>Again since utility is monotone increasing in <span class="math-container">$-L$</span>, the budget constraint will hold with equality, so substituting <span class="math-container">$wL$</span> for <span class="math-container">$c$</span> into our utility function we get</p>
<p><span class="math-container">$$ u(L)= \alpha w L - w^2 L^2 + T - L$$</span></p>
<p>Taking the FOC we get</p>
<p><span class="math-container">$$\alpha w - 2w^2L - 1=0$$</span>
<span class="math-container">$$L^* (w)=\frac{\alpha w -1}{2w^2}=T-l^*$$</span></p>

CC BY-SA 4.0
2024-08-07T23:17:45.273Z
2024-08-07T23:17:45.273Z
Ross K
<p>This result can be derived with CES preferences between consumption and leisure, if consumption and leisure are assumed to be complements.</p>
<p>In particular, suppose <span class="math-container">$u(c,l)=\left[ c^{\rho } +( 1-l)^{\rho }\right]^{\frac{1}{\rho }}$</span></p>
<p>Then the labor supply function is <span class="math-container">$l=\frac{1}{1+w^{\frac{\rho }{\rho -1}}}$</span></p>
<p>If <span class="math-container">$\rho&lt;0$</span>, which corresponds to the case where the two goods are complements, this is decreasing in <span class="math-container">$w$</span></p>

CC BY-SA 4.0
2024-03-26T14:52:02.833Z
Plugging this result into Desmos for graphing it looks like you get a downward sloping labor supply curve under the specification of where $\rho<0$. Not exactly what I'm looking for but interesting.

https://www.desmos.com/calculator/hn9omrre5r
EconJohn
CC BY-SA 4.0
2024-03-25T04:35:05.490Z
2024-03-25T04:35:05.490Z
cruijf
<p>Denoting leisure with <span class="math-container">$l$</span> consumption with <span class="math-container">$c$</span>,the function
<span class="math-container">$$
U(c,l) = \min(15;c) + l
$$</span>
works. Assume <span class="math-container">$c$</span> is a numeraire, there is a total of <span class="math-container">$T$</span> time units, no non-wage income. Optimal <span class="math-container">$l$</span> as a function of <span class="math-container">$w$</span>:
<span class="math-container">\begin{align}
l = \left\{
\begin{array}{ll}
T &amp; \text{ when } w &lt; 1 \\
\left[\max\left(0;T - \frac{15}{w}\right),T\right] &amp; \text{ when } w = 1 \\
\max\left(0;T - \frac{15}{w}\right) &amp; \text{ when } w &gt; 1 \\
\end{array}
\right.
\end{align}</span>
A graph of the labor supply function with <span class="math-container">$T=15$</span>:</p>
<p><a href="https://i.sstatic.net/g2RqX.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/g2RqX.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-03-25T09:22:23.507Z
2024-03-25T15:31:16.977Z
2024-03-25T15:31:16.977Z
Giskard
<p>One way to conceptualize a backwards bending labour supply curve is that leisure is effectively a Giffen good. When the price of leisure (ie. the wage) rises, the worker demands more leisure because the income effect outweighs the substitution effect.</p>
<p>Given this, it should be possible to adapt a utility function that's descriptive of a Giffen good (see examples here: <a href="https://socialsci.libretexts.org/Bookshelves/Economics/Intermediate_Microeconomics_with_Excel_(Barreto)/04%3A_Compartive_Statics/4.05%3A_Giffen_Goods" rel="nofollow noreferrer">https://socialsci.libretexts.org/Bookshelves/Economics/Intermediate_Microeconomics_with_Excel_(Barreto)/04%3A_Compartive_Statics/4.05%3A_Giffen_Goods</a>) to fit what you're looking for. Keep in mind that the binding income constraint is different in this case, instead of <span class="math-container">\begin{equation} p_1x_1+p_2x_2=I \end{equation}</span>
it would be
<span class="math-container">\begin{equation} p_1C=wL \end{equation}</span>
where C is consumption, w is the wage and L is the amount of labour supplied.</p>

CC BY-SA 4.0
2024-03-21T21:35:34.203Z
"*leisure is effectively a Giffen good.*" It is not though. To determine Giffenness, all other parameters, such as income should remain unchanged. Your answer does not take into account the [endowment income effect](https://wiki.ubc.ca/Slutsky_Substitution_and_Income_Effect#Endowment_income_effect). The price of leisure is the wage only for people who sell their leisure time (i.e. work). For these people at least part of their income come from wages. Therefore you do not have a situation where price goes up but income is unchanged.
Giskard
CC BY-SA 4.0
2024-03-21T23:54:16.593Z
To my understanding this is true, but ultimately semantic. Whether leisure is technically a Giffen good or just analogous to one, the utility function for a Giffen good can be converted into a leisure/consumption utility function that leads to a backwards-bending supply curve.
H Rogers
CC BY-SA 4.0
2024-03-22T04:36:54.633Z
I have doubts about your argument, so it would be nice if you would take the time to demonstrate it rather than just claim it.
Giskard
CC BY-SA 4.0
2024-03-22T14:41:21.397Z
After applying some effort I've not found a way to adapt a  Giffen utility function for leisure without including the price of labour as a variable in the direct utility function. So it would seem that you're correct, my mistake.
H Rogers
CC BY-SA 4.0
2024-03-21T20:32:18.473Z
2024-03-21T20:32:18.473Z
H Rogers
<p>The idea of a backwards bending labor supply curve is often discussed in many introductory labor economics classes where a consumer's labor leisure decision at higher wages (due to a diminishing marginal utility from money) will result in them supplying less labor when there is a wage increase.</p>
<p>While this is intuitive, mathematically I have been having a hard time finding a utility function which generates a backwards bending supply curve. While there is a great deal of older theoretical literature out there, I'm interested in a specific functional form of a utility function that will generate a backwards bending supply curve.</p>
<p>What functional forms exist for deriving a backwards bending labor supply curve?</p>

CC BY-SA 4.0
2024-03-21T17:50:45.517Z
Do you need backward bending, or is decreasing enough?
Giskard
<mathematical-economics><labor-economics>
What kind of utility function is needed to derive a "Backwards Bending Labor Supply Curve"
EconJohn
# What kind of utility function is needed to derive a "Backwards Bending Labor Supply Curve"

(Asked by: username_0 on 2024-03-21)

The idea of a backwards bending labor supply curve is often discussed in many introductory labor economics classes where a consumer's labor leisure decision at higher wages (due to a diminishing marginal utility from money) will result in them supplying less labor when there is a wage increase.

While this is intuitive, mathematically I have been having a hard time finding a utility function which generates a backwards bending supply curve. While there is a great deal of older theoretical literature out there, I'm interested in a specific functional form of a utility function that will generate a backwards bending supply curve.

What functional forms exist for deriving a backwards bending labor supply curve?

**Tags:** mathematical-economics, labor-economics
> **Comments:**
> **username_1**: Do you need backward bending, or is decreasing enough?  

---

## 4 answers

### Answer by username_3 on 2024-03-25. Score: 5

This result can be derived with CES preferences between consumption and leisure, if consumption and leisure are assumed to be complements.

In particular, suppose $u(c,l)=\left[ c^{\rho } +( 1-l)^{\rho }\right]^{\frac{1}{\rho }}$

Then the labor supply function is $l=\frac{1}{1+w^{\frac{\rho }{\rho -1}}}$

If $\rho<0$, which corresponds to the case where the two goods are complements, this is decreasing in $w$

> **Comments:**
> **username_0**: Plugging this result into Desmos for graphing it looks like you get a downward sloping labor supply curve under the specification of where $\rho<0$. Not exactly what I'm looking for but interesting.  https://www.desmos.com/calculator/hn9omrre5r  

---

### Answer by username_2 on 2024-08-07. Score: 4

I'm surprised nobody suggested a quadratic (in consumption) and linear (in leisure) utility function! Suppose we have a single price-taking (wrt both the wage and price of the good) consumer with preferences over a consumption good $c$ and leisure.

Let $c$ be a numeraire and let $w$ be the wage, $T$ the consumer's time endowment. This utility function should work whether the second decision variable is the amount of leisure $l$ or the amount of labor $L$.

Let's first consider the case where leisure is the decision variable, and so the budget constraint is $c \le w(T-l)$. The utility function we will be considering is
$$U(c,l)= \alpha c - c^2 +l$$
for some $\alpha>0$

Since utility is monotone increasing in leisure we know the constraint will hold with equality at an optimal bundle, so $c=w(T-l)$. Substituting back into our utility function we obtain

$$U(l)=\alpha w (T-l) - w^2 (T-l)^2 + l$$
Taking the FOC we have
$$U'=-\alpha w + 2w^2(T-l) + 1=0$$

$$T-l^\*=L(w)=\frac{\alpha w -1}{2w^2}$$

Plotting $L(w)$ for $\alpha=10$ we get

![plot](https://github.com/user-attachments/assets/cea5a696-bb7a-4cf5-a38f-5eef104ad108)

In the case where labor $L$ is the decision variable we have that the consumer's problem is

$$\max\_{c,L} \{\alpha c - c^2 + (T-L)\}$$ s.t $$c \le wL $$

Again since utility is monotone increasing in $-L$, the budget constraint will hold with equality, so substituting $wL$ for $c$ into our utility function we get

$$ u(L)= \alpha w L - w^2 L^2 + T - L$$

Taking the FOC we get

$$\alpha w - 2w^2L - 1=0$$
$$L^\* (w)=\frac{\alpha w -1}{2w^2}=T-l^\*$$


---

### Answer by username_1 on 2024-03-25. Score: 1

Denoting leisure with $l$ consumption with $c$,the function
$$
U(c,l) = \min(15;c) + l
$$
works. Assume $c$ is a numeraire, there is a total of $T$ time units, no non-wage income. Optimal $l$ as a function of $w$:
\begin{align}
l = \left\{
\begin{array}{ll}
T & \text{ when } w < 1 \\
\left[\max\left(0;T - \frac{15}{w}\right),T\right] & \text{ when } w = 1 \\
\max\left(0;T - \frac{15}{w}\right) & \text{ when } w > 1 \\
\end{array}
\right.
\end{align}
A graph of the labor supply function with $T=15$:

[![enter image description here](https://i.sstatic.net/g2RqX.png)](https://i.sstatic.net/g2RqX.png)


---

### Answer by username_4 on 2024-03-21. Score: -3

One way to conceptualize a backwards bending labour supply curve is that leisure is effectively a Giffen good. When the price of leisure (ie. the wage) rises, the worker demands more leisure because the income effect outweighs the substitution effect.

Given this, it should be possible to adapt a utility function that's descriptive of a Giffen good (see examples here: <https://socialsci.libretexts.org/Bookshelves/Economics/Intermediate_Microeconomics_with_Excel_(Barreto)/04%3A_Compartive_Statics/4.05%3A_Giffen_Goods>) to fit what you're looking for. Keep in mind that the binding income constraint is different in this case, instead of \begin{equation} p\_1x\_1+p\_2x\_2=I \end{equation}
it would be
\begin{equation} p\_1C=wL \end{equation}
where C is consumption, w is the wage and L is the amount of labour supplied.

> **Comments:**
> **username_1**: "*leisure is effectively a Giffen good.*" It is not though. To determine Giffenness, all other parameters, such as income should remain unchanged. Your answer does not take into account the [endowment income effect](https://wiki.ubc.ca/Slutsky_Substitution_and_Income_Effect#Endowment_income_effect). The price of leisure is the wage only for people who sell their leisure time (i.e. work). For these people at least part of their income come from wages. Therefore you do not have a situation where price goes up but income is unchanged.  
> **username_4**: To my understanding this is true, but ultimately semantic. Whether leisure is technically a Giffen good or just analogous to one, the utility function for a Giffen good can be converted into a leisure/consumption utility function that leads to a backwards-bending supply curve.  
> **username_1**: I have doubts about your argument, so it would be nice if you would take the time to demonstrate it rather than just claim it.  
> **username_4**: After applying some effort I've not found a way to adapt a  Giffen utility function for leisure without including the price of labour as a variable in the direct utility function. So it would seem that you're correct, my mistake.  

---

