# What kind of utility function is needed to derive a "Backwards Bending Labor Supply Curve"

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?

• Do you need backward bending, or is decreasing enough? Mar 21 at 17:50

## 3 Answers

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 $$w$$

• 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. desmos.com/calculator/hn9omrre5r
– EconJohn
Mar 26 at 14:52

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 $$T=15$$:

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 $$$$p_1x_1+p_2x_2=I$$$$ it would be $$$$p_1C=wL$$$$ where C is consumption, w is the wage and L is the amount of labour supplied.

• "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. 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. Mar 21 at 21:35
• 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. Mar 21 at 23:54
• 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. Mar 22 at 4:36
• 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. Mar 22 at 14:41