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Chapter 19 / Labor Supply

# 19.4 Policy Implications

This is a fairly simple model, but we can still use it to answer some important policy questions using comparative statics analysis: that is, analyzing how the solution to the model varies when the underlying parameters change.

### How elastic is labor supply?

This is an important question, especially for analyzing the market power of firms.

Recall that when some variable $Y$ depends on some other variable $X$, the elasticity of $Y$ with respect to $X$ is $$\epsilon_{Y,X} = \frac{\Delta Y}{\Delta X} \times {X \over Y}$$ For the Cobb-Douglas utility function $u(R,C) = R^\alpha C^{1-\alpha}$, we showed that the labor supply was $$L(w) = (1-\alpha)24 - {\alpha M \over w}$$ Therefore the wage elasticity of labor is \begin{aligned} \epsilon_{L,w} &= L^\prime(w) \times {w \over L(w)}\\ &= {\alpha M \over w^2} \times {w \over {(1-\alpha)24 - {\alpha M \over w}}} &= {\alpha M \over {(1-\alpha)24w - \alpha M}} \end{aligned} Note that this is increasing in $M$ and $\alpha$: the more nonwage income you have, and the more you value leisure, the more elastic your labor supply will be. It’s also decreasing in $w$: the higher the wage you face, the less elastic your labor supply will be (at least for this utility function!).

### How does an increase in nonwage income $M$ affect labor supply?

This has been the source of some political controversy recently, during COVID-19, as some argue that unemployment benefits discourage people from returning to work and thereby hurt the economy.

We can see that, in this particular model, an increase in $M$ unambiguously decreases labor supply. One critical way we can analyze this is to ask the question of what the minimum wage at which someone will choose to supply any labor at all. The condition for the labor supply being greater than zero is $$(1-\alpha)24 - {\alpha M \over w} > 0 \Rightarrow {\alpha \over 1 - \alpha} \times {M \over 24} < w$$ Note that the left-hand side of this expression is the MRS at the endowment point of $(24, M)$, and the right-hand side ($w$) is the slope of the constraint: so this is our familiar condition that an agent wants to sell good 1 (in this case, their time) if their MRS is less than the slope of the constraint!

The MRS at the endowment, therefore, reflect the monetary value someone places on their last hour of leisure time. Because the MRS at the endowment is increasing in $M$, we can see that the more nonwage income someone has, the higher the wage needs to be in order for them to supply labor. Intuitively, MRS increasing in good 2 reflects the fact that the more good 2, the less you’re willing to give up good 1 to get additional units of good 2: in this case, the more money you have, the less you’re willing to give up your leisure time to get more of it.

We can also see that, in this case, the MRS is increasing in $\alpha$, which we can interpret as how much you “like” leisure (or “dislike” working). In the current climate, this could reflect how risky you feel going to work would be in the middle of a pandemic. If your job involves coding for Twitter from a sweet home office, you might be perfectly fine continuing to work; but if you’re an Uber driver who people might call to take them to the hospital, you might rightfully fear every hour of work, which would result in an increase in $\alpha$, and mean that you’ll require a much higher wage to spend time working.

This model, therefore would argue that higher non-wage income – for example, in the form of generous unemployment benefits – reduces the labor supply and discourages people from finding a job. In normal times, that might be an undesirable outcome; in a pandemic, it might actually be a useful policy tool to help reduce the spread of a virus by encouraging people to stay home. However, it can also have the consequence of increasing inequality between people who can work safely from home and those whose jobs rely on in-person contact. In short, a relatively simple application of optimization from an endowment can have profound policy implications.

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Copyright (c) Christopher Makler / econgraphs.org