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If the government pays a certain amount $b>0$ to every person who is not working, what is the impact of this subsidy on labour supply? How does it alter the reservation wage? Is your answer modified when payments are restricted to persons who are looking for a job?

My attempt: Suppose $A$ is an individual in the labour force. For an individual who is not in the labour force, there is no impact on labour supply. Therefore, it makes sense to apply the analysis to a person in the labour force.

In the income-leisure ($i$-$l$) diagram (of a week) for an individual $A$, without any government aid, let $w_R$ be the reservation wage. Suppose $G$ represents $A$'s other sources of income. Let $T=168$ be the number of hours in a week. Our analysis is done in the interval $[0,T]$ of the $l$-axis. We have the line $i=G+w(T-l)$ or $$i+wl=(wT+G).$$ At $l=T$ and $w=w_R$, $A$ is most happy (maximum utility) to not work at all. However, $w_R$ is the maximum wage rate at which $A$ doesn't work (by definition).

With the given government intervention, $A$'s original line shifts upward by $b$. $A$ comes to a higher utility point (than in previous case without government intervention) whether he is working or not. It is impossible to tell how the $w_R$ will be altered in this case. It is possible that if $A$ was working before, $A$ is not working now and vice-versa.

Am I correct?

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The individual has a utility function over consumption and leisure

$$U(C, \ell) = U(C, T-L)$$ and to keep this static, it has the no-savings/no-credit budget constraint

$$C = G + wL$$

He wants to maximize utility. Inserting the budget constraint into the utility function the first-order condition is (maximizing with respect to $L$)

$${\rm (f.o.c)} :\;\; U_c(G+wL, T-L)\cdot w - U_{\ell}(G+wL, T-L) = 0 $$

This will hold in all optimal solutions. Set $L=0$ to express the reservation wage, (i.e. ask "what is the wage consistent with zero work?")

$$L=0 \Rightarrow U_c(G, T)\cdot w - U_{\ell}(G, T) =0$$

$$\Rightarrow w_R = \frac {U_{\ell}(G, T)}{U_c(G, T)}$$

Then

$${\rm sign} \left\{\frac {\partial w_R}{\partial G}\right\} = {\rm sign} \left\{U_{\ell c}(G, T)\cdot U_c(G, T) - U_{cc}(G, T)\cdot U_{\ell}(G, T)\right\}$$

We see that

A) If preferences are assumed separable between consumption and leisure, i.e. if $U_{\ell c} = 0$, then under the usual assumption that $U_{cc} <0$ we see that $\frac {\partial w_R}{\partial G} >0$.

B) If separability is not assumed, then it is usually assumed (and reasonably) that $U_{\ell c} >0$, in which case we again obtain $\frac {\partial w_R}{\partial G} >0$.

In both cases, an increase in $G$ (which is what happens when the government pays the amount $b$), increases the reservation wage. If we assume that wages are the same before and after the government subsidy, then if the person did not work, he will continue not to work, while if the person worked previously, he may now stop, given the government subsidy.

Of course, making the problem dynamic by introducing intertemporal utility maximization and a consumption-savings decision, requires an analysis on its own.

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