<p>We solve the utility maximization problem of the individual whose utility function is $u(c, l) = c - \frac{l^{1+\gamma}}{1+\gamma}$ to get the supply function. The problem can be written as:
\begin{eqnarray*} \max_{c,l} &amp; &amp; c - \frac{l^{1+\gamma}}{1+\gamma} \\ \text{s.t.} &amp;&amp; c \leq wl\end{eqnarray*}
In this problem, we are assuming that the only source of income of the consumer is his wage income. When we solve the problem we get the labor supply function as:
\begin{eqnarray*} l(w) = w^{1/\gamma}\end{eqnarray*}
The elasticity of labor supply curve is this case is constant and equal to $\frac{1}{\gamma}$. Supply will be elastic if $0 &lt; \gamma &lt; 1$ and inelastic if $\gamma &gt; 1$.</p>

CC BY-SA 3.0
2017-01-27T13:30:11.477Z
2017-01-27T14:19:29.157Z
2017-01-27T14:19:29.157Z
Amit
<p>The supply function will be perfectly inelastic if it does not depend on the wage. Will this happen with quasilinear? If you substitute out $c=w(1-l)$ you will find the wage is in the labor supply function. You need a first order condition where the wage drops out, like in $u=cl$ . However if you have some exogenous wealth source in addition to your work, think $c=w(1-l)+\pi$, then this will no longer be the case. </p>

<p>The linear good in quasilinear absorbs all the wealth effects, but with labor/leisure, the other good is a source of the wealth so it behaves differently than with a typical consumer problem with exogenous wealth. </p>

<p>So if $c=wl$ then a sufficient condition for perfectly inelastic labor supply would be homogeneity of degree 1 in consumption. </p>

CC BY-SA 3.0
2016-08-30T04:38:17.657Z
Forgot the production function, but i got that $l=(A(1-a))^{1/(a+γ)}$
neto333
CC BY-SA 3.0
2016-08-29T18:46:52.347Z
2016-08-30T00:30:05.170Z
2016-08-30T00:30:05.170Z
VCG
<p>if you have a quasi linear utility function, for example $U(l,c)=c-l^{1+γ}/(1+ γ)$ the supply function of <em>l</em> is inelastic right? But can we extend this argument and say that any quasi linear function is going to give you an inelastic function of the non lineal good?
Forgot to mention the production function. But this is the function $y(l)=A(l)^{1-a}$. A is the productivity 
So if you take the efficiency condition  you get that $l=(A(1-a))^{1/(a+γ)}$
Thank you.</p>

<p>Thank you.</p>

CC BY-SA 3.0
2016-08-29T04:44:36.800Z
What makes you think the supply is inelastic in your case? Can you show a derivation?
VCG
CC BY-SA 3.0
2016-08-30T04:32:44.160Z
i think i forgot to mention the production function. But is one without capital. $y(l)=A(l)^{1-a}$. A is the productivity
neto333
CC BY-SA 3.0
2016-08-30T10:56:15.027Z
So when I think inelastic supply function I think about finding the derivative of the labor supply function in terms of the wage. Why are you subbing that out?
VCG
<microeconomics><mathematical-economics><utility>
quasi linear utility function
neto333
# quasi linear utility function

(Asked by: username_0 on 2016-08-29)

if you have a quasi linear utility function, for example $U(l,c)=c-l^{1+γ}/(1+ γ)$ the supply function of *l* is inelastic right? But can we extend this argument and say that any quasi linear function is going to give you an inelastic function of the non lineal good?
Forgot to mention the production function. But this is the function $y(l)=A(l)^{1-a}$. A is the productivity
So if you take the efficiency condition you get that $l=(A(1-a))^{1/(a+γ)}$
Thank you.

Thank you.

**Tags:** microeconomics, mathematical-economics, utility
> **Comments:**
> **username_1**: What makes you think the supply is inelastic in your case? Can you show a derivation?  
> **username_0**: i think i forgot to mention the production function. But is one without capital. $y(l)=A(l)^{1-a}$. A is the productivity  
> **username_1**: So when I think inelastic supply function I think about finding the derivative of the labor supply function in terms of the wage. Why are you subbing that out?  

---

## 2 answers

### Answer by username_2 on 2017-01-27. Score: 2

We solve the utility maximization problem of the individual whose utility function is $u(c, l) = c - \frac{l^{1+\gamma}}{1+\gamma}$ to get the supply function. The problem can be written as:
\begin{eqnarray\*} \max\_{c,l} & & c - \frac{l^{1+\gamma}}{1+\gamma} \\ \text{s.t.} && c \leq wl\end{eqnarray\*}
In this problem, we are assuming that the only source of income of the consumer is his wage income. When we solve the problem we get the labor supply function as:
\begin{eqnarray\*} l(w) = w^{1/\gamma}\end{eqnarray\*}
The elasticity of labor supply curve is this case is constant and equal to $\frac{1}{\gamma}$. Supply will be elastic if $0 < \gamma < 1$ and inelastic if $\gamma > 1$.


---

### Answer by username_1 on 2016-08-29. Score: 0

The supply function will be perfectly inelastic if it does not depend on the wage. Will this happen with quasilinear? If you substitute out $c=w(1-l)$ you will find the wage is in the labor supply function. You need a first order condition where the wage drops out, like in $u=cl$ . However if you have some exogenous wealth source in addition to your work, think $c=w(1-l)+\pi$, then this will no longer be the case.

The linear good in quasilinear absorbs all the wealth effects, but with labor/leisure, the other good is a source of the wealth so it behaves differently than with a typical consumer problem with exogenous wealth.

So if $c=wl$ then a sufficient condition for perfectly inelastic labor supply would be homogeneity of degree 1 in consumption.

> **Comments:**
> **username_0**: Forgot the production function, but i got that $l=(A(1-a))^{1/(a+γ)}$  

---

