<p>Here is another way we could solve this problem with some simple calculus. We first start by maximizing the utility function subject to the budget constraint and the time constraint. 
$$\underset{L,C}{max}\;\underset{s.t.\;C=100+w(40-L)}{L^{1/2}C^{1/2}}\qquad (1)$$
$$\underset{L}{max}\;L^{1/2}[100+w(40-L)]^{1/2}\qquad (2)$$
$\textbf{FOC:}$
$$\bigg(\frac{100+w(40-L)}{L}\bigg) ^{1/2}=w\bigg(\frac{L}{100+w(40-L)}\bigg) ^{1/2}\qquad (3) $$</p>

<p>With some math
$$L=\frac{50+20w}{w}\qquad (4)$$</p>

<p>Let $N$ be the number of hours worked. If you plug 5 and 8 in for $w$, you get 10 and 13.75 for $N$ respectively. </p>

<p>Now we have two points $(5,10),(8,13.75)$. Solving for slope we get $\frac{13.75-10}{8-5}=1.25$. </p>

<p>Now we must consider the intercept. Because $w=0$ is not defined, we must use the other intercept (where labor hours equal zero).
$$40-L=0\implies 40-\bigg(\frac{50+20w}{w}\bigg)=0\qquad (4)$$</p>

<p>With some math, we get $w=2.5$
$$\implies\qquad N=1.25(w-2.5)\qquad (5)$$ </p>

<p>If we did not want to impose the restriction of linearity on the relationship between labor hours and real wage, we would simply use equation (4): </p>

<p>$$N=40-\bigg(\frac{50+20w}{w}\bigg)\qquad (6)$$</p>

<p>I know your question specified a linear relationship between labor hours and real wages, however this method of solving this question does not impose this restriction, which is nice because, as we can see, the relationship between labor hours and real wage is not actually linear.</p>

CC BY-SA 3.0
2016-03-11T01:19:36.140Z
2016-03-11T05:48:17.163Z
2016-03-11T05:48:17.163Z
DornerA
<p>Okay, so the way you do it by first taking w to be 5 and solving for L using the constraints: C = W*N (where C= Consumption, W= wage rate, N = number of hours worked), T = N + L (where l = leisure hours).</p>

<p>For w = 5, N comes out to be 10.
Repeating the same procedure above for w = 8, we get N = 13.75.</p>

<p>Now, we need to come up with a relationship between w = wage rate and N for our basic labor supply function. All in all, it's going to be a straight line for which we need two variables that is: slope and intercept.</p>

<p>Basically,
N = x + y*w -> a general supply equation relating Supply to wage rate in case of labor.</p>

<p>Putting the values we previously found, we get:
10 = x + y*5 and 13.75 = x +y*8 -> a system of simultaneous linear equations.</p>

<p>Solving these two equations we get: x = 13.75 and y = 1.25.</p>

<p>So, our labor supply function becomes: N (Supply of Labor) = 13.75 + 1.25*w</p>

<p>P.S Pardon me for the typos. I was really in a rush.</p>

CC BY-SA 3.0
2016-03-11T02:52:15.970Z
You neglected to add the investment income in your budget constraint which is a source of income. You should have C=100+wN
DornerA
CC BY-SA 3.0
2016-03-11T02:58:24.447Z
Also, intuitively your answer doesn't make sense. You got 13.75 for when wage is 8, so 13.75 should not be the intercept (aka when wage is 0)
DornerA
CC BY-SA 3.0
2016-03-10T23:29:18.290Z
2016-03-10T23:29:18.290Z
user7595
<p>The supply side of the labour market is given by the following set of
equations:
Utility of worker is given by $$U = L^{\frac{1}{2}}C^{\frac{1}{2}}.$$
Real wage $w = 5$,
T-Max = 40 hours,
Investment Income (Fixed) = 100</p>

<p>Now assume that the real wage increase from 5 to 8; assuming labour
supply to liner line, find the equation of the labour supply function using
two discrete points?</p>

CC BY-SA 3.0
2016-03-10T23:19:06.313Z
Quick clarification, is L labor hours or leisure hours? Either way, you can solve for the worker's desired labor hours in terms of real wage by maximizing the utility function subject to the budget constraint and time constraint, and you can then plug in the different values of real wage to get two points and then connect them. Do you think you can work it out, or would you like me to see if I can?
DornerA
CC BY-SA 3.0
2016-03-10T23:23:55.623Z
It's leisure hours. Well, never mind. I figured it out.
user7595
CC BY-SA 3.0
2016-03-10T23:25:43.563Z
I'm glad to hear it! Welcome to the economics stack exchange!
DornerA
<supply-and-demand><labor-economics>
Finding labor supply function from equation of labor market
user7595
# Finding labor supply function from equation of labor market

(Asked by: username_0 on 2016-03-10)

The supply side of the labour market is given by the following set of
equations:
Utility of worker is given by $$U = L^{\frac{1}{2}}C^{\frac{1}{2}}.$$
Real wage $w = 5$,
T-Max = 40 hours,
Investment Income (Fixed) = 100

Now assume that the real wage increase from 5 to 8; assuming labour
supply to liner line, find the equation of the labour supply function using
two discrete points?

**Tags:** supply-and-demand, labor-economics

> **Comments:**
> **username_1**: Quick clarification, is L labor hours or leisure hours? Either way, you can solve for the worker's desired labor hours in terms of real wage by maximizing the utility function subject to the budget constraint and time constraint, and you can then plug in the different values of real wage to get two points and then connect them. Do you think you can work it out, or would you like me to see if I can?  
> **username_0**: It's leisure hours. Well, never mind. I figured it out.  
> **username_1**: I'm glad to hear it! Welcome to the economics stack exchange!  

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## 2 answers

### Answer by username_1 on 2016-03-11. Score: 2

Here is another way we could solve this problem with some simple calculus. We first start by maximizing the utility function subject to the budget constraint and the time constraint.
$$\underset{L,C}{max}\;\underset{s.t.\;C=100+w(40-L)}{L^{1/2}C^{1/2}}\qquad (1)$$
$$\underset{L}{max}\;L^{1/2}[100+w(40-L)]^{1/2}\qquad (2)$$
$\textbf{FOC:}$
$$\bigg(\frac{100+w(40-L)}{L}\bigg) ^{1/2}=w\bigg(\frac{L}{100+w(40-L)}\bigg) ^{1/2}\qquad (3) $$

With some math
$$L=\frac{50+20w}{w}\qquad (4)$$

Let $N$ be the number of hours worked. If you plug 5 and 8 in for $w$, you get 10 and 13.75 for $N$ respectively.

Now we have two points $(5,10),(8,13.75)$. Solving for slope we get $\frac{13.75-10}{8-5}=1.25$.

Now we must consider the intercept. Because $w=0$ is not defined, we must use the other intercept (where labor hours equal zero).
$$40-L=0\implies 40-\bigg(\frac{50+20w}{w}\bigg)=0\qquad (4)$$

With some math, we get $w=2.5$
$$\implies\qquad N=1.25(w-2.5)\qquad (5)$$

If we did not want to impose the restriction of linearity on the relationship between labor hours and real wage, we would simply use equation (4):

$$N=40-\bigg(\frac{50+20w}{w}\bigg)\qquad (6)$$

I know your question specified a linear relationship between labor hours and real wages, however this method of solving this question does not impose this restriction, which is nice because, as we can see, the relationship between labor hours and real wage is not actually linear.


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### Answer by username_0 on 2016-03-10. Score: 0

Okay, so the way you do it by first taking w to be 5 and solving for L using the constraints: C = W\*N (where C= Consumption, W= wage rate, N = number of hours worked), T = N + L (where l = leisure hours).

For w = 5, N comes out to be 10.
Repeating the same procedure above for w = 8, we get N = 13.75.

Now, we need to come up with a relationship between w = wage rate and N for our basic labor supply function. All in all, it's going to be a straight line for which we need two variables that is: slope and intercept.

Basically,
N = x + y\*w -> a general supply equation relating Supply to wage rate in case of labor.

Putting the values we previously found, we get:
10 = x + y\*5 and 13.75 = x +y\*8 -> a system of simultaneous linear equations.

Solving these two equations we get: x = 13.75 and y = 1.25.

So, our labor supply function becomes: N (Supply of Labor) = 13.75 + 1.25\*w

P.S Pardon me for the typos. I was really in a rush.

> **Comments:**
> **username_1**: You neglected to add the investment income in your budget constraint which is a source of income. You should have C=100+wN  
> **username_1**: Also, intuitively your answer doesn't make sense. You got 13.75 for when wage is 8, so 13.75 should not be the intercept (aka when wage is 0)  

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