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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?
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.
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.
