# Labor Supply- how to do comparative statics?

Consider an economy with a competitive industry where the representative firm's production function takes the form of a Cobb Douglas production function $$Y=z K^{\theta} L^{1-\theta}$$. $$z$$ is an index of TFP (fixed). Derive an expression of a firm's labor demand function as a function of $$w, K, z$$. (w = REAL WAGE)

Sol (My trial) Exogenow var $$=[z, \theta, K, w]$$ assuming short run and $$K$$ fixed. Firm's labor demand: $$Y=z K^{\theta}L^{1-\theta}$$ $$M P_{L}= w$$ (Equlibrium condition) $$w=$$ real wege $$\therefore(1-\theta) \frac{Y}{L}=w$$ $$\Rightarrow(1-\theta) Y=w L \tag{1}$$

$$L=\frac{(1-\theta) Y}{w}$$ Solving for $$L=N^{d}$$ we get

$$N^{d}=\left[\frac{(1-\theta) z K^{\theta}}{w}\right]^{\frac{1}{\theta}}$$

Part (b) Suppose Household preferences is given by $$u(c, l)=\ln c+\beta l, \quad \beta>0$$ Find an expression for consumption, Employment, output leisure, and real wage Assume $$G=\lambda Y, \quad 0<\lambda<1$$ What happens when $$\beta$$ increases?

Here I am getting 2 different solutions.

Method 1: Tangency condition

$$\frac{u_{L}^{\prime}}{u_{c}^{\prime}}=w$$

$$\frac{\beta}{1 / c}=w \Rightarrow \beta C=w \tag{2}$$

or $$\quad c=\frac{w}{\beta}$$

$$C=w[h-N^{s}]$$ [ Income contraint] [h = total hours available]

$$C=w [h-N^{s}]=\frac{w}{\beta}$$

$$\Rightarrow 1 / \beta=h-N^{s}$$

$$\Rightarrow \quad N^{s}=h-1 / \beta \tag{3}$$. $$\Rightarrow l=1/\beta \tag{4}$$

(Labor supply curve is verticat in $$w-N$$ space)

\begin{array}{l} \text { Macro-Economy } \\ \qquad \begin{aligned} Y &=C+G \\ Y &=\frac{w}{\beta}+\lambda Y \\ \Rightarrow \quad Y(1-\lambda) &=\frac{w}{\beta} \end{aligned} \tag{5} \\ \Rightarrow \quad Y=\frac{w}{\beta(1-\lambda)} \end{array} We call this equation (5)

Also notice $$C=Y(1-\lambda)$$ we call this equation (5A) Substituting Equation 3 into Equation 1, we get: We call this Equation (6) \begin{aligned} \therefore w &=(1-\theta) z K^{\theta} L^{-\theta} \\ &=(1-\theta) z K^{\theta} \cdot\left[h-\frac{1}{\beta}\right]^{-\theta} \\ &=(1-\theta) z K^{\theta}\left[\frac{h \beta-1}{\beta}\right]^{-\theta} \end{aligned}\

Substituting Equation (6) in (2) we get, $$C=\frac{(1-\theta) z K^{\theta}}{(h \beta-1)^{\theta}} \beta^{\theta-1}\tag{7}$$

Substituting Equation (7) in (5)

$$Y=\frac{c}{1-\lambda}=\frac{w}{\beta(1-\lambda)}=\frac{(1-\theta) z K^{\theta} \beta^{\theta-1}}{(h \beta-1)^{\theta}(1-\lambda)} \tag{8}$$

Employment $$N^{s}$$ and leisure are given by Equation (3) and (4).

Now, by making some changes we can get simpler expressions which lead to easier Comparative statics for C,Y, w, N, l with respect to $$\beta$$.

I call this Method 2:

Substitute equation 5A in equation 2:

$$\beta Y(1-\lambda)=w \tag{9}$$ Equating Equation (9) and (1) we get:(Equation 10)

$$N^{s}= \frac{(1-\theta)}{\beta(1-\lambda)} \tag{10}$$

Substituting Equation (10) in (1):

$$w=(1-\theta)zK^{\theta}\left(\frac{\beta(1-\lambda)}{1-\theta}\right)^{\theta}$$

It is clear to see we will get extremely different expressions for C, Y and leisure, l from here.

Confession: This was my teacher's method. And from this method, he did some comparative static analysis which went all over my head!

How to reconcile both methods? How can we do comparative statics? from which method would be easier?

Can we do graphical analysis from any method? That is draw the labor demand and labor supply curves?

• by the way you can number equations by using \tag{#} (where # is the number/letter/symbol you want to assign to the eq) – 1muflon1 Jan 26 at 13:17
• @1muflon1 I am not getting a reply. It has been a lot many days. Don't have enough points to put a bounty. Do you have any ideas on how to proceed now? – Karan Kumar Feb 2 at 7:23
• unanswered questions get automatically put at the beginning active question every once in a while - so it won’t just stay buried. However, if you want to bring more attention to the question then just try answer some unanswered questions you can - minimum bounty is 50 points. You just need 31 points - any upvote on answer gives you 10 points and if your answer is accepted by user you get another 15 so if you can find two answers you can answer well enough that they get accepted you will have enough points. If you will make even 1 good answer you will get enough points from other users – 1muflon1 Feb 2 at 10:59