Let's take our basic competitive equilibrium from chapters 4 and 5, and work through it with the following specifications:

- The consumer has preferences described by the utility function \(U(C,l)=\ln(C)+\ln(l)\)
- The production function is Cobb-Douglass: \(zF(K,N) = z K^\alpha N^{1-\alpha} \) with \(\alpha = \frac{1}{2}\).
- Exogenous parameters are as follows: TODO

Then the model looks like this:

Given exogenous \(\left\{ h,z,K,g \right\}\), a competitive equilibrium is a set of allocations \(\left\{ C,l,N_s,N_d,T \right\}\) and a real wage \(\left\{ w \right\}\) which satisfy the following:

**Consumer Optimization**: Taking the real wage as given, the representative consumer chooses an allocation \(\left\{ C,l,N_s \right\}\) such that \(N_s = h-l\) and such that their allocation solves: \[\max_{C,l}\; \ln(C) + \ln(l) \] subject to the constraints: \begin{gather} C\geq 0, \quad \quad h \geq l \geq 0 \tag{NonNeg}\\ C \leq w\cdot(h-l) + \pi - T \tag{Budget}\\ \end{gather}**Firm Optimization**: Taking the real wage as given, the representative firm chooses their labor demand, \(N_d\) to maximize profits: \[\max_{N_d}\; \left[z K^\alpha N_d^{1-\alpha} - w N_d \right]\] subject to the constraint that \(N_d \geq 0\).**Markets Clear:**\begin{gather} N_d = N_s \tag{labor}\\ C + g = z K^\alpha N_d^{1-\alpha} \tag{goods}\\ \end{gather}- The government's budget is balanced: \(T=g\)

First note that \(U\) is strictly increasing, so at the optimum, the budget constraint will hold with equality.

Also note that the constriant \(C \geq 0\) and \(l\geq 0\) will be nonbinding because \[ \lim_{C \to 0} \frac{\partial U}{\partial C} = +\infty \quad \text{and} \quad \lim_{l \to 0} \frac{\partial U}{\partial l} = +\infty \]

Assuming that \(l < h \), we can write the Lagrangian for the consumer's problem like so:

\[\mathcal{L} = \ln(C) + \ln(l) - \lambda\cdot \left[ C - w\cdot(h-l) + \pi - T \right]\]Taking the partial derivatives of this, and setting them equal to zero, we get the First-order conditions:

\begin{align} 0 =& \frac{\partial \mathcal{L}}{\partial C} = {1 \over C} - \lambda \\ 0 =& \frac{\partial \mathcal{L}}{\partial l} = {1 \over l} - \lambda w \\ 0 =& \frac{\partial \mathcal{L}}{\partial \lambda} = - C + w\cdot(h-l) + \pi - T \\ \end{align}The first two FOCs can be combined to get the equation \( {1 \over l} = {1 \over C} w \), and then rearranging things, we get the following system of equations which characterizes the solution to the consumer's problem:

\[\boxed{\begin{gather} {C \over l} = w \\ C = w\cdot(h-l) + \pi - T \\ \end{gather}}\]First note that the constraint \(N_d \geq 0\) will be nonbinding because \[ \lim_{N \to 0} \frac{\partial F}{\partial N} = +\infty \]

Since there is only one choice variable, take the derivative of the profit function and set it equal to zero:

\begin{split} 0 &= {\partial \over \partial N_d}\left[z K^\alpha N_d^{1-\alpha} - w N_d \right] \\ &= (1-\alpha) \cdot z \cdot K^\alpha N_d^{-\alpha} - w \\ &= (1-\alpha) \cdot z \cdot \left( {K \over N_d} \right)^\alpha - w \\ \end{split}Solving for \(N_d\), we get:

\begin{align} w &= (1-\alpha) z K^\alpha N_d^{-\alpha} \\ N_d^\alpha w &= (1-\alpha) z K^\alpha \\ N_d^\alpha &= \left( \frac{(1-\alpha) z K^\alpha }{w} \right) \\ N_d &= \left( \frac{(1-\alpha) z }{w} \right)^{1 \over \alpha} K \\ \end{align}If we combine the characterizing equations from each agent's problem with the other equilibrium conditions, we get the following system of equations:

\begin{align} {C \over l} &= w \tag{MRS=w} \\ C &= w\cdot(h-l) + \pi - T \tag{Budget} \\ N_d &= \left( \frac{(1-\alpha) z }{w} \right)^{1 \over \alpha} K \tag{Firm's Choice}\\ N_d &= N_s = h-l \tag{Labor Market}\\ C + g &= z K^\alpha N_d^{1-\alpha} \tag{Goods Market}\\ g &= T \tag{Gov Budget}\\ \pi = z K^\alpha N_d^{1-\alpha} - w N_d \tag{Profit} \\ \end{align}Any solution to this system of equations will give us a competitive equilibrium for this economy.

First let's solve this system of equations without plugging in our exogenous parameters. We can combine the first and second equations above to get:

\[C = lw \] \[lw = w\cdot(h-l) + \pi - T \] \[l = \frac{wh+\pi-T}{2w} = {h \over 2} + {\pi - T \over 2w} \] \[C = \frac{wh+\pi-T}{2} \]And of course, we can plug \(g=T\)into this to get

\[l = \frac{wh+\pi-T}{2w} = {h \over 2} + {\pi - g \over 2w} \] \[C = \frac{wh+\pi-g}{2} \]From the market-clearing condition for the labor market, we get

\begin{split} N_d = N_s &= h-l \\ &= h - {h \over 2} - {\pi - g \over 2w} \\ &= {h \over 2} - {\pi - g \over 2w} \end{split}Plug the formula for profit into the budget constraint to get:

\[C &= w\cdot(h-l) + z K^\alpha N_d^{1-\alpha} - w N_d - g\]And \(h-l=N_s=N_d\), so

\[C &= z K^\alpha N_d^{1-\alpha} - g\] \[C=lw=(h-N)w\] \[w= (1-\alpha) z K^\alpha N_d^{-\alpha} \] Need to solve for C,w,N \[C &= N^{1-\alpha} -0\] \[C=lw=(h-N)w\] \[w= (1-\alpha) N^{-\alpha} \] --- \[C &= N_d^{1-\alpha}\] \[C=lw=(h-N)w\] \[w= (1-\alpha) N^{-\alpha} \] \[N^{1-\alpha} = (h-N)w\] \[w= {N^{1-\alpha} \over (h-N)\]This time, let's start by plugging in the values given for the exogenous parameters. Then the characterizing system of equations becomes:

\begin{align} {C \over l} &= w \\ C &= w\cdot(h-l) + \pi - T \\ N_d &= \left( \frac{(1-\alpha) z }{w} \right)^{1 \over \alpha} K \\ N_d &= N_s = h-l \\ C + g &= z K^\alpha N_d^{1-\alpha} \\ g &= T \\ \pi = z K^\alpha N_d^{1-\alpha} - w N_d \\ \end{align} h = 1 K = 1 g = 1 z = 1 alpha = 1/2 \begin{align} C &= lw = w(1-N) \\ C &= wN + \pi - 1 \\ N &= \left( {1 \over 2w} \right)^2 = {1 \over 4w^2} \\ C + 1 &= z K^\alpha N_d^{1-\alpha} \\ \pi &= N_d^{1-\alpha} - w N_d \\ \end{align} 2wN +pi-1=w - pi +1 N = w-pi+1 / 2wknjnjnjinij

\[C = lw \] \[lw = w\cdot(h-l) + \pi - T \] \[l = \frac{wh+\pi-T}{2w} = {h \over 2} + {\pi - T \over 2w} \] \[C = \frac{wh+\pi-T}{2} \]And of course, we can plug \(g=T\)into this to get

\[l = \frac{w+\pi-1}{2w} = {1 \over 2} + {\pi - 1 \over 2w} \] \[C = \frac{w+\pi-1}{2} \]From the market-clearing condition for the labor market, we get

\begin{split} N_d = N_s &= h-l \\ &= h - {h \over 2} - {\pi - g \over 2w} \\ &= {1 \over 2} - {\pi - 1 \over 2w} \end{split} Combining these: \[N = {1 \over 2} - {\pi - 1 \over 2w}\] \[\pi = N^{1\over 2} - w N\] \[C = wN + \pi - 1 = N^{1\over 2} - 1\] -- \[wN=N^{1\over 2}-\pi\] \[N^{1\over 2}-\pi + \pi - 1 = N^{1\over 2} - 1\] \[N^{1\over 2} = N^{1\over 2} \]