<p>First, we solve the cost minimisation problem of the two firms to determine the cost functions of the two firms. For the firm producing <span class="math-container">$x$</span>, we solve</p>
<p><span class="math-container">$\min_{(l,k)\in\mathbb{R}^2_+}wl+rk $</span> subject to <span class="math-container">$x=k^\alpha l^{1-\alpha}$</span>, where <span class="math-container">$w&gt;0$</span> and <span class="math-container">$r&gt;0$</span> are given,
and we get the conditional input demands as follows
<span class="math-container">$(l_x^c,k_x^c)(w,r,x)=\left(\left(\frac{(1-\alpha)r}{\alpha w}\right)^\alpha x, \left(\frac{\alpha w}{(1-\alpha)r}\right)^{1-\alpha} x\right)$</span> and the corresponding cost function is <span class="math-container">$c_x(w,r,x)=wl_x^c+rk_x^c=\left(\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha+ \left(\frac{\alpha}{(1-\alpha)}\right)^{1-\alpha}\right)w^{1-\alpha}r^{\alpha}x=Aw^{1-\alpha}r^{\alpha}x$</span>, where <span class="math-container">$A:=\left(\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha+ \left(\frac{\alpha}{(1-\alpha)}\right)^{1-\alpha}\right)$</span>.</p>
<p>Similarly, for the firm producing <span class="math-container">$y$</span>, we solve</p>
<p><span class="math-container">$\min_{(l,k)\in\mathbb{R}^2_+}wl+rk $</span> subject to <span class="math-container">$y=k^\beta l^{1-\beta}$</span>, where <span class="math-container">$w&gt;0$</span> and <span class="math-container">$r&gt;0$</span> are given,
and we get the conditional input demands as follows
<span class="math-container">$(l_y^c,k_y^c)(w,r,y)=\left(\left(\frac{(1-\beta)r}{\beta w}\right)^\beta y, \left(\frac{\beta w}{(1-\beta)r}\right)^{1-\beta} y\right)$</span> and the corresponding cost function is <span class="math-container">$c_y(w,r,y)=wl_y^c+rk_y^c=\left(\left(\frac{(1-\beta)}{\beta}\right)^\beta+ \left(\frac{\beta}{(1-\beta)}\right)^{1-\beta}\right)w^{1-\beta}r^{\beta}y=Bw^{1-\beta}r^{\beta}y$</span>, where <span class="math-container">$B:=\left(\left(\frac{(1-\beta)}{\beta}\right)^\beta+ \left(\frac{\beta}{(1-\beta)}\right)^{1-\beta}\right)$</span>.</p>
<p>Since the two technologies producing <span class="math-container">$x$</span> and <span class="math-container">$y$</span> exhibit CRS, we obtained the cost functions that are linear in output. The profit maximising firms will produce strictly positive and finite amounts of output only in the case when price equals the marginal cost. Also, in such a situation they can supply any amount that is demanded and the optimal profits will be zero.
Given that the consumer's utility function is <span class="math-container">$u(x,y)=xy$</span>, and total amount of capital and labor are strictly positive i.e. <span class="math-container">$K&gt;0$</span>, <span class="math-container">$L&gt;0$</span>, consumer in optimum will demand strictly positive amount of <span class="math-container">$x$</span> and <span class="math-container">$y$</span> and therefore, in equilibrium it must be the case that price of <span class="math-container">$x$</span> must be equal to its marginal cost i.e. <span class="math-container">$p_x=Aw^{1-\alpha}r^{\alpha}$</span> and price of <span class="math-container">$y$</span> must be equal to its marginal cost i.e. <span class="math-container">$p_y=Bw^{1-\beta}r^{\beta}$</span>.</p>
<p>To determine the demand for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, we simply solve the utility maximisation problem of the consumer:</p>
<p><span class="math-container">$\max_{(x,y)\in\mathbb{R}^2_+} xy$</span> subject to <span class="math-container">$p_xx+p_yy=wL+rK$</span></p>
<p>and we get <span class="math-container">$(x^d,y^d)(p_x,p_y,w,r)=\left(\frac{wL+rK}{2p_x},\frac{wL+rK}{2p_y}\right)$</span></p>
<p>Now we can determine the equilibrium prices <span class="math-container">$(p_x,p_y,w,r)$</span> using the following conditions:</p>
<ul>
<li>Price equals marginal cost in the production of <span class="math-container">$x$</span>, <span class="math-container">$p_x=Aw^{1-\alpha}r^{\alpha}$</span></li>
<li>Price equals marginal cost in the production of <span class="math-container">$y$</span>, <span class="math-container">$p_y=Bw^{1-\beta}r^{\beta}$</span></li>
<li>Equilibrium in the labor market, <span class="math-container">$\left(\frac{(1-\alpha)r}{\alpha w}\right)^\alpha\left(\frac{wL+rK}{2p_x}\right)+\left(\frac{(1-\beta)r}{\beta w}\right)^\beta \left(\frac{wL+rK}{2p_y}\right)=L$</span> (Using the demand for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, and conditional input demands of labor)</li>
<li>Equilibrium in the capital market, <span class="math-container">$\left(\frac{\alpha w}{(1-\alpha)r}\right)^{1-\alpha}\left(\frac{wL+rK}{2p_x}\right)+\left(\frac{\beta w}{(1-\beta)r}\right)^{1-\beta} \left(\frac{wL+rK}{2p_y}\right)=K$</span> (Using the demand for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, and conditional input demands of capital)</li>
</ul>
<p>To determine the equilibrium, we can take one of the prices as numeraire, say <span class="math-container">$r=1$</span>, and re-write the problem as finding <span class="math-container">$p_x,p_y$</span> and <span class="math-container">$w$</span> that satisfy:</p>
<ul>
<li><span class="math-container">$p_x=Aw^{1-\alpha}$</span></li>
<li><span class="math-container">$p_y=Bw^{1-\beta}$</span></li>
<li><span class="math-container">$\left(\frac{(1-\alpha)}{\alpha w}\right)^\alpha\left(\frac{wL+K}{2p_x}\right)+\left(\frac{(1-\beta)}{\beta w}\right)^\beta \left(\frac{wL+K}{2p_y}\right)=L$</span></li>
<li><span class="math-container">$\left(\frac{\alpha w}{(1-\alpha)}\right)^{1-\alpha}\left(\frac{wL+K}{2p_x}\right)+\left(\frac{\beta w}{(1-\beta)}\right)^{1-\beta} \left(\frac{wL+K}{2p_y}\right)=K$</span></li>
</ul>
<p>Substituting <span class="math-container">$p_x=Aw^{1-\alpha}$</span>, and <span class="math-container">$p_y=Bw^{1-\beta}$</span> in  <span class="math-container">$\left(\frac{(1-\alpha)}{\alpha w}\right)^\alpha\left(\frac{wL+K}{2p_x}\right)+\left(\frac{(1-\beta)}{\beta w}\right)^\beta \left(\frac{wL+K}{2p_y}\right)=L$</span> we get
<span class="math-container">$\left(\frac{(1-\alpha)}{\alpha w}\right)^\alpha\left(\frac{wL+K}{2Aw^{1-\alpha}}\right)+\left(\frac{(1-\beta)}{\beta w}\right)^\beta \left(\frac{wL+K}{2Bw^{1-\beta}}\right)=L$</span> as an equation in <span class="math-container">$w$</span> which can be solved for <span class="math-container">$w$</span>.
This equation can be further simplified to:
<span class="math-container">$\left(\frac{\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha}{A}+\frac{\left(\frac{(1-\beta)}{\beta}\right)^\beta}{B}\right) \left(wL+K\right)=2wL$</span>. Solving it for <span class="math-container">$w$</span>, we get
<span class="math-container">$w=\dfrac{\theta K}{(2-\theta)L}$</span>, where <span class="math-container">$\theta := \left(\frac{\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha}{A}+\frac{\left(\frac{(1-\beta)}{\beta}\right)^\beta}{B}\right)=(2-(\alpha+\beta)) \in (0,2)$</span>. Therefore, <span class="math-container">$p_x=A\left(\dfrac{\theta K}{(2-\theta)L}\right)^{1-\alpha}$</span> and <span class="math-container">$p_y=B\left(\dfrac{\theta K}{(2-\theta)L}\right)^{1-\beta}$</span>.
These equilibrium prices can now be used to determine the equilibrium allocation. Plugging them in the demand for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> expressions above, we obtain the equilibrium level of output of <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, which can be substituted along with the prices in the conditional input demands obtained above to determine the equilibrium level of employment of labor and capital in the production of <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, respectively.</p>

CC BY-SA 4.0
2024-08-11T11:17:45.257Z
2024-08-12T05:59:29.707Z
2024-08-12T05:59:29.707Z
Amit
<ol>
<li><p>Your equation system may look scary, but it is not, it is linear in <span class="math-container">$x$</span>, so things could still be fine.</p>
</li>
<li><p>There is a lot of information that you are not yet using. The consumer maximizes utility (probably using the profits), the other firm maximizes profit, the amount of available resources are fixed,in equilibrium there is also equilibrium on the resource market.</p>
</li>
<li><p>By combining your two first order conditions into one you are losing some information, note that price has completely left your equations. So be sure to remember that eventually you will probably have to use a first order condition as well.</p>
</li>
</ol>
<hr />
<ol start="4">
<li>Use some formulas? Just memorizing formulas is bad form, and you may also have to 'show your work' but if you have done several Lagrangians for Cobb-Douglas functions you might notice that the formula
<span class="math-container">$$k_1=\frac{\alpha l_1 w}{(1-\alpha)r}$$</span>
also has this (frequently used) form:
<span class="math-container">$$|\text{MRTS}(k_1,l_1)| = \frac{\alpha}{(1-\alpha)}\frac{l_1}{k_1}= \frac{r}{w}$$</span>
which is useful because it is easy to remember, and using these formulas you don't need to do the basic calculations for Cobb-Douglas production functions. (Similarly see MRS for C-D utility functions.)</li>
</ol>

CC BY-SA 4.0
2023-11-15T14:05:19.830Z
hi - what i'm struggling with is that since the $k_1, l_1$ I got still have $x$ in it, I cannot calculate the profit without having $x$ in the expression. Thus I cannot do the utility maximization problem. Or can I?
Ludwig Gershwin
CC BY-SA 4.0
2023-11-16T10:08:09.430Z
@LudwigGershwin Having $x$ in the expression might complicate things but it is not in itself a problem, you can treat as a parameter in the utility maximization. Or you can try to do the other things (firm 2 profitmax and resource equilibrum) which will hopefully resolve $x$ before dealing with utility maximization.
Giskard
CC BY-SA 4.0
2023-11-15T06:53:32.097Z
2023-11-15T07:02:53.553Z
2023-11-15T07:02:53.553Z
Giskard
<p>Consider a production economy with 1 consumer and 2 firms. The consumer's utility is <span class="math-container">$u(x,y)=xy$</span>. The first firm makes <span class="math-container">$x$</span> using capital and labor according to the production function <span class="math-container">$x=k^\alpha l^{1-\alpha}$</span> and the second firm makes <span class="math-container">$y$</span> using <span class="math-container">$k,l$</span> by <span class="math-container">$y=k^\beta l^{1-\beta}$</span>. The total amount of capital and labor available in the economy are fixed at <span class="math-container">$K$</span> and <span class="math-container">$L$</span> respectively and are owned by the consumer. The consumer is also the owner of both firms.</p>
<p>How do I find the equilibrium prices for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>?</p>
<p>Suppose we normalize the price of <span class="math-container">$y$</span> to be 1. The first firm maximizes profits:
<span class="math-container">$$px-rk_1-wl_1\;\;s.t. x=k_1^\alpha l_1^{1-\alpha}\implies \max (pk_1^\alpha l_1^{1-\alpha}-rk_1-wl_1)$$</span>
By first order condition, <span class="math-container">$\alpha p k_1^{\alpha-1}l_1^{1-\alpha}=r$</span> and <span class="math-container">$(1-\alpha)p k_1^\alpha l_1^{-\alpha}=w$</span>. Thus,
<span class="math-container">$$k_1=\frac{\alpha l_1 w}{(1-\alpha)r}$$</span>
substitute this back into <span class="math-container">$x=k_1^\alpha l_1^{1-\alpha}$</span> we have <span class="math-container">$l_1=\frac{x}{\left(\frac{\alpha w}{(1-\alpha r)}\right)^\alpha}$</span>. So <span class="math-container">$k_1=\frac{\alpha l_1 w}{(1-\alpha)r}=(\frac{\alpha}{1-\alpha})^{1-\alpha}(\frac{w}{r})^{1-\alpha}x$</span>. But I cannot get the <span class="math-container">$x$</span> out. How do I proceed?</p>

<walrasian>
How do I find the Walrasian equilibrium price?
Ludwig Gershwin
# How do I find the Walrasian equilibrium price?

(Asked by: username_0 on 2023-11-15)

Consider a production economy with 1 consumer and 2 firms. The consumer's utility is $u(x,y)=xy$. The first firm makes $x$ using capital and labor according to the production function $x=k^\alpha l^{1-\alpha}$ and the second firm makes $y$ using $k,l$ by $y=k^\beta l^{1-\beta}$. The total amount of capital and labor available in the economy are fixed at $K$ and $L$ respectively and are owned by the consumer. The consumer is also the owner of both firms.

How do I find the equilibrium prices for $x$ and $y$?

Suppose we normalize the price of $y$ to be 1. The first firm maximizes profits:
$$px-rk\_1-wl\_1\;\;s.t. x=k\_1^\alpha l\_1^{1-\alpha}\implies \max (pk\_1^\alpha l\_1^{1-\alpha}-rk\_1-wl\_1)$$
By first order condition, $\alpha p k\_1^{\alpha-1}l\_1^{1-\alpha}=r$ and $(1-\alpha)p k\_1^\alpha l\_1^{-\alpha}=w$. Thus,
$$k\_1=\frac{\alpha l\_1 w}{(1-\alpha)r}$$
substitute this back into $x=k\_1^\alpha l\_1^{1-\alpha}$ we have $l\_1=\frac{x}{\left(\frac{\alpha w}{(1-\alpha r)}\right)^\alpha}$. So $k\_1=\frac{\alpha l\_1 w}{(1-\alpha)r}=(\frac{\alpha}{1-\alpha})^{1-\alpha}(\frac{w}{r})^{1-\alpha}x$. But I cannot get the $x$ out. How do I proceed?

**Tags:** walrasian


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## 2 answers

### Answer by username_1 on 2024-08-11. Score: 1

First, we solve the cost minimisation problem of the two firms to determine the cost functions of the two firms. For the firm producing $x$, we solve

$\min\_{(l,k)\in\mathbb{R}^2\_+}wl+rk $ subject to $x=k^\alpha l^{1-\alpha}$, where $w>0$ and $r>0$ are given,
and we get the conditional input demands as follows
$(l\_x^c,k\_x^c)(w,r,x)=\left(\left(\frac{(1-\alpha)r}{\alpha w}\right)^\alpha x, \left(\frac{\alpha w}{(1-\alpha)r}\right)^{1-\alpha} x\right)$ and the corresponding cost function is $c\_x(w,r,x)=wl\_x^c+rk\_x^c=\left(\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha+ \left(\frac{\alpha}{(1-\alpha)}\right)^{1-\alpha}\right)w^{1-\alpha}r^{\alpha}x=Aw^{1-\alpha}r^{\alpha}x$, where $A:=\left(\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha+ \left(\frac{\alpha}{(1-\alpha)}\right)^{1-\alpha}\right)$.

Similarly, for the firm producing $y$, we solve

$\min\_{(l,k)\in\mathbb{R}^2\_+}wl+rk $ subject to $y=k^\beta l^{1-\beta}$, where $w>0$ and $r>0$ are given,
and we get the conditional input demands as follows
$(l\_y^c,k\_y^c)(w,r,y)=\left(\left(\frac{(1-\beta)r}{\beta w}\right)^\beta y, \left(\frac{\beta w}{(1-\beta)r}\right)^{1-\beta} y\right)$ and the corresponding cost function is $c\_y(w,r,y)=wl\_y^c+rk\_y^c=\left(\left(\frac{(1-\beta)}{\beta}\right)^\beta+ \left(\frac{\beta}{(1-\beta)}\right)^{1-\beta}\right)w^{1-\beta}r^{\beta}y=Bw^{1-\beta}r^{\beta}y$, where $B:=\left(\left(\frac{(1-\beta)}{\beta}\right)^\beta+ \left(\frac{\beta}{(1-\beta)}\right)^{1-\beta}\right)$.

Since the two technologies producing $x$ and $y$ exhibit CRS, we obtained the cost functions that are linear in output. The profit maximising firms will produce strictly positive and finite amounts of output only in the case when price equals the marginal cost. Also, in such a situation they can supply any amount that is demanded and the optimal profits will be zero.
Given that the consumer's utility function is $u(x,y)=xy$, and total amount of capital and labor are strictly positive i.e. $K>0$, $L>0$, consumer in optimum will demand strictly positive amount of $x$ and $y$ and therefore, in equilibrium it must be the case that price of $x$ must be equal to its marginal cost i.e. $p\_x=Aw^{1-\alpha}r^{\alpha}$ and price of $y$ must be equal to its marginal cost i.e. $p\_y=Bw^{1-\beta}r^{\beta}$.

To determine the demand for $x$ and $y$, we simply solve the utility maximisation problem of the consumer:

$\max\_{(x,y)\in\mathbb{R}^2\_+} xy$ subject to $p\_xx+p\_yy=wL+rK$

and we get $(x^d,y^d)(p\_x,p\_y,w,r)=\left(\frac{wL+rK}{2p\_x},\frac{wL+rK}{2p\_y}\right)$

Now we can determine the equilibrium prices $(p\_x,p\_y,w,r)$ using the following conditions:

* Price equals marginal cost in the production of $x$, $p\_x=Aw^{1-\alpha}r^{\alpha}$
* Price equals marginal cost in the production of $y$, $p\_y=Bw^{1-\beta}r^{\beta}$
* Equilibrium in the labor market, $\left(\frac{(1-\alpha)r}{\alpha w}\right)^\alpha\left(\frac{wL+rK}{2p\_x}\right)+\left(\frac{(1-\beta)r}{\beta w}\right)^\beta \left(\frac{wL+rK}{2p\_y}\right)=L$ (Using the demand for $x$ and $y$, and conditional input demands of labor)
* Equilibrium in the capital market, $\left(\frac{\alpha w}{(1-\alpha)r}\right)^{1-\alpha}\left(\frac{wL+rK}{2p\_x}\right)+\left(\frac{\beta w}{(1-\beta)r}\right)^{1-\beta} \left(\frac{wL+rK}{2p\_y}\right)=K$ (Using the demand for $x$ and $y$, and conditional input demands of capital)

To determine the equilibrium, we can take one of the prices as numeraire, say $r=1$, and re-write the problem as finding $p\_x,p\_y$ and $w$ that satisfy:

* $p\_x=Aw^{1-\alpha}$
* $p\_y=Bw^{1-\beta}$
* $\left(\frac{(1-\alpha)}{\alpha w}\right)^\alpha\left(\frac{wL+K}{2p\_x}\right)+\left(\frac{(1-\beta)}{\beta w}\right)^\beta \left(\frac{wL+K}{2p\_y}\right)=L$
* $\left(\frac{\alpha w}{(1-\alpha)}\right)^{1-\alpha}\left(\frac{wL+K}{2p\_x}\right)+\left(\frac{\beta w}{(1-\beta)}\right)^{1-\beta} \left(\frac{wL+K}{2p\_y}\right)=K$

Substituting $p\_x=Aw^{1-\alpha}$, and $p\_y=Bw^{1-\beta}$ in $\left(\frac{(1-\alpha)}{\alpha w}\right)^\alpha\left(\frac{wL+K}{2p\_x}\right)+\left(\frac{(1-\beta)}{\beta w}\right)^\beta \left(\frac{wL+K}{2p\_y}\right)=L$ we get
$\left(\frac{(1-\alpha)}{\alpha w}\right)^\alpha\left(\frac{wL+K}{2Aw^{1-\alpha}}\right)+\left(\frac{(1-\beta)}{\beta w}\right)^\beta \left(\frac{wL+K}{2Bw^{1-\beta}}\right)=L$ as an equation in $w$ which can be solved for $w$.
This equation can be further simplified to:
$\left(\frac{\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha}{A}+\frac{\left(\frac{(1-\beta)}{\beta}\right)^\beta}{B}\right) \left(wL+K\right)=2wL$. Solving it for $w$, we get
$w=\dfrac{\theta K}{(2-\theta)L}$, where $\theta := \left(\frac{\left(\frac{(1-\alpha)}{\alpha}\right)^\alpha}{A}+\frac{\left(\frac{(1-\beta)}{\beta}\right)^\beta}{B}\right)=(2-(\alpha+\beta)) \in (0,2)$. Therefore, $p\_x=A\left(\dfrac{\theta K}{(2-\theta)L}\right)^{1-\alpha}$ and $p\_y=B\left(\dfrac{\theta K}{(2-\theta)L}\right)^{1-\beta}$.
These equilibrium prices can now be used to determine the equilibrium allocation. Plugging them in the demand for $x$ and $y$ expressions above, we obtain the equilibrium level of output of $x$ and $y$, which can be substituted along with the prices in the conditional input demands obtained above to determine the equilibrium level of employment of labor and capital in the production of $x$ and $y$, respectively.


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### Answer by username_2 on 2023-11-15. Score: 0

1. Your equation system may look scary, but it is not, it is linear in $x$, so things could still be fine.
2. There is a lot of information that you are not yet using. The consumer maximizes utility (probably using the profits), the other firm maximizes profit, the amount of available resources are fixed,in equilibrium there is also equilibrium on the resource market.
3. By combining your two first order conditions into one you are losing some information, note that price has completely left your equations. So be sure to remember that eventually you will probably have to use a first order condition as well.

---

4. Use some formulas? Just memorizing formulas is bad form, and you may also have to 'show your work' but if you have done several Lagrangians for Cobb-Douglas functions you might notice that the formula
   $$k\_1=\frac{\alpha l\_1 w}{(1-\alpha)r}$$
   also has this (frequently used) form:
   $$|\text{MRTS}(k\_1,l\_1)| = \frac{\alpha}{(1-\alpha)}\frac{l\_1}{k\_1}= \frac{r}{w}$$
   which is useful because it is easy to remember, and using these formulas you don't need to do the basic calculations for Cobb-Douglas production functions. (Similarly see MRS for C-D utility functions.)

> **Comments:**
> **username_0**: hi - what i'm struggling with is that since the $k_1, l_1$ I got still have $x$ in it, I cannot calculate the profit without having $x$ in the expression. Thus I cannot do the utility maximization problem. Or can I?  
> **username_2**: @username_0 Having $x$ in the expression might complicate things but it is not in itself a problem, you can treat as a parameter in the utility maximization. Or you can try to do the other things (firm 2 profitmax and resource equilibrum) which will hopefully resolve $x$ before dealing with utility maximization.  

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