<p>In these problems, you are generally dealing with identical firms - all of whom will supply the market according to their marginal cost curve.  If we were solving for a long run equilibrium the first thing we would do is get the supply curve - which can be done by finding the optimal mix of capital and labor via:</p>

<p>MPL/W = MPK/r</p>

<p>This will get you L = 4K.</p>

<p>In this situation you can skip to:
$$
Q = \sqrt{3}\sqrt{L}\\
L = Q^{2}/3
$$</p>

<p>Plug those into $C(Q) = Kr + Lw$</p>

<p>$$
C(Q) = 108 + 3Q^{2}\,
$$</p>

<p>The fixed cost is \$108 (but that is sunk in the short run, so the firm will ignore it), the variable cost is $3Q^{2}$, making the marginal cost $6Q$.  For the market, this will be $\frac{Q}{1.5}$ (the horizontal sum of all firms' production)</p>

<p>Marginal revenue in this market is just the price since we assume it is perfect competition, so setting market supply = demand gives you
$$
360 - 2Q = \frac{Q}{1.5}\\
Q = 135\,
$$</p>

<p>$Q/n = q$  tells us that each of the 9 firms must produce 15 units for the market to supply 135 total units.  According to the demand curve, the market price is \$90 when 135 units are sold.</p>

<p>Total surplus can be found by calculating the area of two triangles:</p>

<p>$$
CS = (1/2)135*(360-90) = \$18,225\\
PS = (1/2)135*(90) = \$6,075\\
TS = \$24,300\,
$$</p>

CC BY-SA 3.0
2015-06-06T12:18:13.187Z
2015-06-06T12:35:33.727Z
2015-06-06T12:35:33.727Z
Matthew Friedman
<p>I am having a lot of problems trying to find the equilibrium price when we are given a cost function and demand function, but no supply function.</p>

<blockquote>
  <p>All firms have the following production function: $Q = \sqrt{K \cdot L}$</p>
  
  <p>Wages are w = 9, and the rental rate of capital is r = 36.</p>
  
  <p>In the short run, capital is fixed at ̅ = 3 units. Market Demand is given by P = 360 – 2Q </p>
  
  <p>There are 9 firms in the market, find P*, Q*, and Total Surplus</p>
</blockquote>

<p><strong>In solving the production function with a fixed K:</strong></p>

<p>$MPL$ = K$^{1/2}$ \cdot L$^{1/2}$</p>

<p>$MPL$ = ${1/2} \cdot $K$^{1/2}$ $\cdot$ $L^{-1/2}$</p>

<p>= 1/2 \cdot (K$^{1/2}$ $\cdot$ L$^{1/2}$)</p>

<p><em>Substituting 3 for K:</em></p>

<p>= 1/2 \cdot ($\sqrt{3}$ \cdot $\sqrt{L}$)</p>

<p>$\sqrt{3}$/2 =  $\sqrt{L}$</p>

<p>$\sqrt{L}$ = .8660254</p>

<p>$L^*$ = .75</p>

<p><strong>Where C = wL + rK</strong></p>

<p>C(Q) = w($L^*$) + r($K^*$)</p>

<p>C(Q) = (9*.75) + (36*3)</p>

<p>C(Q) = 114.75</p>

<p>C(Q) @ 9 Firms =  (114.75 * 9) = 1032.75 (I think???)</p>

<p>But this doesn't have a slope, so how does this make sense? Obviously you would normally find the equilibrium through Qd = Qs, but what do I equal Qd to now?</p>

CC BY-SA 3.0
2015-06-03T07:08:30.193Z
A syntax matter: you should put backslash before sqrt and use {} brackets instead of (), so "\sqrt{3}". You don't have to but may want to use "\frac{}{}" for fractions.
Giskard
CC BY-SA 3.0
2015-06-03T07:09:29.930Z
I don't see how you get $L^*$. Why do you assume MPL = 1?
Giskard
CC BY-SA 3.0
2015-06-03T07:12:19.957Z
@denesp Sorry that was a typo. MPL was really the production function, Q
Alex McLean
CC BY-SA 3.0
2015-06-03T07:13:00.510Z
Okay, so why do you assume Q = 1?
Giskard
CC BY-SA 3.0
2015-06-03T07:14:52.450Z
@denesp I don't follow... The demand function Q=K*L usually adds up to 1 I thought, in terms of 1/2 + 1/2 in the exponents. Not too sure where you are getting the 1 from
Alex McLean
CC BY-SA 3.0
2015-06-03T07:14:56.170Z
The formula on the right hand side is MPL by the way.
Giskard
CC BY-SA 3.0
2015-06-03T07:17:04.833Z
Can you look over your calculations where you get $L^*$?
After a while you put $ = \frac{1}{2} \frac{\sqrt{3}}{\sqrt{L}}$.
Then you proceed as if the left hand side was one. But it is not one, it is $Q$.
Giskard
CC BY-SA 3.0
2015-06-03T07:17:33.517Z
And it should not be $Q$ either it should be MPL :)
Giskard
CC BY-SA 3.0
2015-06-03T07:18:46.903Z
@denesp ahh I see what you mean now. I'm not sure how to solve for $L*$ if Q is on the left though...
Alex McLean
CC BY-SA 3.0
2015-06-03T07:21:12.133Z
@denesp particularly if Q is unknown... this seems to become more confusing the more I think about it.
Alex McLean
CC BY-SA 3.0
2015-06-03T07:21:31.427Z
Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/24408/discussion-between-denesp-and-mclean25).
Giskard
<microeconomics>
Finding equilibrium of perfect competition in the short run with a cost function
Alex McLean
# Finding equilibrium of perfect competition in the short run with a cost function

(Asked by: username_0 on 2015-06-03)

I am having a lot of problems trying to find the equilibrium price when we are given a cost function and demand function, but no supply function.

> All firms have the following production function: $Q = \sqrt{K \cdot L}$
>
> Wages are w = 9, and the rental rate of capital is r = 36.
>
> In the short run, capital is fixed at ̅ = 3 units. Market Demand is given by P = 360 – 2Q
>
> There are 9 firms in the market, find P\*, Q\*, and Total Surplus

**In solving the production function with a fixed K:**

$MPL$ = K$^{1/2}$ \cdot L$^{1/2}$

$MPL$ = ${1/2} \cdot $K$^{1/2}$ $\cdot$ $L^{-1/2}$

= 1/2 \cdot (K$^{1/2}$ $\cdot$ L$^{1/2}$)

*Substituting 3 for K:*

= 1/2 \cdot ($\sqrt{3}$ \cdot $\sqrt{L}$)

$\sqrt{3}$/2 = $\sqrt{L}$

$\sqrt{L}$ = .8660254

$L^\*$ = .75

**Where C = wL + rK**

C(Q) = w($L^\*$) + r($K^\*$)

C(Q) = (9\*.75) + (36\*3)

C(Q) = 114.75

C(Q) @ 9 Firms = (114.75 \* 9) = 1032.75 (I think???)

But this doesn't have a slope, so how does this make sense? Obviously you would normally find the equilibrium through Qd = Qs, but what do I equal Qd to now?

**Tags:** microeconomics
> **Comments:**
> **username_1**: A syntax matter: you should put backslash before sqrt and use {} brackets instead of (), so "\sqrt{3}". You don't have to but may want to use "\frac{}{}" for fractions.  
> **username_1**: I don't see how you get $L^*$. Why do you assume MPL = 1?  
> **username_0**: @denesp Sorry that was a typo. MPL was really the production function, Q  
> **username_1**: Okay, so why do you assume Q = 1?  
> **username_0**: @denesp I don't follow... The demand function Q=K*L usually adds up to 1 I thought, in terms of 1/2 + 1/2 in the exponents. Not too sure where you are getting the 1 from  
> **username_1**: The formula on the right hand side is MPL by the way.  
> **username_1**: Can you look over your calculations where you get $L^*$? After a while you put $ = \frac{1}{2} \frac{\sqrt{3}}{\sqrt{L}}$. Then you proceed as if the left hand side was one. But it is not one, it is $Q$.  
> **username_1**: And it should not be $Q$ either it should be MPL :)  
> **username_0**: @denesp ahh I see what you mean now. I'm not sure how to solve for $L*$ if Q is on the left though...  
> **username_0**: @denesp particularly if Q is unknown... this seems to become more confusing the more I think about it.  
> **username_1**: Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/24408/discussion-between-denesp-and-mclean25).  

---

## 1 answers

### Answer by username_2 on 2015-06-06. Score: 2 | Accepted answer

In these problems, you are generally dealing with identical firms - all of whom will supply the market according to their marginal cost curve. If we were solving for a long run equilibrium the first thing we would do is get the supply curve - which can be done by finding the optimal mix of capital and labor via:

MPL/W = MPK/r

This will get you L = 4K.

In this situation you can skip to:
$$
Q = \sqrt{3}\sqrt{L}\\
L = Q^{2}/3
$$

Plug those into $C(Q) = Kr + Lw$

$$
C(Q) = 108 + 3Q^{2}\,
$$

The fixed cost is \$108 (but that is sunk in the short run, so the firm will ignore it), the variable cost is $3Q^{2}$, making the marginal cost $6Q$. For the market, this will be $\frac{Q}{1.5}$ (the horizontal sum of all firms' production)

Marginal revenue in this market is just the price since we assume it is perfect competition, so setting market supply = demand gives you
$$
360 - 2Q = \frac{Q}{1.5}\\
Q = 135\,
$$

$Q/n = q$ tells us that each of the 9 firms must produce 15 units for the market to supply 135 total units. According to the demand curve, the market price is \$90 when 135 units are sold.

Total surplus can be found by calculating the area of two triangles:

$$
CS = (1/2)135\*(360-90) = \$18,225\\
PS = (1/2)135\*(90) = \$6,075\\
TS = \$24,300\,
$$


---

