Finding equilibrium of perfect competition in the short run with a cost function 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?
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:

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