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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 𝐾̅ = 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?

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    $\begingroup$ 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. $\endgroup$ – Giskard Jun 3 '15 at 7:08
  • $\begingroup$ I don't see how you get $L^*$. Why do you assume MPL = 1? $\endgroup$ – Giskard Jun 3 '15 at 7:09
  • $\begingroup$ @denesp Sorry that was a typo. MPL was really the production function, Q $\endgroup$ – thefoxrocks Jun 3 '15 at 7:12
  • $\begingroup$ Okay, so why do you assume Q = 1? $\endgroup$ – Giskard Jun 3 '15 at 7:13
  • $\begingroup$ @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 $\endgroup$ – thefoxrocks Jun 3 '15 at 7:14
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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:

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

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