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?

  • 1
    $\begingroup$ 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 $\endgroup$ 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 $\endgroup$ Jun 3 '15 at 7:14

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:


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:

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.