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If the market demand for shoes is given by $QD = 10000-250P$ and the supply is $QS = 5000$, what is the equilibrium price of shoes? How many pairs of shoes will be sold?

Thanks in advance.

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The incoming money from a choice of price $P$ will be $P\cdot QD=P(10^4-250P)$, which is maximized at $P=20$ to $100000$ (I don't know your units), as can be shown with either basic calculus or algebra. For this profit-maximizing choice of price, the number of shoes sold is $10000-250P=5000$. Is this what you're asking? (Also, there's a [homework] tag in case this is indeed homework.) –  anon Aug 29 '11 at 9:34
    
So far, I did 10000-250P = 5000 250P = 5000 P = 20. So I think that is the price. Does that mean that the number of shoes sold is 5000? Thanks –  mr. chan Aug 29 '11 at 10:08

3 Answers 3

As $QD = QS$ at market equilibrium:

$$10000-250P=5000,$$

$$10000-5000=250P,$$

$$\frac{5000}{250}=P.$$

Thus, $P=20$.

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Since you know $Q_S = 5000$, this means that the quantity supplied is 5000, no matter what the price is. So if the market clears, you are correct that the number sold is 5000 (and that, from the demand equation, the market clearing price is $P=20$).

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In this example, supply is a constant, so it really depends on what the demand is.

Equating demand and supply, 10000-250P=5000

Subtracting 5000 from both sides, then adding 250P to both sides,

5000=250P

Then P=5000/250 or 200.

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