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Difficulty: MediumMarket Equilibrium Price and Quantity

In a competitive wholesale market for cement, the daily quantity demanded is given by the linear demand function Qd=450−12PQ_d = 450 - 12P and the daily quantity supplied is given by Qs=−50+13PQ_s = -50 + 13P, where PP is the price per bag in hundreds of Naira and QQ is measured in metric tons. What is the equilibrium quantity of cement traded in metric tons?

Answer: 210 metric tons

Answer

The equilibrium quantity of cement traded daily is 210 metric tons.
Market equilibrium occurs at the point where quantity demanded equals quantity supplied (Qd=QsQ_d = Q_s). Equating 450−12P=−50+13P450 - 12P = -50 + 13P yields 500=25P500 = 25P, which solves to an equilibrium price of P=20P = 20. Substituting P=20P = 20 into the demand function gives Q∗=450−12(20)=210Q^* = 450 - 12(20) = 210 metric tons.

Step-by-Step Solution

1
Set quantity demanded equal to quantity supplied to establish market equilibrium.
450−12P=−50+13P450 - 12P = -50 + 13P
Market equilibrium occurs at the price level where Qd=QsQ_d = Q_s.
2
Group like terms to solve for the equilibrium price (PP).
450+50=13P+12P  ⟹  500=25P  ⟹  P=20450 + 50 = 13P + 12P \implies 500 = 25P \implies P = 20
Adding 5050 and 12P12P to both sides isolates the variable PP.
3
Substitute the equilibrium price (P=20P = 20) back into the demand function to find equilibrium quantity (Q∗Q^*).
Q∗=450−12(20)=450−240=210Q^* = 450 - 12(20) = 450 - 240 = 210 metric tons
Evaluating either the demand or supply function at P=20P = 20 gives the market clearing quantity.

Key Concept

Market Equilibrium Quantity Determination
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Topics

Market Equilibrium Price and Quantity

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