Demand and Supply Curves: Equilibrium Price and Quantity

  • Demand curves typically show the relationship between price and quantity, illustrating that as price decreases, quantity demanded increases.

The Problem with the Standard Demand Curve Representation

  • Demand curve: q=f(p)q = f(p) , where quantity depends on price.

  • Typically plotted with price (dependent variable) on the y-axis and quantity (independent variable) on the x-axis, which is opposite of true functional relationship.

Inverse Demand Curves

  • Demand curves typically drawn are actually inverse demand curves: p=f(q)p = f(q).

  • Crucial to distinguish between normal demand curve q=f(p)q = f(p) and inverse demand curve p=f(q)p = f(q).

Question: Equilibrium Price and Quantity

  • Supply Curve: p=2+qp = 2 + q (inverse supply curve).

  • Demand Curve: p=721+qp = \frac{72}{1 + q} (inverse demand curve).

  • Task: Calculate the equilibrium price and quantity.

Calculating Equilibrium

To find the equilibrium price and quantity, set the supply and demand equations equal to each other:
2+q=721+q2 + q = \frac{72}{1 + q}
(2+q)(1+q)=72(2 + q)(1 + q) = 72
2+2q+q+q2=722 + 2q + q + q^2 = 72
q2+3q+2−72=0q^2 + 3q + 2 - 72 = 0
q2+3q−70=0q^2 + 3q - 70 = 0
Solve the quadratic equation for qq: (q+10)(q−7)=0(q + 10)(q - 7) = 0. Thus, q=−10q = -10 or q=7q = 7. Since quantity cannot be negative, q=7q = 7.
Substitute q=7q = 7 into the supply equation: p=2+q=2+7=9p = 2 + q = 2 + 7 = 9.

Equilibrium

  • Equilibrium Quantity: q=7q = 7

  • Equilibrium Price: p=9p = 9