Complements and Substitutes in General Equilibrium

Christopher Makler

Stanford University Department of Economics

Econ 50Q: Section 10

Demand Effects

Suppose two goods are complements.

What happens in both markets
if there is a supply shift
in the market for one of the goods?

Specifically, what happens to the equilibrium prices and quantity in both markets, if there is an increase in the cost of producing good 1?

pollev.com/chrismakler

Equilibrium in One Market

Market for Good 2

Market for Good 1

Equilibrium in Two Markets with Related Demand

S_1(p_1)

D_1(p_1,p_2)

S_2(p_2)

D_2(p_1,p_2)

S(p)

D(p)

Market for Good 2

Market for Good 1

Equilibrium in Two Markets with Related Demand

S_1(p_1) = {p_1 \over a}

D_1(p_1,p_2) = {m \over p_1 + p_2}

S_2(p_2) = {p_2 \over b}

D_2(p_1,p_2) = {m \over p_1 + p_2}

\displaystyle{Q_1 = Q_2 = \sqrt{m \over a + b}}

\displaystyle{p_1 = a\sqrt{m \over a + b}}

\displaystyle{p_2 = b\sqrt{m \over a + b}}

u(x_1,x_2) = \min\{x_1,x_2\}

(\(a\) and \(b\) are cost shifters)

\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}

\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}

\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}

Which good will have the higher price in equilibrium?

Key insight: remember that firms will produce at the point along the PPF where the MRT is equal to the price ratio. What point along the PPF must be chosen with this utility function?

pollev.com/chrismakler

\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}

\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}

\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}

What is the equilibrium price ratio?

Key insight: remember that firms will produce at the point along the PPF where the MRT is equal to the price ratio. What point along the PPF must be chosen with this utility function?

pollev.com/chrismakler

\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}

\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}

\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}

What is the equilibrium price ratio?

Key insight: remember that firms will produce at the point along the PPF where the MRT is equal to the price ratio. What point along the PPF must be chosen with this utility function?

Equation of the PPF:

Expression for the MRT:

{x_1^3 \over 6^3} + {x_2^3 \over 12^3} = \overline L

{{3x_1^2 \over 6^3} \over {3x_2^2 \over 12^3}}

= \left({12 \over 6}\right)^3 \times \left(x_1 \over x_2\right)^2

= 8 \left(x_1 \over x_2\right)^2

\text{Consumer utility function: }u(x_1,x_2) = \min\{x_1,x_2\}

\text{Good 1 production function: }Y_1(L_1) = 6L_1^{1 \over 3}

\text{Good 2 production function: }Y_2(L_2) = 12L_2^{1 \over 3}

Bonus question to do at home: suppose that in equilibrium, \(p_1 = 80\) and \(Y_1 = 24\). What must the wage rate be?