Appendix: Algebraic Presentation of Supply and Demand

7.7 Appendix: Algebraic Presentation of Supply and Demand

The supply-and-demand framework can be analyzed with algebra. We start with supply and demand and then talk about market equilibrium. This presentation uses some notation rather than only words:

p is the price of a chocolate bar.
I measures the income of individuals in a market.
q^d is the quantity demanded.
q^s is the quantity supplied.
A measures the technology of chocolate bar production.

Market Demand

With this notation, we represent the demand curve as follows:

Equation 7.1

q^d = α_d – β_dp + γ_dI.

α_d, β_d, and γ_d are constants that characterize the effects of prices and income on the quantity demanded. With the restriction that β_d > 0, the demand curve is downward sloping because an increase in p implies a reduction in the quantity demanded. It is natural to assume γ_d > 0, so an increase in income leads to an increase in the quantity demanded. This is represented as a shift in the demand curve.

Market Supply

With this notation, we represent the supply curve as follows:

Equation 7.2

q^s = α_s – β_sp + γ_sA.

α_s, β_s, and γ_s are constants that characterize the effects of prices and income on the quantity demanded. With the restriction that β_s > 0, the supply curve is upward sloping because an increase in p leads to an increase in the quantity supplied by all firms. It is natural to assume γ_s > 0, so an increase in the productivity of the current technology leads to an increase in the quantity produced at a given price. This is represented as a shift in the supply curve.

Market Equilibrium

The market is in equilibrium if there is a price and quantity combination, denoted (p*, q*) such that at the price p*, the quantity demanded, and the quantity supplied equal q*. Equilibrium is the simultaneous solution of supply and demand and can be found using the substitution method outlined in the toolkit.

Using q^s = q^d = q*, we can substitute Equation 7.2 into Equation 7.1 yielding:

Equation 7.3

α_d – β_dp* + γ_dI = α_s – β_sp* + γ_sA.

This is a single equation in a single unknown, p*. Solving the equation for p* implies

Equation 7.4

p^{*} = \frac{α_{d} - α_{s} + γ_{d} I - γ_{s} A}{β_{d} + β_{s}} .

The denominator is positive because we have assumed that both β_d and β_s are positive. The numerator is positive as long as the vertical intercept of the demand curve is greater than the vertical intercept of the supply curve: (α_d + γ_sI) > (α_s + γ_dA). This condition, combined with the restrictions on the slopes of supply and demand, is sufficient to guarantee that an equilibrium price exists in the market.

Using this calculation of p* in, say, the supply curve, we find

Equation 7.5

q^{*} = α_{s} + β_{s} \frac{α_{d} - α_{s} + γ_{d} I - γ_{s} A}{β_{d} + β_{s}} + γ_{s} A .

Grouping the terms into a constant, γ_dI and γ_sA, this becomes

Equation 7.6

q^{*} = α_{s} + β_{s} (\frac{α_{d} - α_{s}}{β_{d} + β_{s}}) + γ_{d} I (\frac{1}{β_{d} + β_{s}}) + γ_{s} A (1 - \frac{1}{β_{d} + β_{s}}) .

Looking at Equation 7.4 and Equation 7.6, these expressions determine the equilibrium price and the equilibrium quantity depending on the two (exogenous) factors that impact supply and demand: income level I and state of technology A. Though income influences only the position of the demand curve, variations in income influence both the equilibrium price and the equilibrium quantity. The same is true for variations in technology that shift only the supply curve.