This is “Appendix: Algebraic Presentation of Supply and Demand”, section 7.7 from the book Theory and Applications of Microeconomics (v. 1.0).
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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:
With this notation, we represent the demand curve as follows:
Equation 7.1
q^{d} = α_{d} – β_{d}p + γ_{d}I.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.
With this notation, we represent the supply curve as follows:
Equation 7.2
q^{s} = α_{s} – β_{s}p + γ_{s}A.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.
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_{d}p* + γ_{d}I = α_{s} – β_{s}p* + γ_{s}A.This is a single equation in a single unknown, p*. Solving the equation for p* implies
Equation 7.4
$${p}^{*}=\frac{{\alpha}_{d}-{\alpha}_{s}+{\gamma}_{d}I-{\gamma}_{s}A}{{\beta}_{d}+{\beta}_{s}}\text{.}$$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_{s}I) > (α_{s} + γ_{d}A). 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
$${q}^{*}={\alpha}_{s}+{\beta}_{s}\frac{{\alpha}_{d}-{\alpha}_{s}+{\gamma}_{d}I-{\gamma}_{s}A}{{\beta}_{d}+{\beta}_{s}}+{\gamma}_{s}A\text{.}$$Grouping the terms into a constant, γ_{d}I and γ_{s}A, this becomes
Equation 7.6
$${q}^{*}={\alpha}_{s}+{\beta}_{s}\left(\frac{{\alpha}_{d}-{\alpha}_{s}}{{\beta}_{d}+{\beta}_{s}}\right)+{\gamma}_{d}I\left(\frac{1}{{\beta}_{d}+{\beta}_{s}}\right)+{\gamma}_{s}A\left(1-\frac{1}{{\beta}_{d}+{\beta}_{s}}\right)\text{.}$$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.