Demand, Supply and Equilibrium

flowchart TD
    A[02 Demand Supply And Equilibrium] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

We define demand as the quantity of a good or service that consumers are willing and able to Purchase at each possible price during a given time period, ceteris paribus.

Where $P$ = price of the good, $Y$ = income, $P_s$ = price of substitutes, $P_c$ = price of Complements, $T$ = tastes, $E$ = expectations, $N$ = population.

The law of demand states that, ceteris paribus, as price rises, quantity demanded falls. This Follows from:

1. Income effect: a higher price reduces real purchasing power
2. Substitution effect: a higher price makes substitutes relatively more attractive

Consider a consumer with utility function $U(x, y)$ facing prices $P_x$, $P_y$ and income $M$. The Consumer solves:

The Lagrangian is:

First-order conditions:

Therefore:

This equates the marginal rate of substitution (the consumer”s internal valuation) with the price Ratio (the market’s valuation). Solving for $x$ as a function of $P_x$ (holding other parameters Constant) yields the individual demand curve $x = d_i(P_x)$.

The market demand curve is derived by horizontal summation of individual demand curves:

At each price, we add up the quantities demanded by all consumers.

• Movement along the demand curve: caused by a change in the good’s own price. We move from one point to another on the same curve.
• Shift of the demand curve: caused by a change in any determinant other than the good’s own price. The entire curve moves left (decrease in demand) or right (increase in demand).

We define supply as the quantity of a good or service that producers are willing and able to Offer for sale at each possible price during a given time period, ceteris paribus.

Where $C$ = costs of production, $T$ = technology, $S$ = subsidies/taxes, $E$ = expectations, $n$ = Number of firms.

The law of supply states that, ceteris paribus, as price rises, quantity supplied rises. This Follows from profit maximisation.

A firm with cost function $C(Q)$ and facing price $P$ maximises profit:

First-order condition:

Where $MC(Q) = C'(Q)$ is marginal cost. Second-order condition requires $C''(Q) \gt 0$ (MC Rising). The supply curve of a competitive firm is the portion of its $MC$ curve above the Average variable cost (AVC) curve.

Horizontal summation of individual firm supply curves.

We define market equilibrium as the price-quantity pair $(P^*, Q^*)$ at which quantity demanded Equals quantity supplied:

Stability proof. Suppose price $P_1 \gt P^*$. Then $Q_S(P_1) \gt Q_D(P_1)$, there is excess Supply (a surplus). Unsold goods pile up, so firms cut prices. As price falls, quantity demanded Rises and quantity supplied falls until equilibrium is restored.

Suppose price $P_2 \lt P^*$. Then $Q_D(P_2) \gt Q_S(P_2)$, there is excess demand (a shortage). Consumers bid up prices. As price rises, quantity supplied rises and quantity demanded falls until Equilibrium is restored.

Therefore, the equilibrium is stable: any deviation sets in motion forces that restore Equilibrium. $\blacksquare$

The price mechanism is the process by which prices adjust to equate demand and supply, thereby Allocating resources without central direction. It performs three functions:

1. Signalling: prices convey information about scarcity (high price = scarce)
2. Incentive: high prices incentivise production, low prices incentivise consumption
3. Rationing: prices ration scarce goods to those willing and able to pay

We define the price elasticity of demand as:

Since the demand curve slopes downward, $\mathrm{PED} \lt 0$. We often state the absolute value $|\mathrm{PED}|$.

Classification:

Total revenue is $TR = P \times Q$.

Since PED < 0:

• If $|\mathrm{PED}| \gt 1$ (elastic): $\frac{d(TR)}{dP} \lt 0$. Price increase $\Rightarrow$ revenue falls.
• If $|\mathrm{PED}| \lt 1$ (inelastic): $\frac{d(TR)}{dP} \gt 0$. Price increase $\Rightarrow$ revenue rises.
• If $|\mathrm{PED}| = 1$ (unit elastic): $\frac{d(TR)}{dP} = 0$. Revenue is maximised.

Proposition: Total revenue is maximised where $|\mathrm{PED}| = 1$.

Proof. We showed $\frac{d(TR)}{dP} = Q(1 + \mathrm{PED})$. Setting $\frac{d(TR)}{dP} = 0$: $1 + \mathrm{PED} = 0$ So $\mathrm{PED} = -1$ I.e., $|\mathrm{PED}| = 1$. The second derivative Confirms this is a maximum (for downward-sloping demand). $\blacksquare$

Proposition: For a linear demand curve $Q = a - bP$ PED varies from $0$ (at the quantity axis) To $-\infty$ (at the price axis), with $|\mathrm{PED}| = 1$ at the midpoint.

Proof. $P = \frac{a - Q}{b}$ So:

At the midpoint, $Q = a/2$: $\mathrm{PED} = -\frac{a}{a/2} + 1 = -1$. As $Q \to 0$ (price axis): $\mathrm{PED} \to -\infty$ (perfectly elastic). As $Q \to a$ (quantity axis): $\mathrm{PED} \to 0$ (perfectly inelastic). $\blacksquare$

1. Availability of substitutes: more substitutes $\Rightarrow$ more elastic (e.g., bottled water vs insulin)
2. Proportion of income spent: larger share $\Rightarrow$ more elastic (e.g., cars vs matches)
3. Time period: longer time horizon $\Rightarrow$ more elastic (consumers can adjust behaviour)
4. Necessity vs luxury: necessities tend to be inelastic, luxuries elastic
5. Definition of the market: narrowly defined markets are more elastic (e.g., “Coca-Cola” vs “soft drinks”)

The magnitude of XED indicates the closeness of the relationship, relevant for competition policy (defining the relevant market).

Determinants of PES:

1. Time period: momentary (perfectly inelastic) < short-run < long-run (more elastic)
2. Spare capacity: excess capacity $\Rightarrow$ more elastic
3. Mobility of factors: reallocated factors $\Rightarrow$ more elastic
4. Ability to store goods: storable goods $\Rightarrow$ more elastic
5. Natural constraints: agricultural supply is inelastic in the short run

Consumer surplus is the difference between what consumers are willing to pay and what they Actually pay:

Where $P_d(Q)$ is the inverse demand function (the maximum price consumers will pay for quantity $Q$).

Producer surplus is the difference between the price received and the minimum price producers Would accept:

Where $P_s(Q)$ is the inverse supply function.

Total surplus = $CS + PS$. At competitive equilibrium, total surplus is maximised, this is the First Theorem of Welfare Economics.

• Provides a powerful, general framework for analysing markets
• Equilibrium concept is robust (stable under reasonable conditions)
• Elasticity provides a quantitative measure of responsiveness
• Consumer/producer surplus allows welfare analysis

• Assumes perfect competition, many markets are not competitive
• Static analysis, doesn’t capture dynamic adjustment processes
• Representative agent assumption, ignores heterogeneity
• Ceteris paribus is unrealistic, many variables change simultaneously
• Doesn’t account for behavioural biases (prospect theory, loss aversion)