Supply-demand model

The demand curve shows how much of a good people are willing to buy at a given price. It is generally downward sloping because people are willing to buy more of a good if it is cheaper.

The supply curve shows how much of a good firms are willing to sell for a given price. It is generally upward sloping.

The market equilibrium occurs where the supply and demand curve meet.

$$ \varepsilon = \frac{\Delta Q / Q_0}{\Delta P / P_0} \leq 0 $$

• $\Delta Q$: change in quantity
• $Q_0$: initial quantity
• $\Delta P$: change in price
• $P_0$: initial price

Perfectly inelastic demand: $\varepsilon = 0$

• No substitute is available
• Demand curve is vertical
• Quantity is fixed

Perfectly elastic demand: $\varepsilon = \infty$

• Perfect substitutes are available
• Demand curve is vertical
• Price is fixed

Elasticity of goods is determined by their substitutability.

$$ \gamma = \frac{\Delta Q / Q_0}{\Delta Y / Y_0} $$

• $Q$: quantity of good
• $Y$: income

Engel curve: curve showing relationship between income and quantity of good.

• $ \gamma > 0 $: normal good
  • $ \gamma > 1 $: luxuries - spending on these goods goes up as a proportion of income as income increases.
  • $ 0 < \gamma < 1 $: necessitities - spending on these goods goes down as a proportion of income as income increases.
• $ \gamma < 0 $: inferior good

When the price of a good changes, it changes the desirability of that good relative to another good (substitution effect), and it also changes your effective income (income effect).

The substitution effect is the change in the quantity of a good as price changes holding utility constant.

$$ \left.\frac{dQ}{dP}\right|_{\bar{U}} $$

Because utility is kept constant, this is graphically represented by a shift along the original indifference curve. At the new point, the curve will have the slope of the new budget constraint curve (i.e. slope = ratio between prices of goods), although it will not be on the old budget constraint curve. The budget constraint curve that this point lies on (which is not really achievable) compensates for the increase in price of the good, so it is known as the compensated demand.

The substitution effect is always negative because:

$$ P_X \uparrow \implies \frac{P_X}{P_Y} = \frac{MU_Y}{MU_X} \uparrow \implies \frac{Q_Y}{Q_X} \uparrow$$

The income effect is the change in the quantity of a good as income changes.

$$ \left.\frac{dQ}{dY}\right|_\bar{P} $$

This is represented by an inward shift of the budget constraint curve from the compensated demand curve down to one that is actually achievable with the same income, although it will have the same slope.

For a Giffen good, the effects of price changes are dominated by the income effect, leading to an increase in quantity demanded with an increase in price and therefore an upward sloping demand curve.