Household Demand and Utility Maximization: Microeconomics Study Notes

Unit 7: Household Demand

A. Utility Maximization

Utility maximization is a central concept in microeconomics, describing how consumers allocate their income to maximize satisfaction from goods and services. This section introduces utility functions, indifference curves, and budget constraints.

• Utility Function: Represents consumer preferences for combinations of goods. For example, $u(x, y) = x^{0.5}y^{0.5}$ shows utility from consuming quantities x and y.

• Indifference Curves: Graphical representations of combinations of goods that yield the same utility. For $u(x, y) = k$, the curve is $x^{0.5}y^{0.5} = k$.

• Marginal Rate of Substitution (MRS): The slope of the indifference curve, showing the rate at which a consumer is willing to substitute one good for another while maintaining the same utility. Equation: $MRS = -\frac{dy}{dx} = -\frac{MU_x}{MU_y}$

• Budget Line: Shows all combinations of goods a consumer can afford given income and prices. For example, if $p_x = 24$, $p_y = 4$, and income $I = 24$, the budget line is $24x + 4y = 24$.

• Opportunity Cost: The slope of the budget line represents the opportunity cost of one good in terms of the other: $-\frac{p_x}{p_y}$.

B. Other Indifference Curves

Different utility functions yield different shapes of indifference curves, reflecting various consumer preferences.

• Perfect Substitutes: Utility function $u(x, y) = x + y$; indifference curves are straight lines.

• Perfect Complements: Utility function $u(x, y) = \min\{x, y\}$; indifference curves are L-shaped.

• Additive Goods: Utility function $u(x, y) = x^2 + y^2$; indifference curves are convex to the origin.

• Economic Bads: Utility function $u(x, y) = (10 - x)^{0.5}(10 - y)^{0.5}$; indifference curves reflect negative utility from increased consumption.

C. Statics of Choice

Consumer choice is affected by changes in prices and income, leading to cross-price and income effects.

• Cross-Price Effects: Changes in the price of one good affect the quantity demanded of another. For example, with $u(x, y) = x^{0.5}y^{0.5}$, $p_x = 10$, $p_y = 1$, $I = 1$.

• Income Effects: Changes in income affect the quantity demanded of goods. For example, with $u(x, y) = x^{0.5}y^{0.5}$, $p_x = 10$, $p_y = 2$, $I = 20$.

• Utility Maximization: Consumers choose the bundle that gives the highest utility, subject to their budget constraint.

D. Income and Substitution Effects

When the price of a good changes, the total effect on quantity demanded can be decomposed into income and substitution effects.

• Substitution Effect: Change in consumption due to a change in relative prices, holding utility constant.

• Income Effect: Change in consumption due to a change in purchasing power.

• Illustration: If the price of good x falls, the consumer moves to a higher indifference curve, increasing consumption of x due to both effects.

• Equation: $\frac{\partial x}{\partial p_x} = \text{Substitution Effect} + \text{Income Effect}$

E. Demand Curves and Utility Maximization

Demand curves are derived from utility-maximizing behavior, showing the relationship between price and quantity demanded.

• Derivation: By varying the price of a good and solving the utility maximization problem, the demand curve can be traced.

• Example: For $u(x, y) = x^{0.5}y^{0.5}$, $p_x$ and $p_y$ are varied to observe changes in $x^*$.

F. Tutorial and Applications

Practice problems and applications reinforce understanding of household demand, utility maximization, and the effects of price and income changes.

• Indifference Curve and Budget Line Diagrams: Used to illustrate consumer choices and quantify effects.

• Marginal Utility: The additional satisfaction from consuming one more unit of a good. Calculated as the change in total utility.

• Table Example: Marginal utility and total utility for different quantities of goods.

Quantity of Good	Total Utility	Marginal Utility
1	10	10
2	18	8
3	24	6
4	28	4

• Optimal Choice: Occurs where the highest indifference curve is tangent to the budget line, i.e., $MRS = \frac{p_x}{p_y}$.

• Comparative Statics: Analysis of how changes in prices and income affect consumer choices.

Additional info:

• These notes cover key concepts from Chapter 6: Consumer Behaviour, including utility functions, indifference curves, budget constraints, and the derivation of demand curves.

• Examples and tables are inferred and expanded for clarity and completeness.