Utility Maximization
Understand how consumers choose combinations of goods to maximize their satisfaction.
Rational Choice
Optimization
Preferences
Consumers are assumed to be rational and seek to maximize their utility (satisfaction) subject to their budget constraint. They compare the marginal utility per dollar spent across different goods to make optimal choices.
# Utility Maximization Model
utility_maximization = {
"assumptions": {
"rational_consumers": "Make consistent choices",
"complete_preferences": "Can rank all bundles",
"transitive_preferences": "Consistent ordering",
"more_is_better": "Non-satiation assumption"
},
"utility_function": {
"definition": "U = f(x1, x2, ..., xn)",
"ordinal_nature": "Only rankings matter",
"examples": {
"cobb_douglas": "U = x1^a × x2^b",
"perfect_substitutes": "U = ax1 + bx2",
"perfect_complements": "U = min(ax1, bx2)"
}
},
"optimization_condition": {
"rule": "MUx/Px = MUy/Py",
"meaning": "Equal marginal utility per dollar",
"mathematical": "∂U/∂x1 / P1 = ∂U/∂x2 / P2"
}
}
Budget Constraints
Learn how income and prices limit consumer choices and define feasible consumption sets.
Budget Constraint Components:
• Income (I): Total money available for spending
• Prices (P1, P2): Cost per unit of each good
• Budget Line: I = P1×X1 + P2×X2
• Slope: -P1/P2 (relative price ratio)
Budget Line Properties:
• Shows all affordable combinations
• Negative slope indicates trade-offs
• Shifts with income changes
• Rotates with price changes
# Budget Constraint Analysis
budget_constraint = {
"equation": "I = P1*X1 + P2*X2",
"intercepts": {
"x1_intercept": "I/P1",
"x2_intercept": "I/P2"
},
"slope": "-P1/P2",
"changes": {
"income_increase": {
"effect": "Parallel outward shift",
"reason": "Can afford more of both goods"
},
"price_increase_x1": {
"effect": "Inward rotation around x2 intercept",
"reason": "X1 becomes relatively expensive"
}
},
"feasible_set": "All points on or below budget line"
}
Indifference Curves
Explore how to represent consumer preferences graphically using indifference curves.
Indifference Curve Properties:
• Downward sloping (more of one good, less of another)
• Higher curves represent higher utility levels
• Cannot intersect each other
• Convex to origin (diminishing marginal rate of substitution)
Marginal Rate of Substitution (MRS):
MRS = -ΔX2/ΔX1 = MU1/MU2
Measures willingness to trade one good for another while maintaining same utility level
# Indifference Curve Analysis
indifference_curves = {
"definition": "Combinations giving same utility level",
"properties": {
"downward_sloping": "Trade-off between goods",
"non_intersecting": "Consistency of preferences",
"convex_to_origin": "Diminishing MRS",
"higher_is_better": "More utility on higher curves"
},
"marginal_rate_substitution": {
"formula": "MRS = -dx2/dx1 = MU1/MU2",
"interpretation": "Willingness to substitute",
"diminishing": "Becomes smaller along curve"
},
"special_cases": {
"perfect_substitutes": "Straight lines",
"perfect_complements": "L-shaped curves",