Budget Constraints in Microeconomics

Introduction to Budget Constraints

In microeconomics, a fundamental problem is how agents maximize their objectives (such as utility or profit) subject to constraints. For consumers, the most common constraint is the budget constraint, which limits the combinations of goods they can purchase given their income and the prices of those goods. This section focuses on the mathematical and graphical representation of budget constraints in a two-good model.

• Key Point 1: The budget constraint represents all combinations of two goods a consumer can afford given their income and the prices of the goods.

• Key Point 2: Changes in income or prices shift or rotate the budget constraint, affecting the consumer's feasible choices.

• Example: An individual maximizing utility subject to a budget constraint; a firm maximizing profit subject to input price constraints.

Assumptions of the Two-Good Consumption Model

The basic two-good model is built on several simplifying assumptions:

• Two Goods: The consumer can purchase two goods, denoted as x and y, with prices and respectively.

• Single Period Decision: The consumer makes decisions for one period; there is no intertemporal choice, borrowing, or lending.

• Fixed Income: Income is given and fixed, denoted as (sometimes in other texts).

Additional info: These assumptions can be relaxed in more advanced models, such as those involving multiple periods or variable income.

Mathematical Representation of Budget Sets and Constraints

Budget Set and Budget Constraint

The budget set is the collection of all affordable combinations of goods x and y given the consumer's income. The budget constraint is the boundary of this set, where the consumer spends all their income.

• Budget Set:

• Budget Constraint:

To graph the budget constraint, solve for y in terms of x:

This is a straight line with:

• Vertical (y) intercept:

• Horizontal (x) intercept:

• Slope:

Economic Meaning: The slope represents the opportunity cost of consuming good x in terms of good y—the rate at which the market allows substitution between the two goods.

Example: Kennedy's Budget Constraint

• Given: Income , ,

• Budget Constraint:

• Intercepts: -intercept: , -intercept:

• Slope:

Effects of Changes in Income and Prices

Income Changes

When income increases (with prices constant), the budget constraint shifts outward, increasing both intercepts but keeping the slope unchanged.

• New Budget Constraint:

• Intercepts: -intercept: , -intercept:

• Slope: Unchanged,

Price Changes

Changing the price of one good rotates the budget constraint, affecting the intercept for that good and the slope.

• Example: If increases from $20 (Kennedy's case), -intercept decreases, -intercept unchanged, slope changes from to .

Summary Table: Effects of Income and Price Changes

Worked Examples

General Model Example

• Given: Income $100p_x = 2$, $p_y = 10$

• Budget Constraint:

• Intercepts: -intercept: $50y

• Slope:

• Interpretation: The slope is the rate at which the market trades y for x; intercepts are the maximum quantities purchasable given income and prices.

Specific Goods Example: Baseball and Movies

• Given: Income $100p_b = 5p_m = 10$ (movies)

• Budget Constraint:

• Intercepts: -intercept: $20m

• Slope:

• Price Change: If increases to $10b, slope becomes

Kinked Budget Constraints

Quantity Discounts and Piecewise Constraints

Sometimes, the price of a good changes after a certain quantity is purchased, resulting in a kinked budget constraint. This is modeled with piecewise equations.

• Example: for first 3 units, then for additional units; ,

• Budget Constraint:

  • If :

  • If :

• Intercepts: Only y: $100 units if all income spent on x (with discount).

• Slope: Changes from to at .

Extensions and Applications

Other Types of Constraints

The budget constraint framework can be applied to other allocation problems, such as time, calories, or other resources. The axes can represent any two competing uses of a limited resource.

• Example: Allocating time between studying and socializing.

Summary

• Budget constraints are determined by income and prices, not by preferences.

• Changes in income or prices shift or rotate the budget constraint, affecting feasible consumption bundles.

• To determine which bundle a consumer will choose, preferences and utility must be considered (covered in subsequent units).

Additional info: The graphical and mathematical analysis of budget constraints is foundational for understanding consumer choice, utility maximization, and demand theory in microeconomics.