Budget Constraints and Consumer Optimization

Budget Constraints

A budget constraint shows the combinations of two goods a consumer can afford given their income and the prices of the goods.

Equation:
Px⋅Qx+Py⋅Qy=IP_x \cdot Q_x + P_y \cdot Q_y = I
Where:

• PxP_x, PyP_y are the prices of goods X and Y
• QxQ_x, QyQ_y are the quantities of X and Y
• II is income

The budget line represents all possible combinations that exhaust the consumer’s income.

Graphical Representation

The slope of the budget line is:
−PxPy-\frac{P_x}{P_y}
This reflects the rate at which the consumer can trade one good for another — aka the opportunity cost.

Example: If pizza is $10 and soda is $5, giving up 1 pizza frees up cash for 2 sodas.

Consumer Optimization

The optimal choice is where the budget line is tangent to the highest possible indifference curve.

At this point:
MUxPx=MUyPy\frac{MU_x}{P_x} = \frac{MU_y}{P_y}
Where:

• MUxMU_x, MUyMU_y = marginal utility of goods X and Y
• PxP_x, PyP_y = prices of goods X and Y

This is known as the equal marginal principle.

Key Takeaways

• Budget constraints define purchasing limits.
• Optimization happens where utility is maximized under a budget.
• Consumers aim for the highest utility within what they can afford.