Budget line equation: $ P_1X_1 + P_2X_2 = M $. Then slope of the budget line is:

Question

Prepp Team · Accepted Answer

Understanding the Budget Line Equation

The budget line represents the various combinations of two goods that a consumer can afford given their income and the prices of the two goods. The standard equation for a budget line with two goods, Good 1 and Good 2, is given as:

$ P_1X_1 + P_2X_2 = M $

Where:

$ P_1 $ is the price of Good 1
$ X_1 $ is the quantity of Good 1
$ P_2 $ is the price of Good 2
$ X_2 $ is the quantity of Good 2
$ M $ is the consumer's total income or budget

This equation states that the total expenditure on Good 1 (price times quantity) plus the total expenditure on Good 2 must be equal to the consumer's total income $ M $.

Calculating the Slope of the Budget Line

The slope of the budget line tells us the rate at which the consumer can trade off one good for another while remaining within their budget. It represents the opportunity cost of Good 1 in terms of Good 2. To find the slope, we need to rearrange the budget line equation to express $ X_2 $ (the quantity of Good 2, typically plotted on the vertical axis) in terms of $ X_1 $ (the quantity of Good 1, typically plotted on the horizontal axis). This is similar to rearranging a linear equation into the form $ y = mx + c $, where $ m $ is the slope.

Starting with the budget line equation:

$ P_1X_1 + P_2X_2 = M $

Subtract $ P_1X_1 $ from both sides:

$ P_2X_2 = M - P_1X_1 $

Now, divide both sides by $ P_2 $ to isolate $ X_2 $:

$ X_2 = \frac{M - P_1X_1}{P_2} $

We can rewrite this equation by separating the terms on the right side:

$ X_2 = \frac{M}{P_2} - \frac{P_1X_1}{P_2} $

Or, arranging it in the standard $ y = mx + c $ form (where $ X_2 $ is $ y $ and $ X_1 $ is $ x $):

$ X_2 = -\frac{P_1}{P_2}X_1 + \frac{M}{P_2} $

In this form, the coefficient of $ X_1 $ is the slope of the budget line. Comparing this to $ y = mx + c $, we see that the slope $ m $ is $ -\frac{P_1}{P_2} $.

Interpreting the Budget Line Slope

The slope of the budget line is $ -\frac{P_1}{P_2} $. The negative sign indicates that the budget line is downward sloping. This means that to consume more of Good 1 (increase $ X_1 $), the consumer must give up some amount of Good 2 (decrease $ X_2 $) to stay within their budget. The magnitude of the slope, $ \frac{P_1}{P_2} $, represents the ratio of the price of Good 1 to the price of Good 2. It tells us how many units of Good 2 the consumer must give up to obtain one additional unit of Good 1.

For example, if the price of Good 1 is $10 and the price of Good 2 is $5, the slope is $ -\frac{10}{5} = -2 $. This means the consumer must give up 2 units of Good 2 to purchase 1 additional unit of Good 1, given their budget.

Comparing with Options

We found the slope of the budget line $ P_1X_1 + P_2X_2 = M $ to be $ -\frac{P_1}{P_2} $.

Let's look at the given options:

$ \frac{P_1}{P_2} $
$ \frac{X_1}{X_2} $
$ -\frac{X_1}{X_2} $
$ -\frac{P_1}{P_2} $

Our calculated slope matches the fourth option.

Revision Table: Budget Line Summary

Concept	Equation/Formula	Description
Budget Line Equation	$ P_1X_1 + P_2X_2 = M $	All combinations of $ X_1 $ and $ X_2 $ affordable given $ M, P_1, P_2 $.
Slope of Budget Line	$ -\frac{P_1}{P_2} $	Rate at which $ X_2 $ must be given up for $ X_1 $; opportunity cost of $ X_1 $ in terms of $ X_2 $.
Vertical Intercept (Max $ X_2 $)	$ \frac{M}{P_2} $	Maximum amount of Good 2 that can be purchased if all income is spent on Good 2.
Horizontal Intercept (Max $ X_1 $)	$ \frac{M}{P_1} $	Maximum amount of Good 1 that can be purchased if all income is spent on Good 1.

Additional Information on Budget Constraints

The budget line is a fundamental concept in consumer theory within microeconomics. Its position and slope are determined by income and the prices of the goods.

Shifts in the Budget Line: If income $ M $ changes, or if both prices $ P_1 $ and $ P_2 $ change proportionally, the budget line will shift parallel to its original position. An increase in income or a proportional decrease in both prices shifts the line outward, expanding the feasible set. A decrease in income or a proportional increase in both prices shifts the line inward, contracting the feasible set.
Pivots in the Budget Line: If the price of only one good changes, the budget line will pivot. For example, if $ P_1 $ decreases while $ P_2 $ and $ M $ remain constant, the budget line pivots outward around the vertical intercept ($ M/P_2 $), becoming flatter because the consumer can now buy more of Good 1. If $ P_1 $ increases, it pivots inward around the vertical intercept, becoming steeper. Similarly, changes in $ P_2 $ cause pivots around the horizontal intercept ($ M/P_1 $).
Opportunity Cost: The slope $ -\frac{P_1}{P_2} $ precisely captures the opportunity cost. Giving up one unit of Good 1 frees up $ P_1 $ amount of money, which can then be used to buy $ P_1/P_2 $ units of Good 2. Thus, the opportunity cost of one unit of Good 1 is $ P_1/P_2 $ units of Good 2. The negative sign indicates this trade-off.

Understanding the budget line and its slope is crucial for analyzing consumer choices and utility maximization.

Prepp Team · Answer

$ \frac{P_1}{P_2} $

Prepp Team · Answer

$ \frac{X_1}{X_2} $

Prepp Team · Answer

$ -\frac{X_1}{X_2} $

Concept	Equation/Formula	Description
Budget Line Equation	\( P_1X_1 + P_2X_2 = M \)	All combinations of \( X_1 \) and \( X_2 \) affordable given \( M, P_1, P_2 \).
Slope of Budget Line	\( -\frac{P_1}{P_2} \)	Rate at which \( X_2 \) must be given up for \( X_1 \); opportunity cost of \( X_1 \) in terms of \( X_2 \).
Vertical Intercept (Max \( X_2 \))	\( \frac{M}{P_2} \)	Maximum amount of Good 2 that can be purchased if all income is spent on Good 2.
Horizontal Intercept (Max \( X_1 \))	\( \frac{M}{P_1} \)	Maximum amount of Good 1 that can be purchased if all income is spent on Good 1.

List-I	List-II
(A) Price changes but no change in demand	(I) Perfectly elastic (e_p = ∞)
(B) Price remains the same but demand changes	(II) Unit elastic (e_p = 1)
(C) Price and demand change in the same proportion	(III) More than elastic
(D) Price changes in less proportion than demand	(IV) Perfectly inelastic (e_p = 0)