Budget Constraint
Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC  Fall 2017
Consumption Choice Sets

A consumption choice set is the collection of all consumption choices available to the consumer.

What constrains consumption choice?
 Budgetary, time and other resource limitations.
Consumption bundle

A consumption bundle containing $ x_1 $ units of commodity 1, $ x_2 $ units of commodity 2 and so on up to $ x_n $ units of commodity n is denoted by the vector $ ( x_1, x_2, … , x_n ) $.

Prices or goods are denoted by: $ p_1, p_2, … , p_n $.
Affordable Bundles  Budget Constraints

Suppose prices are $ p_1, p_2, … , p_n $ and a consumer has $ m $ as income.

Question:
 When is a consumption bundle $ (x_1, … , x_n) $ affordable at those given prices and income?
Affordable Bundles  Budget Constraints

Answer:

when $ p_1 x_1 + … + p_n x_n \leq m $

where $ m $ is the consumer’s (disposable) income.


That is, all the bundles that when purchased do not exhaust the consumer's income.
"Budget line" or "budget constraint"

The bundles that are only just affordable form the consumer’s budget constraint or budget line.

This is the set:
$ \{ ( x_1 ,…, x_n ) \quad:\quad p_1 x_1 + … + p_n x_n = m \} $
 For simplicity we will only work with $ x_1, … , x_n $ are all equal or greater than zero.
Budget Set
 The consumer’s budget set is the set of all affordable bundles;
$ B(p_1, … , p_n, m) = \{ (x_1, … , x_n) \quad:\quad p_1 x_1 + … + p_n x_n \leq m \} $
 The budget constraint (or budget line) is the upper boundary of the budget set.
Budget for Two Commodities
 $ p_1 x_1 + p_2 x_2 = m $. Affordable set, intercepts, slope.
Budget for Three Commodities
Finding the slope of the BC

Budget line: $ p_1 x_1 + p_2 x_2 = m $

Solve for $ x_2 $ :

$ p_2 x_2 = m  p_1 x_1 $

$ x_2 = \frac{m}{p_2}  \frac{p_1}{p_2} x_1 $


Therefore the slope is: $  \frac{p_1}{p_2} $

What is the interpretation: relative price.
Example of BC

Good one is beer (good 1) and orange juice (good 2).

Suppose $ p_1 = 3 $ and $ p_2 = 1 $.

Income = 100

slope =  3: Consumer need to give up (buy less) 3 oz. of orange juice to afford (be able to buy) 1 additional oz of beer.

You can use the market to transform three units of OJ into one unit of beer, at the current prices. Therefore the term of relative price
Changes in the BC

The budget constraint and budget set depend upon prices and income. What happens as prices or income change?

Income change?

Prices change?

Board  Doc Camera

Makler's EconGraphs
Introducing EconGraphs
Income Changes
 What bundles become unaffordable or newly affordable?
Income Increases

Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.

Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.

Which one is "good" for consumer?
Price Changes
 What bundles become unaffordable or newly affordable?
$ p_1 $ increases

$ p_1 $ increases from $ p_1 $ to $ p_1' $

Budget constraint pivots: slope get steeper from $ p_1 / p_2 $ to $ p_1'/p_2 $

Increasing the price of one commodity pivots the constraint inward.

Some old choices are lost, so increasing one price could make the consumer worse off.
Ad Valorem Sales Taxes

An ad valorem sales tax levied at a rate of 5% increases all prices by 5%, from $ p $ to $ 1.05 p $ .

An ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+t)p.

BC under a uniform sales tax: $ (1+t) p_1 x_1 + (1+t) p_2 x_2 = m $

Do the graph!
Exercise: In kind gifts

Consumer receives $ g_1 $ units of good one as a gift.

Case 1: you can sell (trade) the gift if you want to.

Case 2: you cannot sell the gift.

Draw the budget line.
Exercise: The Food Stamp Program

Coupons that can be exchanged only for food.

How does a food stamp alter a family’s budget constraint?

Suppose $ m = {$}400 $ , $ p_F = {$}1 $ and the price of “other goods” is $ p_G = {$}1 $.

The budget constraint is then $ F + G = 400 $

Draw the budget line.
The Food Stamp Program
The Food Stamp Program
 What if food stamps can be traded on a black market for $0.50 each?
Other important cases

What if both, prices and income, double?

What if there are bulk discounts for units beyond a threshold?

What if there are quantity penalties for units beyond a threshold?