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?