Budget Constraint

Intermediate Microeconomics (Econ 100A)

UCSC

Consumption Choice Sets

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

  • What does constrain actual consumption?

    • Budget, time and other resource that are limited.

Consumption bundle

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

  • Prices of goods are denoted by $ p_1, p_2, … , p_n $, respectively.

  • We often use just $ p $ to denote the vector of prices.

Affordable Bundles - Budget Constraints

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

  • We assume all prices and income are strictly positive.

  • Question:

    • When is a consumption bundle $ (x_1, … , x_n) $ affordable at those given prices and with the given income?

Affordable Bundles - Budget Constraints

  • Answer:

    • when $ p_1 x_1 + … + p_n x_n \leq m $

    • That is, the bundle is affordable when the consumer purchases it and does not exhaust her income.

"Budget line" or "budget constraint"

  • The set of bundles that are just affordable in that they exhaust the consumer's income, conform the consumer’s budget constraint or budget line.

  • Formally, we represent this set as:

$ BL(p,m) = \{ ( x_1 ,…, x_n ) \quad:\quad p_1 x_1 + … + p_n x_n = m \} $

  • For simplicity, we will only work with consumption quantities $ x_1, … , x_n $ that 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 \} $

  • Notice that 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.

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 of the slope? It is the relative price.

Budget for Three Commodities

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?

  • Makler's EconGraphs

Introducing EconGraphs

Income Increases

  • Increases in income $m$ shift the constraint outward in a parallel manner, thereby enlarging the budget set and making more bundles affordable.

  • Decreases in income $m$ shift the constraint inward in a parallel manner, thereby shrinking the budget set and making fewer bundles affordable.

  • Which one is good for the consumer?

Income Changes

  • What bundles become unaffordable or newly affordable?

$ p_1 $ increases

  • If $ 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.

Price Changes

  • What bundles become unaffordable or newly affordable?

Ad Valorem Sales Taxes

  • In ad valorem taxes, we pay an additional percentage of price per-unit in 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 $

  • BC under a sales tax only on good 1 $ (1+t) p_1 x_1 + 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?