## 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

• 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.

## 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

## 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?

## 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?