Chapter Two Budgetary and Other Constraints on Choice

Budget Constraints u Q: When is a consumption bundle (x 1, …, x n ) affordable at given prices p 1, …, p n ?  A: When the cost of the bundle p 1 x 1 + … + p n x n  m where m is the (disposable) income level of the consumer.

Budget Constraints  The consumer’s budget set is the set of all affordable consumption bundles; B(p 1, …, p n, m) = { (x 1, …, x n ) | x 1  0, …, x n  0 and p 1 x 1 + … + p n x n  m } u The budget constraint is the upper boundary of the budget set.

Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 m /p 2

Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Affordable Just affordable Not affordable m /p 2

Budget Set and Constraint for Two Commodities x2x2 x1x1 Budget constraint is p 1 x 1 + p 2 x 2 = m. m /p 1 Budget Set The budget set is the collection of all the affordable bundles. m /p 2

Budget Constraints u For n = 2 and x 1 on the horizontal axis, the slope of the budget constraint is -p 1 /p 2. What is the interpretation of this slope? u So increasing x 1 by 1 must reduce x 2 by p 1 /p 2.

Budget Constraints x2x2 x1x1 1 p 1 /p 2 Slope is -p 1 /p 2

Budget Constraints x2x2 x1x1 +1 -p 1 /p 2 So the opp. cost of an extra unit of commodity 1 is p 1 /p 2 units foregone of commodity 2.

How do the budget set and budget constraint change as income m increases? Original budget set x2x2 x1x1

Higher income gives more choice Original budget set New affordable consumption choices x2x2 x1x1 The original and the new budget constraints are parallel (same slope).

How do the budget set and budget constraint change as income m decreases? x2x2 x1x1 New, smaller budget set Consumption bundles that are no longer affordable. Old and new constraints are parallel.

Budget Constraints - Income Changes u Since no original choices are lost and new choices are added when income increases, an income increase cannot make the consumer worse off. u But an income decrease may (typically will) make the consumer worse off.

How do the budget set and budget constraint change as p 1 decreases from p 1 ’ to p 1 ”? Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” New affordable choices -p 1 ’/p 2

How do the budget set and budget constraint change as p 1 decreases from p 1 ’ to p 1 ”? Original budget set x2x2 x1x1 m/p 2 m/p 1 ’ m/p 1 ” New affordable choices Budget constraint pivots; slope flattens from -p 1 ’/p 2 to -p 1 ”/p 2 -p 1 ’/p 2 -p 1 ”/p 2

Budget Constraints - Price Changes u So reducing the price of just one commodity pivots the budget constraint outward. No old choices are lost and new choices are added, so reducing one price cannot make the consumer worse off and typically makes her better off.

Budget Constraints - Price Changes u Similarly, increasing just one price pivots the budget constraint inwards, reduces choice and may (typically will) make the consumer worse off.

Uniform Ad Valorem Sales Taxes  An ad valorem sales tax levied at a rate of 5% increases all prices by 5%, from p to (1+0  05)p = 1  05p. u An ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+t)p. u A uniform sales tax is applied uniformly to all commodities.

Uniform Ad Valorem Sales Taxes u So a uniform sales tax levied at rate t changes the budget constraint from p 1 x 1 + p 2 x 2 = m to (1+t)p 1 x 1 + (1+t)p 2 x 2 = m which is the same as p 1 x 1 + p 2 x 2 = m/(1+t).

Uniform Ad Valorem Sales Taxes x2x2 x1x1 p 1 x 1 + p 2 x 2 = m p 1 x 1 + p 2 x 2 = m/(1+t)

The Food Stamp Program u Food stamps are coupons that can be legally exchanged only for food. u These coupons are given to eligible families. u How does a commodity-specific gift such as a food stamp alter a family’s budget constraint?

The Food Stamp Program u Suppose m = $100, p F = $1 and the price of “other goods” is p G = $1. u The budget constraint is then F + G =100.

The Food Stamp Program G F 100 F + G = 100: before stamps.

The Food Stamp Program G F 100 F + G = 100: before stamps. Budget set after 40 food stamps issued. 140 The family’s budget set is enlarged. 40

The Food Stamp Program u What if food stamps can be traded on a black market for $0.50 each?

The Food Stamp Program G F 100 F + G = 100: before stamps. Budget constraint after 40 food stamps issued Budget constraint with black market trading. 40

Budget Constraints - Relative Prices u “Numeraire” means “unit of account”. u Suppose prices and income are measured in dollars. Say p 1 =$2, p 2 =$3, m = $12. Then the budget constraint is 2x 1 + 3x 2 = 12.

Budget Constraints - Relative Prices u If prices and income are measured in cents then p 1 =200, p 2 =300, m=1200 so the budget constraint is 200x x 2 = 1200 which is the same as 2x 1 + 3x 2 = 12. u Changing the numeraire changes neither the budget constraint nor the budget set.

Budget Constraints - Relative Prices u The constraint for p 1 =2, p 2 =3, m=12 2x 1 + 3x 2 = 12 is also x 1 + (3/2)x 2 = 6 which is the constraint for p 1 =1, p 2 =3/2, m=6. Setting p 1 =1 makes commodity 1 the numeraire and defines all prices relative to p 1 ; e.g. 3/2 is the price of commodity 2 relative to the price of commodity 1.

Budget Constraints - Relative Prices u So any commodity can be chosen as the numeraire without changing either the budget set or the budget constraint.

Shapes of Budget Constraints - Quantity Discounts  Suppose p 2 is constant at $1 but that p 1 =$2 for 0  x 1  20 and p 1 =$1 for x 1 >20. Then the slope of the budget constraint is - 2 for 0  x 1  20 -p 1 /p 2 = - 1 for x 1 > 20 and the budget constraint looks like {

Shapes of Budget Constraints with a Quantity Discount m = $ Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 x2x2 x1x1

Shapes of Budget Constraints with a Quantity Discount m = $ x2x2 x1x1 Budget Set Budget Constraint

Shapes of Budget Constraints with a Quantity Penalty x2x2 x1x1 Budget Set Budget Constraint

Shapes of Budget Constraints - One Price Negative u Suppose commodity 1 is stinky garbage and that you are paid $2 per unit to accept it; i.e. p 1 = - $2. Commodity 2 is the numeraire, so p 2 = $1. Income, other than from accepting commodity 1, is m = $10. u Then the budget constraint is - 2x 1 + x 2 = 10 or x 2 = 2x

Shapes of Budget Constraints - One Price Negative 10 Budget constraint’s slope is -p 1 /p 2 = -(-2)/1 = +2 x2x2 x1x1 x 2 = 2x

Shapes of Budget Constraints - One Price Negative 10 x2x2 x1x1 Budget set is all bundles for which x 1  0, x 2  0 and x 2  2x

More General Choice Sets u The set of bundles from which a consumer may choose is usually constrained by more than a budget -- there may be time constraints and other resources constraints. u A bundle is available for consumption only if it meets all of these constraints.

More General Choice Sets Food Other Stuff 10 Choice is constrained by the requirement that at least 10 units of food must be eaten to survive

More General Choice Sets Food Other Stuff 10 Budget Set Choice is also constrained by the budget.

More General Choice Sets Food Other Stuff 10 And choice is further restricted by a time constraint.

More General Choice Sets Food Other Stuff 10