Budget Constraints. Consumption Choice Sets. Chapter Two. Budget Constraints. Budget Constraints. Budget Constraints. Budget Constraints

Transcription

1 Consuption Choice Sets Chapter Two A consuption choice set is the collection of all consuption choices available to the consuer. What constrains consuption choice? Budgetary, tie and other resource liitations. A consuption bundle containing units of coodity 1, units of coodity 2 and so on up to x n units of coodity n is denoted by the vector (,,, x n ). Coodity prices are p 1, p 2,, p n. Q: When is a consuption bundle (,, x n ) affordable at given prices p 1,, p n? Q: When is a bundle (,, x n ) affordable at prices p 1,, p n? A: When p p n x n where is the consuer s (disposable) incoe. The bundles that are only just affordable for the consuer s budget constraint. This is the set { (,,x n ) 0,, x n 0and p p n x n = }. 1

2 The consuer s is the set of all affordable bundles; B(p 1,, p n, ) = { (,, x n ) 0,, x n 0 and p p n x n } The budget constraint is the upper boundary of the. Budget Set and Constraint for Two Coodities /p 2 Budget constraint is p 1 + p 2 =. /p 1 Budget Set and Constraint for Two Coodities /p 2 Budget constraint is p 1 + p 2 =. Budget Set and Constraint for Two Coodities /p 2 Budget constraint is p 1 + p 2 =. Just affordable /p 1 /p 1 Budget Set and Constraint for Two Coodities /p 2 Budget constraint is p 1 + p 2 =. Not affordable Just affordable Budget Set and Constraint for Two Coodities /p 2 Budget constraint is p 1 + p 2 =. Not affordable Just affordable Affordable /p 1 /p 1 2

3 Budget Set and Constraint for Two Coodities /p 2 Budget Set Budget constraint is p 1 + p 2 =. the collection of all affordable bundles. Budget Set and Constraint for Two Coodities /p 2 Budget Set p 1 + p 2 = is = -(p 1 /p 2 ) + /p 2 so slope is -p 1 /p 2. /p 1 /p 1 If n = 3 what do the budget constraint and the look like? /p 2 Budget Constraint for Three Coodities p 1 + p 2 + p 3 x 3 = /p 3 x 3 /p 1 /p 2 Budget Set for Three Coodities { (,,x 3 ) 0, 0, x 3 0 and p 1 + p 2 + p 3 x 3 } /p 3 x 3 For n = 2 and on the horizontal axis, the constraint s slope is -p 1 /p 2. What does it ean? p x p x 2 = 1 + p /p 1 3

4 For n = 2 and on the horizontal axis, the constraint s slope is -p 1 /p 2. What does it ean? p x p x 2 = 1 + p Increasing by 1 ust reduce by p 1 /p 2. Slope is -p 1 /p 2 -p 1 /p 2 +1 Opp. cost of an extra unit of coodity 1 is p 1 /p 2 units foregone of coodity 2. -p 1 /p 2 +1 Opp. cost of an extra unit of coodity 1 is p 1 /p 2 units foregone of coodity 2. And +1 -p 2 /p 1 the opp. cost of an extra unit of coodity 2 is p 2 /p 1 units foregone of coodity 1. Budget Sets & Constraints; Incoe and Price Changes The budget constraint and budget set depend upon prices and incoe. What happens as prices or incoe change? How do the and budget constraint change as incoe increases? Original 4

5 Higher incoe gives ore choice New affordable consuption choices Original and new budget constraints are parallel (sae Original slope). How do the and budget constraint change as incoe decreases? Original How do the and budget constraint change as incoe decreases? New, saller Consuption bundles that are no longer affordable. Old and new constraints are parallel. - Incoe Changes Increases in incoe shift the constraint outward in a parallel anner, thereby enlarging the and iproving choice. - Incoe Changes Increases in incoe shift the constraint outward in a parallel anner, thereby enlarging the and iproving choice. Decreases in incoe shift the constraint inward in a parallel anner, thereby shrinking the and reducing choice. - Incoe Changes No original choice is lost and new choices are added when incoe increases, so higher incoe cannot ake a consuer worse off. An incoe decrease ay (typically will) ake the consuer worse off. 5

6 - Price Changes What happens if just one price decreases? Suppose p 1 decreases. How do the and budget constraint change as p 1 decreases fro p 1 to p 1? /p 2 -p 1 /p 2 Original /p 1 /p 1 How do the and budget constraint change as p 1 decreases fro p 1 to p 1? /p 2 -p 1 /p 2 New affordable choices Original /p 1 /p 1 How do the and budget constraint change as p 1 decreases fro p 1 to p 1? /p 2 -p 1 /p 2 New affordable choices Original Budget constraint pivots; slope flattens fro -p 1 /p 2 to -p -p 1 /p 2 1 /p 2 /p 1 /p 1 - Price Changes Reducing the price of one coodity pivots the constraint outward. No old choice is lost and new choices are added, so reducing one price cannot ake the consuer worse off. - Price Changes Siilarly, increasing one price pivots the constraint inwards, reduces choice and ay (typically will) ake the consuer worse off. 6

7 Unifor Ad Valore Sales Taxes An ad valore sales tax levied at a rate of 5% increases all prices by 5%, fro p to (1+0 05)p = 1 05p. An ad valore sales tax levied at a rate of t increases all prices by tp fro p to (1+t)p. A unifor sales tax is applied uniforly to all coodities. Unifor Ad Valore Sales Taxes A unifor sales tax levied at rate t changes the constraint fro p 1 + p 2 = to (1+t)p 1 + (1+t)p 2 = Unifor Ad Valore Sales Taxes A unifor sales tax levied at rate t changes the constraint fro p 1 + p 2 = to (1+t)p 1 + (1+t)p 2 = i.e. p 1 + p 2 = /(1+t). Unifor Ad Valore Sales Taxes p 2 p 1 + p 2 = p 1 Unifor Ad Valore Sales Taxes p 2 ( 1+ t) p2 p 1 + p 2 = p 1 + p 2 = /(1+t) Unifor Ad Valore Sales Taxes p 2 ( 1+ t) p2 Equivalent incoe loss is t 1+ t = 1+ t ( 1+ t) p1 p 1 ( 1+ t) p1 p 1 7

8 Unifor Ad Valore Sales Taxes p 2 ( 1+ t) p2 A unifor ad valore sales tax levied at rate t is equivalent to an incoe tax levied at rate t 1+ t. The Food Stap Progra Food staps are coupons that can be legally exchanged only for food. How does a coodity-specific gift such as a food stap alter a faily s budget constraint? ( 1+ t) p1 p 1 The Food Stap Progra Suppose = $, p F = $1 and the price of other goods is p G = $1. The budget constraint is then F + G =. G The Food Stap Progra F + G = ; before staps. F G The Food Stap Progra F + G = : before staps. G The Food Stap Progra F + G = : before staps. Budget set after 40 food staps issued. F F 8

9 G The Food Stap Progra F + G = : before staps. Budget set after 40 food staps issued. The faily s budget set is enlarged. The Food Stap Progra What if food staps can be traded on a black arket for $0.50 each? F G 120 The Food Stap Progra F + G = : before staps. Budget constraint after 40 food staps issued. Budget constraint with black arket trading. G 120 The Food Stap Progra F + G = : before staps. Budget constraint after 40 food staps issued. Black arket trading akes the budget set larger again F F - Relative Prices Nueraire eans unit of account. Suppose prices and incoe are easured in dollars. Say p 1 =$2, p 2 =$3, = $12. Then the constraint is = Relative Prices If prices and incoe are easured in cents, then p 1 =200, p 2 =300, =1200 and the constraint is = 1200, the sae as = 12. Changing the nueraire changes neither the budget constraint nor the. 9

10 - Relative Prices The constraint for p 1 =2, p 2 =3, = = 12 is also 1. + (3/2) = 6, the constraint for p 1 =1, p 2 =3/2, =6. Setting p 1 =1 akes coodity 1 the nueraire and defines all prices relative to p 1 ; e.g. 3/2 is the price of coodity 2 relative to the price of coodity 1. - Relative Prices Any coodity can be chosen as the nueraire without changing the or the budget constraint. - Relative Prices p 1 =2, p 2 =3 and p 3 =6 price of coodity 2 relative to coodity 1 is 3/2, price of coodity 3 relative to coodity 1 is 3. Relative prices are the rates of exchange of coodities 2 and 3 for units of coodity 1. Shapes of Q: What akes a budget constraint a straight line? A: A straight line has a constant slope and the constraint is p p n x n = so if prices are constants then a constraint is a straight line. Shapes of But what if prices are not constants? E.g. bulk buying discounts, or price penalties for buying too uch. Then constraints will be curved. Shapes of - Quantity Discounts Suppose p 2 is constant at $1 but that p 1 =$2 for 0 20 and p 1 =$1 for >20.

11 Shapes of - Quantity Discounts Suppose p 2 is constant at $1 but that p 1 =$2 for 0 20 and p 1 =$1 for >20. Then the constraint s slope is - 2, for p 1 /p 2 = {- 1, for > 20 and the constraint is Shapes of with a Quantity Discount 20 Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) 50 = $ Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 Shapes of with a Quantity Discount 20 Slope = - 2 / 1 = - 2 (p 1 =2, p 2 =1) 50 = $ Slope = - 1/ 1 = - 1 (p 1 =1, p 2 =1) 80 Shapes of with a Quantity Discount = $ Budget Set Budget Constraint Shapes of with a Quantity Penalty Budget Set Budget Constraint Shapes of - One Price Negative Coodity 1 is stinky garbage. You are paid $2 per unit to accept it; i.e. p 1 = - $2. p 2 = $1. Incoe, other than fro accepting coodity 1, is = $. Then the constraint is -2 + = or =

12 Shapes of - One Price Negative = 2 + Budget constraint s slope is -p 1 /p 2 = -(-2)/1 = +2 Shapes of - One Price Negative Budget set is all bundles for which 0, 0 and 2 +. Choices are usually constrained by ore than a budget; e.g. tie constraints and other resources constraints. A bundle is available only if it eets every constraint. Other Stuff At least units of food ust be eaten to survive Food Other Stuff Choice is also budget constrained. Other Stuff Choice is further restricted by a tie constraint. Budget Set Food Food 12

13 Other Stuff So what is the choice set? Food Other Stuff Other Stuff The choice set is the intersection of all of the constraint sets. Food Food 13