Varian_Chapter02_Budget Constraint

Click to edit Master title style,Click to edit Master text styles,Second Level,Third Level,Fourth Level,Fifth Level,*,*,*,Chapter Two,Budgetary and Other Constraints on Choice,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.,Budget Constraints,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,).,Commodity prices are p,1, p,2, , p,n,.,Budget Constraints,Q: When is a consumption bundle (x,1, , x,n,) affordable at given prices p,1, , p,n,?,Budget Constraints,Q: When is a bundle (x,1, , x,n,) affordable at prices p,1, , p,n,?,A: When p,1,x,1,+ + p,n,x,n,m,where,m,is the consumers (disposable) income.,Budget Constraints,The bundles that are only just affordable form the consumers,budget constraint,. This is the set (x,1,x,n,) | x,1,0, , x,n, 0,and p,1,x,1,+ + p,n,x,n,=,m,.,Budget Constraints,The consumers,budget set,is the set of all affordable 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,The budget constraint is the upper boundary of the budget set.,Budget Set and Constraint for Two Commodities,x,2,x,1,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,x,2,x,1,Budget constraint is,p,1,x,1,+ p,2,x,2,=,m.,m,/p,2,m,/p,1,Budget Set and Constraint for Two Commodities,x,2,x,1,Budget constraint is,p,1,x,1,+ p,2,x,2,=,m.,m,/p,1,Just affordable,m,/p,2,Budget Set and Constraint for Two Commodities,x,2,x,1,Budget constraint is,p,1,x,1,+ p,2,x,2,=,m.,m,/p,1,Just affordable,Not affordable,m,/p,2,Budget Set and Constraint for Two Commodities,x,2,x,1,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,x,2,x,1,Budget constraint is,p,1,x,1,+ p,2,x,2,=,m.,m,/p,1,Budget,Set,the collection of all affordable bundles.,m,/p,2,Budget Set and Constraint for Two Commodities,x,2,x,1,p,1,x,1,+ p,2,x,2,=,m,is,x,2,= -(p,1,/p,2,)x,1,+,m,/p,2,so slope is -p,1,/p,2,.,m,/p,1,Budget,Set,m,/p,2,Budget Constraints,If n = 3 what do the budget constraint and the budget set look like?,Budget Constraint for Three Commodities,x,2,x,1,x,3,m,/p,2,m,/p,1,m,/p,3,p,1,x,1,+ p,2,x,2,+ p,3,x,3,=,m,Budget Set for Three Commodities,x,2,x,1,x,3,m,/p,2,m,/p,1,m,/p,3, (x,1,x,2,x,3,) | x,1,0, x,2,0, x,3,0 and,p,1,x,1,+ p,2,x,2,+ p,3,x,3,m,Budget Constraints,For n = 2 and x,1,on the horizontal axis, the constraints slope is -p,1,/p,2,. What does it mean?,Budget Constraints,For n = 2 and x,1,on the horizontal axis, the constraints slope is -p,1,/p,2,. What does it mean?,Increasing x,1,by 1 must reduce x,2,by p,1,/p,2.,Budget Constraints,x,2,x,1,Slope is -p,1,/p,2,+1,-p,1,/p,2,Budget Constraints,x,2,x,1,+1,-p,1,/p,2,Opp,. cost of an extra unit of commodity 1 is p,1,/p,2,units foregone of commodity 2.,Budget Constraints,x,2,x,1,Opp,. cost of an extra unit of commodity 1 is p,1,/p,2,units foregone of commodity 2. And the opp. cost of an extra unit of commodity 2 is p,2,/p,1,units foregone of commodity 1.,-p,2,/p,1,+1,Budget Sets Income and Price Changes,The budget constraint and budget set depend upon prices and income. What happens as prices or income change?,How do the budget set and budget constraint change as income,m,increases?,Original,budget set,x,2,x,1,Higher income gives more choice,Original,budget set,New affordable consumptionchoices,x,2,x,1,Original and,new budget,constraints are,parallel (same,slope).,How do the budget set and budget constraint change as income,m,decreases?,Original,budget set,x,2,x,1,How do the budget set and budget constraint change as income,m,decreases?,x,2,x,1,New, smaller,budget set,Consumption bundles,that are no longer,affordable.,Old and new,constraints,are parallel.,Budget Constraints - Income Changes,Increases in income,m,shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.,Budget Constraints - Income Changes,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.,Budget Constraints - Income Changes,No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off.,An income decrease may (typically will) make the consumer worse off.,Budget Constraints - Price Changes,What happens if just one price decreases?,Suppose p,1,decreases.,How do the budget set and budget constraint change as,p,1,decreases from,p,1,to,p,1,”,?,Original,budget set,x,2,x,1,m/p,2,m/p,1,m/p,1,”,-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,x,2,x,1,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,x,2,x,1,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,Reducing the price of one commodity pivots the constraint outward,. No old choice is lost and new choices are added, so reducing one price cannot make the consumer worse off.,Budget Constraints - Price Changes,Similarly, increasing one price pivots the 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.,An,ad valorem,sales tax levied at a rate of,t,increases all prices by,t,p from p to (1+,t,)p.,A uniform sales tax is applied uniformly to all commodities.,Uniform,Ad Valorem,Sales Taxes,A uniform sales tax levied at rate,t,changes the 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,Uniform,Ad Valorem,Sales Taxes,A uniform sales tax levied at rate,t,changes the 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,i.e. p,1,x,1,+ p,2,x,2,=,m,/(1+,t,).,Uniform,Ad Valorem,Sales Taxes,x,2,x,1,p,1,x,1,+ p,2,x,2,=,m,Uniform,Ad Valorem,Sales Taxes,x,2,x,1,p,1,x,1,+ p,2,x,2,=,m,p,1,x,1,+ p,2,x,2,= m/(1+,t,),Uniform,Ad Valorem,Sales Taxes,x,2,x,1,Equivalent income lossis,Uniform,Ad Valorem,Sales Taxes,x,2,x,1,A uniform,ad valorem,sales tax levied at rate,t,is equivalent to an incometax levied at rate,The Food Stamp Program,Food stamps are coupons that can be legally exchanged only for food.,How does a commodity-specific gift such as a food stamp alter a familys budget constraint?,The Food Stamp Program,Suppose m = $100, p,F,= $1 and the price of “other goods” is p,G,= $1.,The budget constraint is then F + G =100.,The Food Stamp Program,G,F,100,100,F + G = 100; before stamps.,The Food Stamp Program,G,F,100,100,F + G = 100: before stamps.,The Food Stamp Program,G,F,100,100,F + G = 100: before stamps.,Budget set after 40 foodstamps issued.,140,40,The Food Stamp Program,G,F,100,100,F + G = 100: before stamps.,Budget set after 40 foodstamps issued.,140,The familys budgetset is enlarged.,40,The Food Stamp Program,What if food stamps can be traded on a black market for $0.50 each?,The Food Stamp Program,G,F,100,100,F + G = 100: before stamps.,Budget constraint after 40 food stamps issued.,140,120,Budget constraint with black market trading.,40,The Food Stamp Program,G,F,100,100,F + G = 100: before stamps.,Budget constraint after 40 food stamps issued.,140,120,Black market trading makes the budget set larger again.,40,Budget Constraints - Relative Prices,“Numeraire” means “unit of account”.,Suppose prices and income are measured in dollars. Say p,1,=$2, p,2,=$3,m,= $12. Then the constraint is 2x,1,+ 3x,2,= 12.,Budget Constraints - Relative Prices,If prices and income are measured in cents, then p,1,=200, p,2,=300,m,=1200 and the constraint is 200x,1,+ 300x,2,= 1200,the same as 2x,1,+ 3x,2,= 12.,Changing the numeraire changes neither the budget constraint nor the budget set,.,Budget Constraints - Relative Prices,The constraint for p,1,=2, p,2,=3,m,=12 2x,1,+ 3x,2,= 12 is also 1.x,1,+ (3/2)x,2,= 6,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,Any,commodity can be chosen as the numeraire without changing the budget set or the budget constraint.,Budget Constraints - Relative Prices,p,1,=2, p,2,=3 and p,3,=6,price of commodity 2 relative to commodity 1 is 3/2,price of commodity 3 relative to commodity 1 is 3.,Relative prices are the,rates of exchange,of commodities 2 and 3 for units of commodity 1.,Shapes of Budget Constraints,Q: What makes a budget constraint a straight line?,A: A straight line has a constant slope and the constraint is p,1,x,1,+ + p,n,x,n,=,m,so if prices are constants then a constraint is a straight line.,Shapes of Budget Constraints,But what if prices are not constants?,E.g.,bulk buying discounts, or price penalties for buying “too much”.,Then constraints will be curved.,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.,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 constraints slope is - 2, for 0,x,1,20-p,1,/p,2,= - 1, for x,1, 20and the constraint is,Shapes of Budget Constraints with a Quantity Discount,m,= $100,50,100,20,Slope = - 2 / 1 = - 2 (p,1,=2, p,2,=1),Slope = - 1/ 1 = - 1 (p,1,=1, p,2,=1),80,x,2,x,1,Shapes of Budget Constraints with a Quantity Discount,m,= $100,50,100,20,Slope = - 2 / 1 = - 2 (p,1,=2, p,2,=1),Slope = - 1/ 1 = - 1 (p,1,=1, p,2,=1),80,x,2,x,1,Shapes of Budget Constraints with a Quantity Discount,m,= $100,50,100,20,80,x,2,x,1,Budget Set,Budget Constraint,Shapes of Budget Constraints with a Quantity Penalty,x,2,x,1,Budget Set,Budget Constraint,Shapes of Budget Constraints - One Price Negative,Commodity 1 is stinky garbage. You are paid $2 per unit to accept it;,i.e.,p,1,= - $2. p,2,= $1. Income, other than from accepting commodity 1, is,m,= $10.,Then the constraint is - 2x,1,+ x,2,= 10 or x,2,= 2x,1,+ 10.,Shapes of Budget Constraints - One Price Negative,10,Budget constraints slope is,-p,1,/p,2,= -(-2)/1 = +2,x,2,x,1,x,2,= 2x,1,+ 10,Shapes of Budget Constraints - One Price Negative,10,x,2,x,1,Budget set is all bundles for which x,1,0,x,2,0 and,x,2,2x,1,+ 10.,More General Choice Sets,Choices are usually constrained by more than a budget; e.g. time constraints and other resources constraints.,A bundle is available only if it meets,every,constraint.,More General Choice Sets,Food,Other Stuff,10,At least 10 units of food,must be eaten to survive,More General Choice Sets,Food,Other Stuff,10,Budget Set,Choice is also budgetconstrained.,More General Choice Sets,Food,Other Stuff,10,Choice is further restricted by a time constraint.,More General Choice Sets,So what is the choice set?,More General Choice Sets,Food,Other Stuff,10,More General Choice Sets,Food,Other Stuff,10,More General Choice Sets,Food,Other Stuff,10,The choice set is the,intersection of all of,the constraint sets.,