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1、Chapter TwoBudget Constraint预算约束预算约束StructureuDescribe budget constraintAlgebraGraphuDescribe changes in budget constraintuGovernment programs and budget constraintsuNon-linear budget linesConsumption SetsuA consumption set （消费集）消费集）is the collection of all physically possible consumption bundles （消
2、费束）消费束）to the consumer.Consumption BundleuA consumption bundle containing x1 units of commodity 1, x2 units of commodity 2 and so on up to xn units of commodity n is denoted by the vector (x1, x2, , xn).Physical ConstraintsuNon-negative: Consumption set: X= (x1, , xn) | x1 0, , xn 0 uYou only have 2
3、4 hours a dayuSubsistence needuEtc.Budget ConstraintuWhat constrains consumption choice?Budgetarytime other resource limitations.Budget ConstraintsuCommodity prices are p1, p2, , pn.uQ: When is a bundle (x1, , xn) affordable at prices p1, , pn?uA: When p1x1 + + pnxn mwhere m is the consumers (dispos
4、able) income.Budget ConstraintsuThe bundles that are only just affordable form the consumers budget constraint. This is the set (x1,xn) | x1 0, , xn 0 0 and p1x1 + + pnxn = = m .Budget ConstraintsuThe consumers budget set （预算集预算集）is the set of all affordable bundles;B(p1, , pn, m) = (x1, , xn) | x1
5、0, , xn 0 and p1x1 + + pnxn m uThe budget constraint is the upper boundary of the budget set.Budget Set and Constraint for Two Commoditiesx2x1Budget constraint isp1x1 + p2x2 = m. m /p1m /p2Budget Set and Constraint for Two Commoditiesx2x1Budget constraint isp1x1 + p2x2 = m.m /p2m /p1Budget Set and C
6、onstraint for Two Commoditiesx2x1Budget constraint isp1x1 + p2x2 = m.m /p1Just affordablem /p2Budget Set and Constraint for Two Commoditiesx2x1Budget constraint isp1x1 + p2x2 = m.m /p1Just affordableNot affordablem /p2Budget Set and Constraint for Two Commoditiesx2x1Budget constraint isp1x1 + p2x2 =
7、 m.m /p1AffordableJust affordableNot affordablem /p2Budget Set and Constraint for Two Commoditiesx2x1Budget constraint isp1x1 + p2x2 = m.m /p1BudgetSet the collection of all affordable bundles.m /p2Budget Set and Constraint for Two Commoditiesx2x1p1x1 + p2x2 = m is x2 = -(p1/p2)x1 + m/p2 so slope is
8、 -p1/p2.m /p1BudgetSetm /p2Meaning of the SlopeuuIncreasing x1 by 1 must reduce x2 by p1/p2.uOpportunity cost of consuming x1uOr, the rate of exchange that market allows.xppxmp21212= = Budget Constraintsx2x1Slope is -p1/p2+1-p1/p2Budget Constraintsx2x1+1-p1/p2Opp. cost of an extra unit of commodity
9、1 is p1/p2 units foregone of commodity 2.Budget Constraintsx2x1Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. Opp. cost of an extra unit of commodity 2 is p2/p1 units foregone of commodity 1. -p2/p1+1Budget Sets & Constraints; Income and Price ChangesuThe budget co
10、nstraint 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?Originalbudget setx2x1Higher income gives more choiceOriginalbudget setNew affordable consumptionchoicesx2x1Original andnew budgetco
11、nstraints areparallel (sameslope).How do the budget set and budget constraint change as income m decreases?Originalbudget setx2x1How do the budget set and budget constraint change as income m decreases?x2x1New, smallerbudget setConsumption bundlesthat are no longeraffordable.Old and newconstraintsar
12、e parallel.Budget Constraints - Income ChangesuIncreases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.uDecreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.
13、Budget Constraints - Income ChangesuNo original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off.uAn income decrease may (typically will) make the consumer worse off.Budget Constraints - Price ChangesuWhat happens if just one price dec
14、reases?uSuppose p1 decreases.How do the budget set and budget constraint change as p1 decreases from p1 to p1”?Originalbudget setx2x1m/p2m/p1m/p1”-p1/p2How do the budget set and budget constraint change as p1 decreases from p1 to p1”?Originalbudget setx2x1m/p2m/p1m/p1”New affordable choices-p1/p2How
15、 do the budget set and budget constraint change as p1 decreases from p1 to p1”?Originalbudget setx2x1m/p2m/p1m/p1”New affordable choicesBudget constraint pivots; slope flattens from -p1/p2 to -p1”/p2-p1/p2-p1”/p2Budget Constraints - Price ChangesuReducing the price of one commodity pivots the constr
16、aint outward. No old choice is lost and new choices are added, so reducing one price cannot make the consumer worse off.Budget Constraints - Price ChangesuSimilarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.Uniform Ad Val
17、orem Sales TaxesuAn ad valorem sales tax （从价营业税）从价营业税）levied at a rate of 5% increases all prices by 5%, from p to (1+0.05)p = 1.05p.uAn ad valorem sales tax levied at a rate of t increases all prices by tp from p to (1+t)p.uA uniform sales tax is applied uniformly to all commodities.Uniform Ad Valo
18、rem Sales TaxesuA uniform sales tax levied at rate t changes the constraint from p1x1 + p2x2 = mto (1+t)p1x1 + (1+t)p2x2 = mUniform Ad Valorem Sales TaxesuA uniform sales tax levied at rate t changes the constraint from p1x1 + p2x2 = mto (1+t)p1x1 + (1+t)p2x2 = mi.e. p1x1 + p2x2 = m/(1+t).Uniform Ad
19、 Valorem Sales Taxesx2x1mp2mp1p1x1 + p2x2 = mp1x1 + p2x2 = m/(1+t)mt p()11 mt p()12 Uniform Ad Valorem Sales Taxesx2x1mt p()12 mp2mt p()11 mp1Equivalent income lossismmtttm = = 11Uniform Ad Valorem Sales Taxesx2x1mt p()12 mp2mt p()11 mp1A uniform ad valoremsales tax levied at rate tis equivalent to
20、an incometax levied at ratett1 .The Food Stamp ProgramuFood stamps are coupons that can be legally exchanged only for food.uHow does a commodity-specific gift such as a food stamp alter a familys budget constraint?The Food Stamp ProgramuSuppose m = $100, pF = $1 and the price of “other goods” is pG
21、= $1.u“Other goods” is a composite good (复合商品复合商品)It simplifies the analysis to a 2-good model.uThe budget constraint is then F + G =100.The Food Stamp ProgramGF100100F + G = 100: before stamps.The Food Stamp ProgramF + G = 100: before stamps.Budget set after 40 foodstamps issued.GF10010014040(F-40)
22、 + G = 100 for F 40G=100 for F40The Food Stamp ProgramGF100100F + G = 100: before stamps.Budget set after 40 foodstamps issued.140The familys budgetset is enlarged.40(F-40) + G = 100 for F 40G=100 for F40The Food Stamp ProgramuWhat if food stamps can be traded on a black market for $0.50 each?uF+G=1
23、00+0.5 (40-F) for F40u(F-40)+G=100 for F 40The Food Stamp ProgramGF100100F + G = 100: before stamps.Budget constraint after 40 food stamps issued.140120Budget constraint with black market trading.40uF+G=100+0.5 (40-F) for F20.Shapes of Budget Constraints - Quantity DiscountsuSuppose p2 is constant a
24、t $1 but that p1=$2 for 0 x1 20 and p1=$1 for x120. uThen the constraints slope is - 2, for 0 x1 20-p1/p2 = - 1, for x1 20Shapes of Budget Constraints with a Quantity Discountm = $1005010020Slope = - 2 / 1 = - 2 (p1=2, p2=1)Slope = - 1/ 1 = - 1 (p1=1, p2=1)80 x2x1Shapes of Budget Constraints with a
25、Quantity Discountm = $1005010020Slope = - 2 / 1 = - 2 (p1=2, p2=1)Slope = - 1/ 1 = - 1 (p1=1, p2=1)80 x2x1Budget Constraints with a Quantity DiscountuThe constraint is 2x1+x2=m for 0 x1 20 2 20+(x1-20)+x2=m for x1 20Shapes of Budget Constraints with a Quantity Discountm = $100501002080 x2x1Budget Se
26、tBudget ConstraintShapes of Budget Constraints with a Quantity Penaltyx2x1Budget SetBudget ConstraintShapes of Budget Constraints - One Price NegativeuCommodity 1 is stinky garbage. You are paid $2 per unit to accept it; i.e. p1 = - $2. p2 = $1. Income, other than from accepting commodity 1, is m = $10.uThen the constraint is - 2x1 + x2 = 10 or x2 = 2x1 + 10.Shapes of Budget Constraints - One Price Negative10Budget constraints slope is-p1/p2 = -(-2)/1 = +2x2x1x2 = 2x1 + 10Shapes of Budget Constraints - One Price Negative10 x2x1 Budget set is all bundles for which x1 0,x2 0 andx2 2x1 + 10.