2 A Utility Function Mathematically Representing Preferences U=U(x, y)U(A)YU(B)Utility functions haveAIndifference curvesdescribe bundle-orderingpreferencesBX
3 Consumption Opportunities: The Budget Constraint Assume that an individual has I dollars to allocate between good x and good ypxx + pyy MyThe individual can affordto choose only combinationsof x and y in the shadedtrianglex
4 The Budget ConstraintMC of consuming one more unit of x, the amount of y that must be foregone.The slope of the budget line is this MC.yThe slope is the changein y for a one unit increasein the consumption of x.If Px = 10 and Py = 5, thenconsuming one more x meansconsuming two less y.x
5 Not an indifference curve! Maximizing UtilityKeep buying x until the MB(x) = MC(x)Interaction of…Preferences, diminishing MB because of diminishing MRS. MB = MRSMB in terms of y willing to be given upIn dollars, MB = py*MRSMC of x = px/pyMC in terms of Y given upIn dollars, MC = px.MB, MCXX*Not an indifference curve!MCMB
6 Optimization Principle To maximize utility, given a fixed amount of income, an individual will buy the goods and services:That exhaust total incomeSavings or borrowing is allowed (if we modify the budget constraint to include a temporal component)So long as MB(x) ≥ MC(x), MB(y) ≥ MC(y), etc.Or, until MB(x) = MC(x), MB(y) = MC(y)
7 IntuitionMRS is the maximum amount of y the person is willing to give up to consume more x; the definition of MB.px/py tells us the number of units of y that must be given up to consume one more x; the definition of MC.yxABAt “A”, MRS>Px/Py (MB > MC),You are willing to pay more than you have to, consumer surplus increases.Utility and consumer surpluscan be increased by consuming more x.At “B”, MRS<Px/Py, (MB < MC)Utility and consumer surpluscan be increased by consumingless x.Px = 10Py = 5Slope of budget line = -2U0
8 IntuitionAt “C”, the MB = MC for the last unit of both goods consumed.That is, at “C”, MRS = px/py, oryACU1BU0x
9 OptimizationUnconstrained optimization is a lot easier to solve than constrained optimization.Substitution: maximize the cross section of U along the budget lineLagrange method
10 SubstitutionThis turns the constrained optimization into an unconstrained problem.Find the equation for the cross section of the U=U(x,y) above the budget line and maximize it -- i.e. find the top of the parabolaUyy*x*x
11 SubstitutionSubstitute and maximizeAnd substituteagain
12 Problem with this method It can get very mathematically complicated very quickly.Even U=xαyβ gets very tricky to solve.
13 LaGrange MethodLaGrange knew that unconstrained optimization (like profit max) is relatively simple compared to constrained optimization.Taking what he knew unconstrained optimization he attempted to simplify the constrained maximization problem by making it mimic the unconstrained problem.
14 Unconstrained Optimization Example Profit maximization is an example of this. We maximize the difference between two functions: π=R(q) – C(q).
15 Lagrange MethodLaGrange wanted to find a way that was simpler than constrained optimization and more workable than simple substitution.He wanted to make constrained optimization take the form of the simpler unconstrained problem.First, let’s look at a simpler problem.
16 Maximize Utility - Expenditure Maximize utility minus the cost of buying bundles. Think about a one-good world.UU=U(x)slope = UxExpenditure = E = pxxslope = Ex= pxProblems:Ux not measured in $ like E.E is not constrained, we can spend as much as we like.xx*
17 Maximizing Utility - Expenditure First change the expenditure function by multiplying px by λ. Now call that function EU.We want λ to measure the marginal utility of $1.In that case, units of x consumed would cost us utility and both U(x) and EU(x) would be measured in the same units.UU=U(x)slope = UxEU=λpxxSlope = EUx= λpxxx*
18 Maximizing Utility - Expenditure Problem, an infinite number of λ choices that will each solve this with a different x*EUx = λ1 pxUU=U(x)EUx= λ2 pxEUx = λ3 pxxx*x*x*Now the slope of the expenditure function and expenditure aremeasured in utils, not dollars. But we are not constraining x yet.
19 LaGrange MethodSo first subtract λM from the expenditure function to getEL = λpxx - λMUU=U(x)slope = UxEU = λpxxEL = λpxx - λMx*xslope = ELx= λpx-λM
20 LaGrange MethodWe know we want to find the x* such that that distance between U(x) and EL(x*) = U(x*). That is, where EL(x*) = 0So we maximize v = U(x) - 0Substitute λpxx – λM = 0 in for 0, to constrain x* to our budget.UU=U(x)slope = UxU=U(x*)-0EL = λpxx - λMxx*slope = ELx= λ px-λI
21 LaGrange MethodOur optimization becomes an unconstrained problem by including the requirement that λpxx = λM.λ is chosen along with x to maximize utility so that λ = the marginal utility of $1. That is, λpx = Ux.UU=U(x)slope = UxU=U(x*)-0EL = λ(pxx – M)xx*slope = ELx= λ px-λM
25 Basic Demand AnalysisUsing Lagrangian to generate ordinary (Marshallian) demand curves.FOCs necessarySOCs sufficient (check that they hold)Ordinary (Marshallian) demand curvesInverse demand curvesMeaning of λIndirect UtilityExpenditure FunctionComparative Statics General Results
26 Demand Functions using Lagrange’s Method Set up and maximize:λ* chosen so that the constraint plane isparallel to the utility function.Any x* and y* that maximizes utilitywill also have to exhaust income.
27 FOCs for an OptimumFor utility to be maximized, it is necessary that the indifference curve is tangent to the budget constraint (as above).But it is not sufficient, we also need a diminishing MRS.FOCs satisfiedUtility MaximizedyyyUtility MaximizedFOCs satisfiedxxx
28 SOCs for an Optimum Sufficient condition for a maximum to exist If the MRS is non-increasing (utility function quasi concave) for all x, that is sufficient for a maximum to exist – but it may not be unique.If the MRS is diminishing (utility function strictly quasi concave) for all x, that is sufficient for a unique maximum to exist. Need this to satisfy second order conditions for maximization.SOCs do not holdUtility MaximizedyyySOCs satisfiedxxx
29 Expenditure Minimization: SOC The FOC ensure that the optimal consumption bundle is at a tangency.The SOC ensure that the tangency is a minimum, and not a maximum by ensuring that away from the tangency, along the budget line, utility falls.YU*>U’U=U*U=U’X
30 Checking SOC: utility function strictly quasi-concave The second order conditions will hold if the utility function is strictly quasi-concaveA function is strictly quasi-concave if its bordered Hessian is negative definite. That is:A function is strictly quasi-concave if:-UxUx < 02UxUxyUy - Uy2Uxx - Ux2Uyy > 0Binger and Hoffman, page 115
31 Checking SOC: Constrained Maximization The second order condition for constrained maximization will hold if the following bordered Hessian matrix is negative definite:Note:So this Hessian andThe last only differby
33 Inverse Demand CurvesStarting with the ordinary demand curves:
34 Utility and Indirect Utility Maximum Utility, a function of quantitiesIndirect Utility a function of price and income
35 Optimization : Expenditure Function Start with indirect utilitySolve for M:This equation determines the expenditure needed to generate Ū, the expenditure function:
36 Digression: Envelope Theorem Say we know that y = f(x; ω)We find y is maximized at x* = x(ω)So we know that y* = y(x*=x(ω),ω)).Now say we want to find outSo when ω changes, the optimal x changes, which changes the y* function.Two methods to solve this…
37 Digression: Envelope Theorem Start with: y = f(x; ω) and calculate x* = x(ω)First option:y = f(x; ω), substitute in x* = x(ω) to get y* = y(x(ω); ω):Second option, turn it around:First, take then substitute x* = x(ω)into yω(x ; ω) to getAnd these two answers are equivalent:
38 Envelope Result, 1st option to get Plug the optimal values into the LaGrangianSilberburg, first edition, page has a similar treatment.
39 Optimization: Envelope Result Plug the optimal values into the LaGrangianSilberburg, first edition, page has a similar treatment.
40 Optimization: Comparative Statics If we have a specified utility function and we derive the equations for the demand functions, the comparative statics are easy.Take the derivatives to calculate the changes in x and y when prices or income change.However, what if all we know is U = U(x, y) and we feel safe only assuming:Ux > 0 Uy > 0 Uxx < 0 Uyy < 0Can we get anything from that?
43 Tells us that if income increases by $1, so will total expenditure. Comparative Statics: Effect of a change in M Differentiate (1), (2), (3) w.r.t. MSide note:Tells us that if income increases by $1, so will total expenditure.
44 Comparative Statics: Effect of a change in M Put in Matrix Notation Solve for
45 Comparative Statics: Effect of a change in I Put in Matrix Notation Solve for
46 Comparative Statics: Effect of a change in px Differentiate (1), (2), (3) w.r.t. px
47 Comparative Statics: Effect of a change in px Put in Matrix Notation Solve for
48 Comparative Statics: Effect of a change in px Put in Matrix Notation… AGAIN Solve for
49 Comparative Statics: Preview of income and substitution effects RearrangetheseSub intheseIncomeeffectmatters
50 Specific Utility Functions Cobb-DouglasCESPerfect Compliments
51 Cobb-Douglas: Utility Max Problem:Set up the LaGrangianFOC
52 Cobb-Douglas: Demand FOC Imply, to maximize utility, these must hold. Plug into the budget constraint to get the ordinary (Marshallian) demand functions:Note, demand only a function of own price changes (one Cobb-Douglas weakness)
53 Cobb-Douglas: DemandPreferences are homothetic (only a function of the ratio of y:x). When income rises, optimal bundle along a ray from the origin.Expenditure a constant proportion of incomeIncome elasticities are = 1
54 Cobb-Douglas: Indirect Utility Plug x* and y* into the utility function
55 Cobb-Douglas: Expenditure Function Start with indirect utility functionSolve for M, and then rename it E
56 CES: Utility MaxProblem:Set up the LagrangianFOC
57 CES: DemandFOC ImplyPlug into the budget constraint and solve:
58 CES: Indirect UtilityPlug x* and y* into the utility function
59 CES: Expenditure Function Solve for M, then rename E.
60 CES: Expenditure Function (cont) Solve for M, then rename E.
61 Perfect Compliments: Utility Max Problem:No Lagrangian, just exhaust income such that:So plug this condition into the budget equationEssentially, we substitute the expansion path into the budget line equation.
63 Perfect Compliments: Indirect Utility Since utility from x = utility from y at utility max:
64 Perfect Compliments: Expenditure Function Since the utility from consumption of each must be equal,
65 Bonus Topics Money Metric Utility Function Homogeneity Corner Solutions (Kuhn-Tucker)Lump-Sum PrincipleMRS and MRT (Marginal Rate of Transformation – slope of PPF)
66 Money Metric Utility Function Start with an expenditure function and replace with Ū with U=U(x,y)Now we know the minimum expenditure to get the same utility as the bundle x’, y’.That is, for an x’, y’, this function tells us the cost of being at the tangency on the same indifference curve. As all bundles on the same curve get the same min. expenditure and prefernece ordering is preserved, this is a utility function.
67 Evaluating Housing Policy Assume you find the poor generally expend 1/3 of income on housing.The government wants to double the quality of housing the poor consume at the same 1/3 their income.How to evaluate?
68 Pre-Public HousingYIf expenditure on housing is generally 1/3 of income, assumeU=h1/3y2/3ph=$1px=$1M=$1,000667H333667
69 Public Housing Y ph=$1 px=$1 M=$1,000 Qualified citizens get 667 housing units for 1/3 of income ($333)667H333667
70 Public Housing: Money Metric Utility ph=$1px=$1M=$1,000Qualified citizens get 667housing units for 1/3 of income ($333)667UEPH=1,261UE=1,000H333667The extra housing has a value to the poor of $261. Depending on the cost to thegovernment of providing the housing, the program can be evaluated.
71 xi* = xi(p1,p2,…,pn,M) = xi(tp1,tp2,…,tpn,tM) HomogeneityIf all prices and income were doubled, the optimal quantities demanded will not changethe budget constraint is unchangedxi* = xi(p1,p2,…,pn,M) = xi(tp1,tp2,…,tpn,tM)Individual demand functions are homogeneous of degree zero in all prices and incomeTo test, replace all prices and income with t*p and t*M. The quantity demanded should be unaffected (all the “t” should cancel out).
72 Corner SolutionsNon-corner solutions: Optimal bundle will be where x > 0, y > 0 and MRS = px/pyYTwo corner solutions: Optimal bundle will be where x =0X
73 Corner SolutionsYAXAt “A”, MRS = px/py, but the optimal quantity of X = 0
74 Corner SolutionsYBXAt “B”, the tangency condition holds where x* < 0. Given the budget line, the optimal feasible x is where x = 0 and MRS < px/py
75 Corner SolutionTo develop these conditions as part of a Lagrangian equation, we add non-negativity constraints: 𝑥= 𝑠 2 (we could add 𝑦= 𝑡 2 if we wanted to be really thorough).Lagrangian:L=𝑈(𝑥,𝑦)+𝜆(M− 𝑝 𝑥 𝑥− 𝑝 𝑦 𝑦)+𝜇(𝑥− 𝑠 2 )Requiring x = s2 is simply a way of ensuring that x ≥ 0.
86 Kuhn-Tucker ResultIn this example, at x = 0, y = 40, 0 :
87 How about a feasible optimum? Optimum where x=0, y=40YX
88 Solving Kuhn-TuckerIf you find that the optimal bundle is not on the budget constraint, check all corners for a maximum utility.
89 Lump Sum Tax Taxing a good vs taxing income Tax on x only Lump sum tax (income tax)
90 Lump Sum TaxDifference in the budget lines: sales tax
91 Lump Sum Principle A tax on x rotates the budget line to have: Y y*
92 Lump Sum TaxDifference in the budget lines: income tax
93 Lump Sum PrincipleTax paid = x*τ. Alternatively, an income tax of that same amount would shift the budget line so that the consumer can just afford the same bundle they chose under the tax on x.Yy*Xx*
94 Lump Sum PrincipleWhenIndirect Utility function is:
95 Lump Sum PrincipleSet, I=60, py = 2, px=1Utility is:
96 Lump Sum Principle With a $1 tax on x, Utility is: And tax revenue is $20.
97 Lump Sum PrincipleWith a $20 income tax,Utility is:68.85 > 63.24
98 The Lump Sum Principle: Perfect Compliments If the utility function is U = Min(x,4y) then plug x=4y into the budget equation to get:The indirect utility function is then:
99 The Lump Sum Principle: Perfect Compliments Set, I=60, py = 2, px=1Utility is:
100 The Lump Sum Principle: Perfect Compliments With a $1 tax on x,Utility is:
101 The Lump Sum Principle: Perfect Compliments With a $24 income tax,Utility is:Since preferences do not allow substitution, the consumer makes the same choice either way and the tax does not affect utility.
102 MRS=MRT Using Varian’s example about milk and butter B* and M* may be different for all consumers.However, the MRS at the tangency is the same for ALL consumers, no matter the income or preferences.MilkPb=3; Pm=1MRS=3MRS=3MRS=3Butter
103 Marginal Rate of Transformation (a.k.a Rate of Product Transformation) Varian is sort of implicitly assuming the PPF is linear. So there is a constant trade of in producing butter or milk as resources are reallocated.MilkMRS=3MRT=3Social Indifference curveButter
104 Marginal Rate of Transformation With a more realistic PPF, the MRT rises as more butter and less milk is produced.MilkMRT=3Butter
105 MRS=MRTIn the long run (π=0), the cost of producing butter must be 3 times the cost of producing milk. That is, the tradeoffs in consumption = the tradeoff in productionMilkpx/py=3Butter
106 Back to Varian’s treatment A new technology that allows you to produce butter with 4 gallons of milk is not going to be a winner as everyone would choose the cheaper butter previously offered.MilkMRT=4Pb=4; Pm=1Butter
107 However, a new technology that allows you to produce 1 pound of butter with 2 gallons of milk IS going to be a winner!MilkMRS=MRT=2Pb=2; Pm=1Butter
108 MRS=MRTHere is what improved butter making technology does with a more standard PPF.Milkpx/py=2px/py=3Butter