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# Utility Maximization.

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Utility Maximization

A Utility Function Mathematically Representing Preferences
U=U(x, y) U(A) Y U(B) Utility functions have A Indifference curves describe bundle-ordering preferences B X

Consumption Opportunities: The Budget Constraint
Assume that an individual has I dollars to allocate between good x and good y pxx + pyy  M y The individual can afford to choose only combinations of x and y in the shaded triangle x

The Budget Constraint MC of consuming one more unit of x, the amount of y that must be foregone. The slope of the budget line is this MC. y The slope is the change in y for a one unit increase in the consumption of x. If Px = 10 and Py = 5, then consuming one more x means consuming two less y. x

Not an indifference curve!
Maximizing Utility Keep buying x until the MB(x) = MC(x) Interaction of… Preferences, diminishing MB because of diminishing MRS. MB = MRS MB in terms of y willing to be given up In dollars, MB = py*MRS MC of x = px/py MC in terms of Y given up In dollars, MC = px. MB, MC X X* Not an indifference curve! MC MB

Optimization Principle
To maximize utility, given a fixed amount of income, an individual will buy the goods and services: That exhaust total income Savings 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)

Intuition MRS 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. y x A B At “A”, MRS>Px/Py (MB > MC), You are willing to pay more than you have to, consumer surplus increases. Utility and consumer surplus can be increased by consuming more x. At “B”, MRS<Px/Py, (MB < MC) Utility and consumer surplus can be increased by consuming less x. Px = 10 Py = 5 Slope of budget line = -2 U0

Intuition At “C”, the MB = MC for the last unit of both goods consumed. That is, at “C”, MRS = px/py, or y A C U1 B U0 x

Optimization Unconstrained optimization is a lot easier to solve than constrained optimization. Substitution: maximize the cross section of U along the budget line Lagrange method

Substitution This 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 parabola U y y* x* x

Substitution Substitute and maximize And substitute again

Problem with this method
It can get very mathematically complicated very quickly. Even U=xαyβ gets very tricky to solve.

LaGrange Method LaGrange 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.

Unconstrained Optimization Example
Profit maximization is an example of this. We maximize the difference between two functions: π=R(q) – C(q).

Lagrange Method LaGrange 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.

Maximize Utility - Expenditure
Maximize utility minus the cost of buying bundles. Think about a one-good world. U U=U(x) slope = Ux Expenditure = E = pxx slope = Ex= px Problems: Ux not measured in \$ like E. E is not constrained, we can spend as much as we like. x x*

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. U U=U(x) slope = Ux EU=λpxx Slope = EUx= λpx x x*

Maximizing Utility - Expenditure
Problem, an infinite number of λ choices that will each solve this with a different x* EUx = λ1 px U U=U(x) EUx= λ2 px EUx = λ3 px x x* x* x* Now the slope of the expenditure function and expenditure are measured in utils, not dollars. But we are not constraining x yet.

LaGrange Method So first subtract λM from the expenditure function to get EL = λpxx - λM U U=U(x) slope = Ux EU = λpxx EL = λpxx - λM x* x slope = ELx= λpx -λM

LaGrange Method We know we want to find the x* such that that distance between U(x) and EL(x*) = U(x*). That is, where EL(x*) = 0 So we maximize v = U(x) - 0 Substitute λpxx – λM = 0 in for 0, to constrain x* to our budget. U U=U(x) slope = Ux U=U(x*)-0 EL = λpxx - λM x x* slope = ELx= λ px -λI

LaGrange Method Our 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. U U=U(x) slope = Ux U=U(x*)-0 EL = λ(pxx – M) x x* slope = ELx= λ px -λM

LaGrange Method

Two Goods: Lagrange’s Manufactured Plane
To maximize utility, maximize the height of the utility function above the plane EL = λpxx + λpyy – λM Such that λpxx + λpyy – λM = 0 U U = U(x,y) y When x = 0 and y = 0, U = - λM ELy= λ py LaGrange Plane EL=g(x,y) EL= λpxx+ λpyy- λM g’x=ELx= λ px g’y=ELy= λ py ELx= λ px x

Lagrange Method U U = U(x,y) y UL=g(x,y) = 0 λ(pxx+pyy – M) = 0 x

Basic Demand Analysis Using Lagrangian to generate ordinary (Marshallian) demand curves. FOCs necessary SOCs sufficient (check that they hold) Ordinary (Marshallian) demand curves Inverse demand curves Meaning of λ Indirect Utility Expenditure Function Comparative Statics General Results

Demand Functions using Lagrange’s Method
Set up and maximize: λ* chosen so that the constraint plane is parallel to the utility function. Any x* and y* that maximizes utility will also have to exhaust income.

FOCs for an Optimum For 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 satisfied Utility Maximized y y y Utility Maximized FOCs satisfied x x x

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 hold Utility Maximized y y y SOCs satisfied x x x

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. Y U*>U’ U=U* U=U’ X

Checking SOC: utility function strictly quasi-concave
The second order conditions will hold if the utility function is strictly quasi-concave A function is strictly quasi-concave if its bordered Hessian is negative definite. That is: A function is strictly quasi-concave if: -UxUx < 0 2UxUxyUy - Uy2Uxx - Ux2Uyy > 0 Binger and Hoffman, page 115

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 and The last only differ by 

Ordinary Demand Curves
And from the FOC:

Inverse Demand Curves Starting with the ordinary demand curves:

Utility and Indirect Utility
Maximum Utility, a function of quantities Indirect Utility a function of price and income

Optimization : Expenditure Function
Start with indirect utility Solve for M: This equation determines the expenditure needed to generate Ū, the expenditure function:

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 out So when ω changes, the optimal x changes, which changes the y* function. Two methods to solve this…

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 get And these two answers are equivalent:

Envelope Result, 1st option to get
Plug the optimal values into the LaGrangian Silberburg, first edition, page has a similar treatment.

Optimization: Envelope Result
Plug the optimal values into the LaGrangian Silberburg, first edition, page has a similar treatment.

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 < 0 Can we get anything from that?

Optimization: Comparative Statics
Start with:

Comparative Statics: Utility Maximizing x*, y*, λ*

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. M Side note: Tells us that if income increases by \$1, so will total expenditure.

Comparative Statics: Effect of a change in M Put in Matrix Notation
Solve for

Comparative Statics: Effect of a change in I Put in Matrix Notation
Solve for

Comparative Statics: Effect of a change in px Differentiate (1), (2), (3) w.r.t. px

Comparative Statics: Effect of a change in px Put in Matrix Notation
Solve for

Comparative Statics: Effect of a change in px Put in Matrix Notation… AGAIN
Solve for

Comparative Statics: Preview of income and substitution effects
Rearrange these Sub in these Income effect matters

Specific Utility Functions
Cobb-Douglas CES Perfect Compliments

Cobb-Douglas: Utility Max
Problem: Set up the LaGrangian FOC

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)

Cobb-Douglas: Demand Preferences 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 income Income elasticities are = 1

Cobb-Douglas: Indirect Utility
Plug x* and y* into the utility function

Cobb-Douglas: Expenditure Function
Start with indirect utility function Solve for M, and then rename it E

CES: Utility Max Problem: Set up the Lagrangian FOC

CES: Demand FOC Imply Plug into the budget constraint and solve:

CES: Indirect Utility Plug x* and y* into the utility function

CES: Expenditure Function
Solve for M, then rename E.

CES: Expenditure Function (cont)
Solve for M, then rename E.

Perfect Compliments: Utility Max
Problem: No Lagrangian, just exhaust income such that: So plug this condition into the budget equation Essentially, we substitute the expansion path into the budget line equation.

Perfect Compliments: Demand
Demand equations

Perfect Compliments: Indirect Utility
Since utility from x = utility from y at utility max:

Perfect Compliments: Expenditure Function
Since the utility from consumption of each must be equal,

Bonus Topics Money Metric Utility Function Homogeneity
Corner Solutions (Kuhn-Tucker) Lump-Sum Principle MRS and MRT (Marginal Rate of Transformation – slope of PPF)

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.

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?

Pre-Public Housing Y If expenditure on housing is generally 1/3 of income, assume U=h1/3y2/3 ph=\$1 px=\$1 M=\$1,000 667 H 333 667

Public Housing Y ph=\$1 px=\$1 M=\$1,000 Qualified citizens get 667
housing units for 1/3 of income (\$333) 667 H 333 667

Public Housing: Money Metric Utility
ph=\$1 px=\$1 M=\$1,000 Qualified citizens get 667 housing units for 1/3 of income (\$333) 667 UEPH=1,261 UE=1,000 H 333 667 The extra housing has a value to the poor of \$261. Depending on the cost to the government of providing the housing, the program can be evaluated.

xi* = xi(p1,p2,…,pn,M) = xi(tp1,tp2,…,tpn,tM)
Homogeneity If all prices and income were doubled, the optimal quantities demanded will not change the budget constraint is unchanged xi* = xi(p1,p2,…,pn,M) = xi(tp1,tp2,…,tpn,tM) Individual demand functions are homogeneous of degree zero in all prices and income To test, replace all prices and income with t*p and t*M. The quantity demanded should be unaffected (all the “t” should cancel out).

Corner Solutions Non-corner solutions: Optimal bundle will be where x > 0, y > 0 and MRS = px/py Y Two corner solutions: Optimal bundle will be where x =0 X

Corner Solutions Y A X At “A”, MRS = px/py, but the optimal quantity of X = 0

Corner Solutions Y B X At “B”, the tangency condition holds where x* < 0. Given the budget line, the optimal feasible x is where x = 0 and MRS < px/py

Corner Solution To 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.

Corner Solutions FOC

Corner Solutions Kuhn-Tucker Condition This tell us that either:
μ =0 (the optimum is at a tangency) s = 0 (the optimum is where x = 0) μ =0 and s = 0 (the optimum is at a tangency where x = 0)

Corner Solutions If s > 0 and μ = 0 Y X The usual assumption is that the optimal bundle will be where x>0, y>0 and

Corner Solutions At “A”, , but the optimal quantity of x = 0
If s = 0 and μ = 0 Y A X At “A”, , but the optimal quantity of x = 0

Corner Solutions At B’, the tangency condition holds where x* < 0.
If s = 0 and μ > 0 Y B B’ X At B’, the tangency condition holds where x* < 0. The optimum is where x=0 and

Kuhn-Tucker Example Utility: U=xy+20y, M = 40, px = \$4 and py = 1.

Looks Like Tangency where x=-5, y=60 Y X

How about a feasible optimum?
Tangency where x=-5, y=60 Y X

Kuhn-Tucker Set-up Utility: U=xy+20y, M = 40, px = \$4 and py = 1

Kuhn-Tucker Result

Kuhn-Tucker Result In this example, at x = 0, y = 40,   0 :

How about a feasible optimum?
Optimum where x=0, y=40 Y X

Solving Kuhn-Tucker If you find that the optimal bundle is not on the budget constraint, check all corners for a maximum utility.

Lump Sum Tax Taxing a good vs taxing income Tax on x only
Lump sum tax (income tax)

Lump Sum Tax Difference in the budget lines: sales tax

Lump Sum Principle A tax on x rotates the budget line to have: Y y*

Lump Sum Tax Difference in the budget lines: income tax

Lump Sum Principle Tax 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. Y y* X x*

Lump Sum Principle When Indirect Utility function is:

Lump Sum Principle Set, I=60, py = 2, px=1 Utility is:

Lump Sum Principle With a \$1 tax on x, Utility is:
And tax revenue is \$20.

Lump Sum Principle With a \$20 income tax, Utility is: 68.85 > 63.24

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:

The Lump Sum Principle: Perfect Compliments
Set, I=60, py = 2, px=1 Utility is:

The Lump Sum Principle: Perfect Compliments
With a \$1 tax on x, Utility is:

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.

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. Milk Pb=3; Pm=1 MRS=3 MRS=3 MRS=3 Butter

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. Milk MRS=3 MRT=3 Social Indifference curve Butter

Marginal Rate of Transformation
With a more realistic PPF, the MRT rises as more butter and less milk is produced. Milk MRT=3 Butter

MRS=MRT In 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 production Milk px/py=3 Butter

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. Milk MRT=4 Pb=4; Pm=1 Butter

However, a new technology that allows you to produce 1 pound of butter with 2 gallons of milk IS going to be a winner! Milk MRS=MRT=2 Pb=2; Pm=1 Butter

MRS=MRT Here is what improved butter making technology does with a more standard PPF. Milk px/py=2 px/py=3 Butter

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