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Utility Maximization

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

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

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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. x y The slope is the change in y for a one unit increase in the consumption of x. If P x = 10 and P y = 5, then consuming one more x means consuming two less y.

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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 = p y *MRS – MC of x = p x /p y MC in terms of Y given up In dollars, MC = p x. Maximizing Utility MB, MC X X* Not an indifference curve! MC MB

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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) Optimization Principle

7 MRS is the maximum amount of y the person is willing to give up to consume more x; the definition of MB. p x /p y tells us the number of units of y that must be given up to consume one more x; the definition of MC. Intuition Px = 10 Py = 5 Slope of budget line = -2 y x A B At “B”, MRS

P x /P y (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. U0U0

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At “C”, the MB = MC for the last unit of both goods consumed. That is, at “C”, MRS = p x /p y, or Intuition y C U1U1 x A BU0U0

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

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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 Substitution x y U x* y*

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Substitute and maximize Substitution And substitute again

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Problem with this method It can get very mathematically complicated very quickly. Even U=x α y β gets very tricky to solve.

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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.

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

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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. Lagrange Method

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

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First change the expenditure function by multiplying p x by λ. Now call that function E U. 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 E U (x) would be measured in the same units. Maximizing Utility - Expenditure x U U=U(x) EU=λpxxEU=λpxx x* Slope = E U x = λp x slope = U x

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Problem, an infinite number of λ choices that will each solve this with a different x* Maximizing Utility - Expenditure U U=U(x) x* E U x = λ 1 p x x* E U x = λ 2 p x E U x = λ 3 p x x Now the slope of the expenditure function and expenditure are measured in utils, not dollars. But we are not constraining x yet.

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So first subtract λM from the expenditure function to get E L = λp x x - λM LaGrange Method U U=U(x) E L = λp x x - λM x* slope = U x -λM-λM E U = λp x x slope = E L x = λp x x

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

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Our optimization becomes an unconstrained problem by including the requirement that λp x x = λM. λ is chosen along with x to maximize utility so that λ = the marginal utility of $1. That is, λp x = U x. LaGrange Method U U=U(x) E L = λ(p x x – M) x* slope = E L x = λ p x slope = U x -λM-λM U=U(x * )-0 0 x

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LaGrange Method

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To maximize utility, maximize the height of the utility function above the plane E L = λp x x + λp y y – λM Such that λp x x + λp y y – λM = 0 Two Goods: Lagrange’s Manufactured Plane x y U U = U(x,y) LaGrange Plane E L =g(x,y) E L = λp x x+ λp y y- λM g’ x =E L x = λ p x g’ y =E L y = λ p y When x = 0 and y = 0, U = - λM E L x = λ p x E L y = λ p y

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Lagrange Method U U L =g(x,y) = 0 λ(p x x+p y y – M) = 0 x y U = U(x,y)

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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

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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.

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

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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 for an Optimum y y x x SOCs satisfied Utility Maximized y x SOCs do not hold

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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. X Y U=U’ U=U* U*>U’

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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: 1. -U x U x < U x U xy U y - U y 2 U xx - U x 2 U yy > 0 Checking SOC: utility function strictly quasi-concave

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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

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Ordinary Demand Curves And from the FOC:

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Inverse Demand Curves Starting with the ordinary demand curves:

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

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

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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…

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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, takethen substitute x* = x(ω) into y ω (x ; ω) to get And these two answers are equivalent:

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Plug the optimal values into the LaGrangian Envelope Result, 1st option to get

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Optimization: Envelope Result Plug the optimal values into the LaGrangian

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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: U x > 0U y > 0U xx < 0U yy < 0 Can we get anything from that?

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Optimization: Comparative Statics Start with:

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

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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:

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

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

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

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

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

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

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Specific Utility Functions Cobb-Douglas CES Perfect Compliments

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

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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)

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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

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

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

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CES: Utility Max Problem: Set up the Lagrangian FOC

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

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

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

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

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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.

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Perfect Compliments: Demand Demand equations

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

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

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

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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.

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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?

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Pre-Public Housing H Y If expenditure on housing is generally 1/3 of income, assume U=h 1/3 y 2/3 p h =$1 p x =$1 M=$1,

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

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Public Housing: Money Metric Utility H Y p h =$1 p x =$1 M=$1,000 Qualified citizens get 667 housing units for 1/3 of income ($333) U E =1,000 U E PH =1,261 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.

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Homogeneity If all prices and income were doubled, the optimal quantities demanded will not change – the budget constraint is unchanged x i * = x i (p 1,p 2,…,p n,M) = x i (tp 1,tp 2,…,tp n,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).

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

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Corner Solutions At “A”, MRS = p x /p y, but the optimal quantity of X = 0 X Y A

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

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Corner Solution

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Corner Solutions FOC

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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)

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

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

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

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

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

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

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

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Kuhn-Tucker Result

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In this example, at x = 0, y = 40, 0 :

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

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

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

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Lump Sum Tax Difference in the budget lines: sales tax

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

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Lump Sum Tax Difference in the budget lines: income tax

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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. X Y y* x*

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Lump Sum Principle When Indirect Utility function is:

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Lump Sum Principle Set, I=60, p y = 2, p x =1 Utility is:

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

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

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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

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Set, I=60, p y = 2, p x =1 Utility is: The Lump Sum Principle: Perfect Compliments

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

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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. The Lump Sum Principle: Perfect Compliments

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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. Butter Milk P b =3; P m =1 MRS=3

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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 MRT=3 Butter MRS=3 Social Indifference curve

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

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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 p x /p y =3 Butter

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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. Butter Milk MRT=4 P b =4; P m =1

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

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

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