Intermediate Microeconomics Optimal Choice Tin Cheuk (Tommy) Leung CUHK

Tin Cheuk (Tommy) Leung (CUHK)

Intermediate Microeconomics

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Optimal Choice

Ultimate goal: explain changes in behavior when constraint (like price) changes

Combine two things in our analysis tastes/preference: utility/indifference curves constraint

Taste/preference covered before

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Intermediate Microeconomics

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Constraint

Suppose you hit the jackpot and have one million dollars to spend, what kind of things do you have to consider?

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Intermediate Microeconomics

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Budget Constraint

Use changes in constraint to explain behavior change ∵ taste unobservable constraint observable

Two main things covered in budget constraint income prices

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Utility Maximization

A rational consumer is assumed to solve max U(x1 , x2 ) x

subject to p1 x1 + p2 x2 ≤ M Question: How do we solve for the optimal consumption bundle?

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Constrained Maximization– Lagrange Lagrangian: L = U(x1 , x2 ) − λ(p1 x1 + p2 x2 − M) Lagrange’s theorem says (x1∗ , x2∗ ) satisfies ∂U(x1∗ , x2∗ ) ∂L = − λp1 = 0 ∂x1 ∂x1 ∂L ∂U(x1∗ , x2∗ ) = − λp2 = 0 ∂x2 ∂x2 ∂L = p1 x1∗ + p2 x2∗ − M = 0 λ

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Intermediate Microeconomics

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Constrained Maximization– Lagrange

First two equations ∂U(x ∗ ,x ∗ )/∂x

⇒ ∂U(x1∗ ,x2∗ )/∂x21 = pp12 1 2 |MRS| = price ratio

Third equation: optimal point on the budget constraint

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Intermediate Microeconomics

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Graphical Illustration of the Optimal Choice

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Example

Two endogenous variables: x1 , x2 Income: 2000 p1 = 5 and p2 = 10 Cobb-Douglas utility: U(x1 , x2 ) = x10.5 x20.5 Question: What is the optimal consumption bundle?

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Intermediate Microeconomics

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Answer

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Intermediate Microeconomics

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Special Preferences

So far, utility has nice properties ⇒ interior solution What if perfect substitutes perfect complements concave indifference curves

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Intermediate Microeconomics

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Perfect Substitutes

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Intermediate Microeconomics

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Perfect Complements

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Intermediate Microeconomics

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Concave Indifference Curves

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Intermediate Microeconomics

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Income vs Quantity tax

Quantity tax: a tax depends on consumption of goods tax rate τ on x1 tax revenue: τ x1

Income tax: a tax on income, R Question: which one is better if R = τ x1 ?

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Intermediate Microeconomics

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Answer

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Intermediate Microeconomics

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Demand

Previous model: how agent choose optimal consumption What is demand? How do we derive demand from the previous model

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Intermediate Microeconomics

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Example

Two endogenous variables: x1 , x2 Income: 2000 p2 = 10 Cobb-Douglas utility: U(x1 , x2 ) = x10.5 x20.5 Question: What is the demand for x1 ?

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Intermediate Microeconomics

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Answer

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Intermediate Microeconomics

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Example: Labor-Leisure Choice

C - income/other goods L - leisure U(C , L) = C 0.5 L0.5 T = 100 - maximum working hours per week Π - non-labor income Suppose hourly wage is w , what is the labor supply as a function of w?

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Answer

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Inverse Demand Curve and MRS Demand curve: x1 a function of p1 holding other things constant downward sloping

Inverse demand: p1 as a function of x1 From Lagrangian analysis, we have p1 p2 = p2 |MRS|

|MRS| = p1

If x2 is income/other goods, and p2 = 1 p1 measures the willingness to pay

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