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Economics 202: Intermediate Microeconomic Theory

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1. Student Information Sheets

2. Any questions?

3. For next time, finish reading Chapter 5

4. HW #2 due Thursday in class (it’s on the website)

Budget Constraints

- Shifts in the Budget Line
- Change in Income
- Change in Prices
- Double I, Triple PX & PY?

Yo-yos

I/PY

2I/3PY

- NB: slope measures the real price, the purchasing power of one good in terms of the other
- so if both prices rise by the same % (or fall by same %), their ratio is the same the slope of budget line stays same!

- Composite consumption good x2: p1x1 + 1x2 = m
- What if the price changes with quantity purchased?

2I/3PX

I/PX

Xylophones

Budget Constraints

- Change in Income
- lump-sum tax or subsidy (grant)
- “earmarked” grant

- Change in Prices
- Per-unit tax or subsidy (quantity tax or subsidy)
- Ad valorem tax or subsidy (value/proportional tax or subsidy)

- Two criteria:
(1) slope of IC = slope of budget line

(2) we have to be on the budget line

- This will give us 2 equations in 2 unknowns, and we can solve for optimal values
- General Lagrangian & MRS = PX/PY is only a necessary condition for utility maximization. Assumption of dim. MRS (strict convexity) gives sufficiency for utility maximization.
- U = GT Income = $100 Ptennis racquet= $10
Pgatorade = $5

- What is Roger’s optimal consumption bundle of Gatorade and tennis racquets?

- Three approaches

- Approach #1
- Write down & solve the MRS condition and budget constraint

- Approach #2
- Create unconstrained utility maximization problem

- Approach #3
- Constrained utility maximization problem (use the Lagrangian)
- has an economic interpretation
- Check
- Let I = $101 and calculate the resulting increase in utility …
- We had U = GT, I = $100, PG = $5 per bottle, PT = $10 per racquet
- This gave us G* = 10 bottles, T* = 5 tennis racuets for U* = 50 “utils”

Pretzels

a

U2

U1

U0

Final four tickets

Exception: Corner Solution

- At point “a”, MRS < slope of the budget line
- But that is our final point since we can’t consume less than 0 Final Four tickets
- NB: the optimality condition (MRS = slope of budget line) only holds for cases in which we consume positive amounts of BOTH goods.
- FOC’s must be modified with a sign, rather than = sign.

- When, e.g., U/X - PX < 0, then X* = 0.
- PX > MUX / which says ?

- Derive the demand functions for the quasi-linear function
U(X,Y) = ln X + Y

X* = dX(PX, PY, I; tastes)Y* = dY(PX, PY, I; tastes)

- Green is a foreshadowing to emphasize now that this is Marshallian demand (uncompensated demand) which holds income fixed.
Numerical example:

U = ln X + YIncome = $10 PX = PY = $1

- Are these homothetic preferences?
- What is optimal consumption bundle (X*,Y*)?
- What is utility at the optimum?
- What is the marginal utility of income at the optimum?