Week Five: Choice

Question 1

How to bundles and indifference curves relate to chioce?

Answer 1

• Bundle Set: Bundles Afford
• Indifference Curves: Value of these bundles

So, which to buy?

1. Maximize Utility
2. Constraint budget

“Rational Chioce”

Question 2

What is the optimal choice? (Generally)

Answer 2

Find the bundle in budget set that is on the highest indifference curve.

Assume well-behaved (therefore cobb-douglas) → only consider those on budget line

Bundle of goods highest indifference curve: (x_1^, x_2^) ⇒ optimal choice

Note:

• Buget tanget to indifference curve

Note that some has “kinky tastes” and budget lines arent tanget (Tho generally ignored)

Question 3

Conditions:

Answer 3

1. Optimality: Requires consumer to equate MRS to relative price (x_1 just as much as market)
2. Feasibility: Consumer purchases bundle that exhausts income

Question 4

Chioce

What is Interior Optimum?

Answer 4

When x_1^>0 and x_2^>0: Demand bundle is “interior”

• Interior: Optimal choice bundle not located on the budget constraint
• If $(x_1^,x_2^)$ costs more than $m$ ⇒ exhausted

Therefore, bundle exhuasted if on budget constraint

Optimal point occurs when consumption of some good = 0

• Represent boundary optimum (represents interior optimum)

(x_1^,x_2^) satisfies two conditions:

1. (x_1^,x_2^) exhausts budget: p_1x_1^+p_2x_2^=m
2. The slope of the indifference curve at (x_1^,x_2^) equals slope of the budget constraint at -p_1/p_2

These conditions hold for well-behaved preferences

Question 5

What is Marshalian Demand?

Answer 5

Demanded Bundle: optimal choice of good one and two at set price and income

Demand Function, then, the function relates the optimal chioce:

(Dependent both price and income)

x1(p1,p2,m) and x2(p1, p2, m)

Cases Utility Maximation:

1. Tangecy Solution: When well-behavedThen optimal: MRS=-p_1/p_2 (Cobb-Douglas)
2. Corner/Boundary Solution:if MRS> -p_1/p_2 or MRS<-p_1/p_2 always (Perfect Substitutes)
3. L-Shaped Solution:if preferences L-Shaped → kink solution (Perfect Complements)

Question 6

Mashalian Demand

Cases for Utility Maximination

Answer 6

Cases Utility Maximation:

1. Tangecy Solution: When well-behavedThen optimal: MRS=-p_1/p_2 (Cobb-Douglas)
2. Corner/Boundary Solution:if MRS> -p_1/p_2 or MRS<-p_1/p_2 always (Perfect Substitutes)
3. L-Shaped Solution:if preferences L-Shaped → kink solution (Perfect Complements)

Question 7

Marshallian Demand

Well-Behaved Curves with Tangential Solution

Answer 7

or Cobb-Douglas (smooth. convex, monotone)

General Form:

u(x_1,x_2)=x_1^a x_2^b

Want to buy bundle at the point where the market value of x_1 is the same (relative to) x_2

Therefore, first optimality conditions: MRS =-p_1/p_2

Steps:

1. Identify clearly the utility function
2. Calculate MRS (funciton of x_1,x_2 and whatever parameters)
3. Set tangency condition: MRS=-p_1/p_2 (1)
4. Identify budget constraint (2)
5. Set (1) = (2) and solve system of equations

Optimal Chioces: (derive using Lagrange Multiples and Extrema with Constraints)

Question 8

Marshallian Demand

Corner Solution

Answer 8

Question 9

Marshalian Demand

Perfect Complements

Answer 9

Steps:

1. Identify clearly utility function: U=min{ ax_1, bx_2} for a,b >0
2. Calculate the optimal solution parth: $ax_1=bx_2$ (1)
3. Identify budget constraint (2)
4. Solve for equations (1) and (2)

Optimal choice always lie on the diagonal, where consumer purchasing equal amounts of both goods (regardless of prices), so satisfy budget constraint