Opt Graph/Math 5&6

Question 1

Hicksian demand

Answer 1

Utility is kept constant after the price change so that the consumer is not better/worse off despite the price change. Only considers the substitution effect

Question 2

Slutsky demand

Answer 2

Purchasing power is the same after the price change as it was before so the consumer has just enough income to buy the original bundle. Only considers substitution effect

Question 3

Graphically Hicksian

Answer 3

Shift the new BC to be tangential to the original IC, this point will be the substitution effect. Normal good sub and income will go in the same direction. Inferior and Giffen will be opposing

Question 4

Graphically Slutsky

Answer 4

The new BC shifts down to intersect at the original optimal point where the original BC intersects the original IC, draw a new IC which is tangential to the shifted BC for the sub effect.

Question 5

Is tangency necessary/sufficient for mathematical optimisation

Answer 5

Tangency is necessary, not sufficient. Only sufficient if preferences are convex

Question 6

The Lagrange multiplier

Answer 6

λ is the marginal utility of an extra £ of expenditure, it’s the marginal utility of income so £1 of extra income will increase utility by λ

Question 7

Mathematical method to using the Lagrange

Answer 7

U = f(X, Y) for utility and BC is M = PxX + PyY. L = f(X, Y) + λ(M - PxX - PyY). Then differentiate to each variable and set equal to 0.

Question 8

What is the general rule for optimal consumption

Answer 8

(MUx/Px) = (MUy/Py)

Question 9

Hicksian expenditure minimisation

Answer 9

For the substitution effect, we want the consumer to receive a given level of utility at the lowest cost. Solutions to the problem gives Hicksian (compensated) demands

Question 10

Indirect utility function

Answer 10

Where optimal level of utility depends indirectly on prices and income
v(p1,p2,M) = U[x1(p1,p2,M)], x2(p1,p2,M)

Question 11

Properties of an indirect utility function

Answer 11

Non-increasing in every price, decreasing in at least one price. Increasing in income. Homogeneous of degree zero in price and income

Question 12

Properties of the Expenditure Function

Answer 12

It is non-decreasing in every price, increasing in at least one. Increasing in utility. Homogeneous of degree 1 in all prices

Question 13

Slutsky Mathematical

Answer 13

Using the lagrange but instead the multiplier affects the utility rather than the budget constraint

Question 14

The Slutsky identity

Answer 14

∆𝒙𝟏/ ∆𝑷𝟏 = (∆𝒙𝟏𝒔/∆𝑷𝟏) − (∆𝒙𝟏𝒎/∆𝑴)𝒙𝟏