Term 1 Week 6: Optimising Mathematically

Question 1

How do we use the lagrangian to optimise mathematically (3)

Answer 1

-L = U(x1, x2) + λ(M - p1x1 - p2x2)
-Then partially differentiate with respect to X, Y and λ (FOC’s), equal the first 2 to eachother then plug this in the third one to get Marshallian demands: x*(p1, p2, m)
-The primal is utility maximisation with a given budget

Question 2

What does tangency/λ imply (2)

Answer 2

-λ* is the marginal utility of income
-Tangency implies (λ* = MU/p) you keep on spending where the marginal utility is equal to the price

Question 3

What is the general rule for optimal consumption (1)

Answer 3

-MUx/Px = MUy/Py etc

Question 4

What is the Dual problem (3)

Answer 4

-Find the initial bundle (utility maximisation (highest IC subject to BC))
-Find change in demand due to substitution (expenditure minimisation (lowest BC subject to initial IC))
-Find new bundle at new price ratio (utility max (also accounts for income effect))

Question 5

What is the hicksian method of isolating the substitution effect (5)

Answer 5

-Optimise mathematically with old information
-Find the compenated budget (cheapest way to reach original utility with new prices), by setting up a lagrangian with min exp = p1x1 + p2x2 - λ(Utility - u(x1, x2))
-Take the FOC’s with respect to each variable, then equal the first 2 to find a formula for x2 in terms of x1
-sub this into the λ FOC to find the new consumption values (only substitution effect)
-Substitute the new prices into the marshallian demand functions to find the final consumption bundle (sub + income effect)

Question 6

What is the indirect utility function (3)

Answer 6

-v(p1, p2, m) = U(x1(p1, p2, m), x2(p1, p2, m))
-This is increasing in income, homogenous of degree 0, non-increasing in every price (price fall = q rise = utility rise)
-This shows optimal level of utility depends indirectly on prices and income

Question 7

What is the expenditure function (2)

Answer 7

-M = E(p1, p2, v) = p1H1(p1, p2, v) + p2H2(p1, p2, v)
-We take our budget constraint, plug in the Hicksian demands, to get the equation for the compensated budget constraint

Question 8

How do we optimise mathematically using the Slutsky method (3)

Answer 8

-Slutsky compensates the consumer such that at the new price, the original bundle is still just affordable
-This happens by increasing income for a rise in price, or decreasing income for a fall in price
-From here, we then set up our lagrangian with the utility function - λ(new income - p1x1 …), work out our FOC’s etc

Question 9

What is the slutsky identity ()

Answer 9

-The substitution effect Δx1s = x1 (p’1, M’) - x1(P1, M) (the change due to substitution = change with the new price and compensated income - old price and income)
-The income effect Δx1n = x1(P’1, M) - x1(p’1, M’) (income effect change = change with new price and current income - change with new price and compensated income)
-total effect = substitution + income

-You can make this a rate of change by defining change due to income as -Δx1n, dividing each side by ΔP1 to get (Δx1/ΔP1) = (Δx1s/ΔP1) - (Δx1m/ΔP1), recalling ΔM = x1ΔP1, then replacing the denominator i term 3
-Δx1/ΔP1 = Δx1s/ΔP1 - (Δx1m/ΔM)x1