Term 2 Lecture 4

Question 1

Summary of this lecture,
- if the consumer consistently chooses the most preferred among affordable bundles, then

Answer 1

It is possible to give a complete description of the properties demands must satisfy, to see this, it is illuminating to think of consumers as expenditure minimisers subject to utility constraints.

Question 2

What is the consumer optimisation problem

Answer 2

Max u(q), s.t. p’q < y
- then choices satisfy WARP, in fact, the assumption of consumer optimisation is equivalent to SARP
- therefore, utility maximising choices satisfy negativity, homogeneity and assuming non-satiation - adding up.

Question 3

If a consumer’s preferences are monotonic and convex

Answer 3

The optimal choice occurs at a tangency point between an IC and the budget line
- at MRS = p1/p2
- if the budget set is linear, this can occur either in the interior of the budget set or at a corner

Question 4

Interior vs Corner solutions

Answer 4

• if the tangency condition holds, the consumer consumes positive amounts off both goods
• if there is no tangency point, consumer may only consume one good, this happens when MRS never equals the prive ratio

Question 5

Unique vs multiple tangencies

Answer 5

• if the budget set is linear and convex, then there is a unique tangency
• if the budget set is non convex, there may be multiple tangency points, so in these cases compare the utility levels to find the true optimum

Question 6

Whats a marshalian demand curve?

Answer 6

Also known as uncompensated demand curve
- if you solve the utility maximisation problem for quantities x and y, you get these demand functions.

Question 7

Graphically why cant there be multiple points of tangency on a linear budget set compared to non-linear

Answer 7

• on linear, the ICs would intersect, which is not possible
• on non-linear, you can find multiple tangencies without any crossing of ICs

Question 8

Impact of income increasing on slope of budget constraints
- relation to IEP

Answer 8

Slope of budget constraint doesn’t change
- IEPs traced out by the tangencies as incomes are increased therefore all occur at points with the same MRS

Question 9

What is homotheticity for a utility function characterised by?

Answer 9

• homothetic if the MRS remains constant along rays from the origin
• IEPs are straight lines through the origin
• so as income increases, the consumer’s budget shares remain constant

Question 10

Marshallian demand under homotheticity:
- budget shares

Answer 10

Fi(y,p) = ai(p)y
- fi(y,p) = demand for good i as a function of income y and prices
- ai(p) = a function that depends only on prices, not on income
- this means demand is proportional to income

Budget shares: wi = piai(p)

Question 11

A utility function is quasilinear if

Answer 11

The MRS is constant along lines parallel to one of the axes
- QL preferences imply that income only affects the consumption of one good
- if income increases, extra income is spent only on the QL good
- since demand for other goods dont change with income, IEPs are parallel to one of the axes

Question 12

QL Marshallian Demand Functions take the form:

Answer 12

F1(y,p) = (y-B(p))/p1, this is the demand for the QL good
Fi(y,p) = bi(p), this is demand for all other good, which depend only on prices, not income

Y is income, B(p) is total expenditure on all goods apart from f1, and as you can see demand for all other goods is not affected by income

Question 13

Solving: min p’q s.t. U(q) >/ v

Answer 13

Gives you the expenditure minimisation problem, which, leads to the same tangency condition as the utility maximisation problem

Question 14

What are Hicksian demand functions, also known as compensated demand functions?

Answer 14

Arise from the expenditure minimisation problem:
- q = g(v,p), v being the fixed level of utility
- since common scaling of all prices implies no change in the slopes of any budget lines, it also implies no change in expenditure minimising demands at any utility.
- therefore, Hicksian demands are homogenous of degree 0

Question 15

Hicksian demands under homotheticity

Answer 15

• just as Marshallian demand scales proportionally with income under homothetic preferences, Hicksian demand scales proportionally with utility
• gi(v,p) = ai(p).k(v), where k(v) is an increasing function of utility
• as v increases, demand scales up proportionally

Question 16

Hicksian demands under QL

Answer 16

• an increase in utility affects only one good’s demand, leaving the others unchanged.
• g1(v,p) = k(v) + B(p)
• gi(v,p) = bi(p)

Question 17

How to get the indirect utility function

Answer 17

• substitute the marshallian demand functions back into the utility function gives the indirect utility function:
  V(y,p) = u(f(y,p))

Question 18

How to get the expenditure function

Answer 18

Min p’q s.t. U(q) >/ v
- leads to the Hicksian demand function
- substitute this demand function into the expenditure equation to get the expenditure function:

C(v,p) = p’g(v,p)

Question 19

Duality of the marshallian and Hicksian demands

Answer 19

Subbing the expenditure function into the marshallian demand function gives the Hicksian demand

• subbing the expenditure function into y in the indirect utility function gives you v
• subbing the indirect utility function into v in the expenditure function gives you y

Question 20

The expenditure function

Answer 20

Tells us the minimum cost required to reach a utility level v given p
- increasing in every price in p and in v
- homogenous of degree 1, Hicksian demands are homogenous of degree 1
- concave in prices

Question 21

Indirect utility function

Answer 21

• increasing in y and decreasing in each element of p
• homogenous of degree zero in y and p

Question 22

Shepherd’s Lemma
- how to get from expenditure function to the Hicksian demands

Answer 22

By differentiating c(v,p)
- Dc(v,p)/Dpi = …. = gi(v,p), using the first order condition for solving the cost min problem and the fact that utility is held constant

You can go from expenditure function to Hicksian demands

Question 23

Roys identity
- how to go from the Marshallian demand functions straight from the indirect utility function

Answer 23

• v(c(v,p),p) = v
• differentiate both sides with respect to pi
• rearrange
• by Shepards Lemma, we know (Dc(v,p))/Dpi = g(v,p)

Roys identity: fi(y,p) = -((Dv(y,p)/Dpi)/(Dv(y,p)/Dy))

Question 24

Slutsky Equation

Answer 24

• differentiate with respect to p
• compensated price effect (Hicksian Demand) = uncompensated price effect (Marshallian demand) + (marshallian demand for good j).(income effect)
• also negativity applies, so Hicksian demand curves always slope downward, as an increase in price always reduces demand when utility is held constant

Question 25

Compensated cross price derivative relations

Answer 25

Symmetric - holding utility constant, the effect of increasing the price of one good on the quantity chosen of another is numerically identical to the effect of increasing the price of the other good on the q chosen of the 1st good

Question 26

Adding up condition

Answer 26

Demands must lie on the budget constraint - p’f(y,p) = y - p’g(v,p) = c(v,p)

Question 27

Homogeneity

Answer 27

Increasing all incomes and prices in proportion leaves the budget constraint and therefore demands unaffected - fi(y,p) = fi(ky,kp) - gi(v,p) = gi(v,kp)

Question 28

Negativity of compensated own price effects

Answer 28

in particular, a compensated increase in any good’s price can only reduce demand for that good - slutsky equation.

Question 29

Symmetry of compensated cross price effects

Answer 29

Effect of change in price of good 1 on good 2 same as change in price of good 2 on good 1

Question 30

We know a system of demands is intégrable if any of the following hold:

Answer 30

- they were derived as solutions to the utility max or expenditure min problem given a well specified direct utility function - derived by Shephard’s Lemma from a well specified cost function or they were derived by RI from a well specified indirect utility function - they satisfy adding up, homogeneity, symmetry and negativity