Market Demand & Welfare

Question 1

How is aggregate demand obtained from individual demand curves?

Answer 1

Horizontal summation on p against q axes to get total amount demanded at each price

Question 2

What is the price elasticity of demand?

Answer 2

PED = ε(p) = p_i/x_i* dx_i*/dp_i
The percentage change in quantity for some (marginal) change in price, or how easy it is to substitute one good for another

Question 3

How can PED be categorised for a good with market price p?

Answer 3

|ε(p)| > 1: elastic demand at price p
|ε(p)| < 1: inelastic demand at price p
ε(p) = -1: unit elastic demand at price p
PED can vary across a demand curve, e.g. for linear demand

Question 4

What is the PED for each good in the optimal choice under Cobb-Douglas preferences?

Answer 4

-1 so Cobb-Douglas demand is unit elastic for each good at every price
(work through formula in your head!)

Question 5

How does revenue relate to elasticity?

Answer 5

R = pq for price p and market demand q = q(p)
dR/dp = q + p dq/dp = q(1 + pdq/qdp) = q(1 + ε) with both q and ε depending on p
Marginal increase in price from p would lower revenue if PED elastic, increase revenue if PED inelastic

Question 6

What is Income Elasticity of Demand?

Answer 6

ε_m = mdx_i/x_idm

Question 7

How can a good be characterised by its income elasticity of demand?

Answer 7

ε_m < 0: inferior good
ε_m > 0: normal good
ε_m > 1: luxury good (also normal)

Question 8

What is Cross-Price Elasticity of Demand?

Answer 8

ε_{i, j} = p_jdx_i/x_idp_j

Question 9

What are the income and cross-price elasticities of Cobb-Douglas demand?

Answer 9

ε_m = 1
ε_{i, j} = 0

Question 10

What is consumer surplus?

Answer 10

The difference between the total amount a consumer would have been willing to pay for an amount of the good and the amount they actually paid

Question 11

How is the demand function usually graphed?

Answer 11

Plotting P = P(q), the inverse demand function, on P against q axes

Question 12

How is consumer surplus found from the demand curve?

Answer 12

When x units are bought, CS = ∫₀^x( P(q)dq) - xp(x)

Question 13

What are the two ways to represent the change in consumer surplus from a change in price?

Answer 13

On the demand curve in (x, p) space, the area between the original price level and the previous price level
On the demand curve in (p, x) space, the area between the original and new price level = ∫_p’’^p’ x(p, m)dp for a fall in price from p’ to p’’

Question 14

What is duality?

Answer 14

The fact that the same bundle will solve the maximal utility given some prices and income (tangency between BL and Marshallian demand curve) and the mimimum cost to achieve some level of utility given some prices (tangency between IC and Hicksian demand curve)

Question 15

What is the difference between Marshallian and Hicksian demand?

Answer 15

Marshallian demand, denoted x*(p, m), solves max u subject to budget constraint
Hicksian demand, denoted h(p, ū), solves min x ⋅ p subject to u(x) ≥ ū

Question 16

How can duality be expressed symbolically?

Answer 16

h(p, u(x(p, m))) = x(p, m) and x*(p, h(p, ū) ⋅ p) = h(p, u)

Question 17

How does Hicksian demand relate to the substitution effect?

Answer 17

The substitution effect moves from the original choice to h(p’’, u’) where p’’ is the new price bundle and u’ is the original utility
The original choice can be written h(p’, u’) so the substitution effect is h(p’’, u’) - h(p’, u’)

Question 18

What is the Slutsky equation?

Answer 18

∂x_i/dp_i = ∂h_i/∂p_i - x_i∂x_i/∂m
The total change in (Marshallian) demand from a change in price is the change in the Hicksian demand from the price (substitution effect) less the change in Marshallian demand from the change in effective income

Question 19

What is the expenditure function and how does it relate to duality?

Answer 19

The expenditure function e(p, ū) = h(p, ū) ⋅ p gives the cost of the cheapest bundle which achieves some utility
Duality can also be expressed h(p, ū) = x*(p, e(p, ū))

Question 20

What is Shephard’s lemma?

Answer 20

Shephard’s lemma gives the change in cost of achieving some utility as the price of a good changes
∂e(p, ū)/∂p_i = h_i(p, ū)

Question 21

How could you derive Shephard’s lemma for the case of two goods?

Answer 21

Considering the cost minimisation problem subject to achieving some utility level, assuming an interior optimum, the tangency condition must be met (ratio of MUs equal to ratio of prices) and solving this gives the Hicksian demand (h_x, h_y) associated with the expenditure function e(p, ū) = p ⋅ h*
Using the fact that the utility target is not changing, equate the partial derivative of utility wrt p_x (MUs dot multiplied with partial derivatives of Hicksian wrt p_x) to zero, then substitute into this the tangency condition to replace one of the MUs so the remaining one can be cancelled out
Substitute this expression into the partial derivative of the expenditure function wrt p_x to get the result

Question 22

How can the Slutsky equation be derived from duality and Shephard’s lemma?

Answer 22

Duality means h_i(p, ū) = x_i(p, e(p, ū))
Differentiating this wrt p_i using chain rule gives ∂h_i/∂p_i = ∂x_i/∂p_i + ∂x_i∂e/∂m∂p_i
Using Shephard’s lemma and replacing the resulting h_i with x_i gives the Slutsky equation

Question 23

When is CS = u?

Answer 23

The quasi-linear utility function u(x, y) = v(x) + y has consumer surplus equivalent to consumer’s utility as long as x is some good and y is the amount of money spent on all other consumption

Question 24

Why are EV and CV used to measure welfare changes?

Answer 24

The change in income equivalent to moving from one indifference curve to another is often the best available metric of change in utility (welfare)

Question 25

What is the indirect utility function and how can it be used to express duality?

Answer 25

The indirect utility function gives the utility of the bundle found with the Marshallian demand function
v(p, m) = u(x*(p, m))
Duality can be written e(p, v(p, m)) = m

Question 26

What is Equivalent Variation?

Answer 26

The change in income required to reach the new utility level under old prices
EV = e(p’, v(p’’, m’’)) - e(p’, v(p’, m’))
= e(p’, v(p’’, m’’)) - m
= e(p’, u’’) - e(p’, u’)
= e(p’, u’’) - m
= ∫_p1’’^p₁’ h₁(p, u’‘)dp₁ (as long as only one price change)
= e(p’, u’’) - e(p’’ - u’’) (as long as only one price change)

Question 27

What is Compensating Variation?

Answer 27

The change in income required to return to the old utility level under new prices
EV = e(p’’, v(p’’, m’’)) - e(p’’, v(p’, m’))
= m - e(p’’, v(p’, m’))
= e(p’’, u’’) - e(p’’, u’)
= m - e(p’’, u’)
= ∫_p1’’^p₁’ h₁(p, u’)dp₁ (as long as only one price change)
= e(p’, u’) - e(p’’ - u’) (as long as only one price change)

Question 28

How could the difference between EV and CV be illustrated after p₁ falls?

Answer 28

The budget line will pivot anti-clockwise around the y-intercept to move from being tangent to one IC to another
EV is the vertical difference between the original BL and the compensated BL with the original slope that would intersect the new IC
EV is the vertical difference between the new BL and the compensated BL with the new slope that would intersect the original IC

Question 29

How can you find the relationship between CS, EV, and CV for a normal good when price decreases?

Answer 29

CS is the integral from new to old price of Marshallian demand wrt the changing price, CV is this integral of Hicksian demand with original utility, EV is this integral of Hicksian demand with new utility, so the relationship will depend on the relationship between these demand functions
Duality shows that the Marshallian demand function must go from meeting one Hicksian function to the other which produces the result that CV < CS < EV

Question 30

What is the diagram that can be used to find the integral expression for EV and CV and to show the relationship between the three measures of welfare?

Answer 30

Setting p₂ = 1, consider a decrease in p₁ so that x(p, m) goes from X’ on IC(u’) to X’’ on IC(u’’), with EV being the vertical distance from the original BL and the line with the same slope intersecting H’’ on IC(u’’) and CV being the vertical distance from the new BL and the line with the same slope intersecting H’ on IC(u’)
As price changes from p₁’ to p₁’’, Hicksian demand with new utility slides along IC(u’’) from H’’ to X’’, Hickian demand with old utility slides along IC(u’) from X’ to H’, and Marshallian demand moves from X’ to X’’ along the curve between these from the BL pivoting