A6_Comparative statistics and duality theory

maximum utility achievable with a given income

V(p,m) = U(x(p,m))

maximization problem

two optimality conditions:

find the MRS.

MRS = p1=p2

fit that into the budget line m for optimal demand x1 and x2

holds m constant

λ=dV/dm how much more utility if one unit more income

expenditure minimization e(p, U_head)

the solution to the min problem is χ(p, U_head)

first optimality condition: MRS=MOC

second: fix a utility level (a specific indifference curve) and find minimal expenditure necessary

χ is the quantity consumed

holds U_head constant

μ= de(p,U_head)/dU_head increasing U_head by one unit, how much more expenditure

μ=1/λ

x(ap,am)=x(p,m)

V(ap,am)=V(p,m)

χ(ap,Uf)=χ(p,Uf) Uf= U max

e(ap,Uf)=ae(p,Uf)

HH cant do any better, even if his m would increase a lot

continous utility function

local nonsatiation, price vector p is positive (= monotonicity)

formulas in book p. 177ff

increase the price of one good in the goods vector

if the HH consumes the same bundle as before (same utility) he needs to increase expenditure

HH can reshuffle his expenditure minimizing bundle.

Implication: doesnt make sense if small price increase

it is concave p.188 in book not ipad

if you have f(a,x) with a as parameter and x as variable.

x^R(a) is your best response fct.

f_head(a,x^R(a)) is called a reduced fct. A change in a changes f_head through a and indirectly through x^R(a).

-> envelop theorem claims that we can forget about the indirect part

imagine f(a,x). x has to obey the constraint g(a,x)=0.

f_head(a, x^R(a)). A change in a influence f directly and indirectly through x^R(a).

now with lagrangian: L(a,x,λ):=f(a,x) + λg(a,x)

if you dL/da, we get the effect of the parameter on the optimal value and can ignore the effect to the optimal response x^R(a).

proofs in the book

how does minimal expenditure vary with a change in p of a good x_g?

• f(a,x) is overtaken by e(p_g,x)=p*x

• f_head = f(a,x^R(a)) is translated into e_head(p_g):=p*χ(p_g)

• equality constraint is U(x)-U_head=0

the equality constraint does not depend on p_g

forget about indirect effects

de_head/dp_g=χ_g (shephards lemma)

χ_g is the expenditure minimizung bundle

Roys identity: using U_head=V(p,e(p,U_head))

differentiate both sides by p_g:

0=dV/ …

this yields: dV/dpg=dV/dm(-χ_g)

a price increase increases necessary expenditure to keep the utility level constant by χ_g (shephards lemma).

if the budget is given, the budget for the other good decreases by χ_g. Then the budget restriction becomes a reduction of utility.

h. law of demand: demand moves inversely to prices

dχ_g/dp_g ≤ 0

two ways to explain concavity

if the convex combination kx+(1-k)y is greater than the convex combination of the values f(x) and f(y).

or f’’(x)≤0 <- Hesse matrix (l x l matrix whose entries are all f’’(x)

eine steigung kann zwar noch positiv sein, aber wenn sie immer weniger positiv wird, ist sie concave

fij(x):=d(df(x)/dx_i)/dx_j)

symmetric if all f’’ are continous

(strictly) concave if its hesse matrix negative semidefinite (definite)

(strictly) convex if its hesse matrix is positive semidefinite (definite)

its concavity has some implications

a change in p_g changes χ_g negatively

-> the hesse matrix of e is negative semidefinite

a change in p_g changes χ_k equally as a a change in p_k changes χ_g.

χ_g is just the hicksian demand curve (kinda like x_g for marshallian)

if the change is positive goods g and k are substitutes

if change negative they are called complements

two substitution effects for marshallian demand if price changes:

first: how much are you willing to pay for a bahncard 50 in terms of good x_2 to reduce price of good x_1 by 50%?

second: how much are you willing to change bundle to have higher utility

if good g is normal (budget increase leads to more demand), it is also ordinary (price increase leads to less demand). Then a price increase has stronger effect on marshalllian demand than hicks demand.

if income effect (dx_g/dm)=0, hicksian demand does not depend on the utility level attained.

remember x_money(p,p*w)=x_endo(p,w)

if a good is normal and HH consumes more than his endowment, it is also ordinary.

variation (sum of money) is equivalent to an event if it leads to the same indifference curve.

you get money if air quality is reduced or vice versa.

two kinds of variations: equivalent and compensating

equivalent (willingness to pay) from b -> a (lower air quality) would mean reduction of income: m_2-m_1.

compensation (compensation cost) would be: m_3-m_2

willingness to pay that something doesnt happen

something occurs, EV indicates the same loss of utility if it was to occur on the other good

one good given to you in exchange for reducing another good to achieve the same utility