Neary Flashcards
Rationality conditions
- Completeness
2. Transitivity
WARP
If x chosen from any B containing x and y, then y never chosen from any B containing both x and y
p.x(p’,w’)=w’
Revealed preference relation
x is at least once chosen when y is available
need not be complete or transitive
rational preferences imply WARP will be observed but not viseversa
Which implies which:
utility function exists, rationality, WARP
U(.) exists means preferences must be rational which means WARP will be satisfies
But WARP is necessary but insufficient for rationality which itself almost always (but not always) implies the utility fn exists
Engel curve
For fixed prices, the bundle demanded as a function of wealth
Gradient=∂x(p, w)/∂w
Walras’ law in budget shares
budget share of product l: bl(p,w)=pl*xl(p,w)/w
W.L.: sum of bl=1
Normal vs inferior products
income effect positive vs negative
income elasticity positive vs negative
Luxury good vs necessity
budget share increases with wealth (so will also be normal) vs decreases with wealth (all inferior are necessities)
income elasticity > 1 vs < 1
Giffen good
negative own price effect
f is homogeneous of degree k if
x · ∇ f(x) = k f(x)
Engel Aggregation
sum of budget shares times income elasticities for all goods is 1
Cournot Aggregation
budget share of k plus sum of budget shares of all times cross-price elasticities of all with k = 0
Compensated price changes
Slutsky wealth compensation
to identify the substitution effect
change the agent’s wealth so that the original bundle is still just affordable
change in w=(p’-p).x(p,w)
What does Law of Demand say about Giffen goods
Impossible when price rises are compensated (assuming rationality ie exhausted budget set)
SARP
if x1 is revealed preferred to xn via a chain of pair-wise comparisons, then xn cannot be revealed preferred to x1 in a pair-wise comparison
Essential WARP with transitivity of weak preference relation
If u(.) is quasi-concave
Then preferences are convex (think indifference curves)
Homothetic preferences
if indifferent between x and y then also indifferent between kx and ky
utility function is h.o.d. 1
Quasi-linear preferences
take 2 indifferent bundles. If you add x units of good 1 to both bundles, then still indifferent
Utility functions: U=x1+u(x2…xL)
x1 often used as money
Quasi-concavity implies
The solution to the consumer’s problem is unique
through convexity of preferences
Indirect utility
The maximum utility obtainable with prices p and wealth w
Convex shape of price indifference curves for indirect utility means
Prefer extreme prices to average prices
expenditure function
minimum p.x s.t. u(x)>=u
h(p,u)
Compensated demand function. Demand for each good given prices and utility to achieve.
Compensated means own price effect must be negative (no income effect here)
e(p,u) is convex or concave in p?
Concave
Shephard’s Lemma
partial of e(p,u) wrt pl=hl(p,u)
Hicksian Substitutes vs Complements
Partial of hl wrt pk>0 vs <0
An increase in the price of good k increases vs decreases demand for good l
Each product always has at least one substitute
Since own price effect is negative and sum of price effects =0 as h is hod0
Gross subs
partial of xl wrt pk>0
Edgeworth Subs
second partial of u wrt xk, xl <0
Roy’s identity
xl(p,w)=-partial of v wrt pl/partial of v wrt w
Proof starts by differentiation u=v(p,e(p,u))
The Slutsky Equation
partial of xl wrt pk= substitution effect and wealth effect
=partial of hl wrt pk - partial of xl wrt w * xk(p,w)
If a good is normal and gross substitutes with k then
the two products must also be Hicksian substitutes
With homothetic preferences:
e(p,u)=
v(p,w)=
x(p,w)=
u(.) is hod1 in x:
e=e(p)u
v=v(p)w
x=x(p)*w
With quasi-linear preferences, Hicksian and Walrasian demand functions for x
Coincide as there is no wealth effect
Slutsky wealth compensation after a price change
Gives the consumer enough wealth that they can buy the old bundle
change in wealth=(p’-p)*x(p,w)
But can usually do better with this wealth with the new prices and will choose a different bundle and be happier than they were
Hicksian wealth compensation
Gives the consumer enough wealth so their utility is unchanged.
w’=e(p’,v(p,w))
If yl is positive vs negative
it is an output vs input
Hotelling’s Lemma
the partial of the profit function wrt pl = yl(p)
1st step of Hotelling’s Lemma proof
let G(p)=p.y(p*)-profit function of p
Law of supply
Prices and quantities move in the same direction
Firms version of Shephard’s Lemma
partial of the cost function c(w,q) wrt wl=zl(w,q) which is the conditional factor demand function (w is input prices)
Production is efficient if
Can’t produce more output with same inputs or same output with fewer inputs
The long run cost curve
The lower envelope of the short-run cost curves (the locus of minimum points)
Certainty equivalent of a lottery
C satisfies u(C)=E(u(x))
Risk premium
P satisfies u(E(x)-P)=E(u(x))
p=E(x)-C
Coefficient of ARA
Ra=-u’‘(x)/u’(x)>0
CARA
constant absolute risk aversion
No income effect in the demand for insurance
DARA
decreasing absolute risk aversion
Insurance is an inferior good, pay less to eliminate a small risk
If the utility function is quadratic, expected utility only depends on
Mean and variance of the distribution of the shock
Coefficient of relative risk aversion
Rr(x)=-xu’‘(x)/u’(x)
First step of proof that maximised profit function is convex
let p’‘=ap+(1-a)p’
