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Utility fn.
x preferred y <=> u(x)>=u(y)

If U. fn. exists, then preference is rational

WARP
Weak Axim of Revealed Preference = Consistency among choices

Rational choice structure satisfies WARP
Converse may not be true

Walasian demand
 X is convex
 Competitive budget => Price takers
denoted x(p,w)

Properties of Walasian Demands
 1. homogeneous of degree 0: x(p,w)=x(ap,aw)
 2. Walas' law: px=w

Walasian demand: wealth effects
 Wealth expansion path=Engel path
 ={x(p,w):w>0}
 Normal good: income increases, then consumption inc.

Walasian demand: price effects
 =price expansion path
 ={x(p_k,p_l,w0): p_k>>0}
 price higher, consume lower
 => cf. Giffen good: price higher, consume higher
 substitution matrix=D_px(p,w)
 diagoal=ownprice effects
 offdiagoal=crossprice effects

Walasian demand 3 characteristics
 Homogeneity condition (home. degree of 0)
 Cournot Aggregation (Walas' law)
 Engel Aggregation (Walas' law)
cf. WART, rationality assumption not needed

x(p,w) satisfies WARP if px(p',w')<=w & x(p,w) is not same with x(p',w') implies p'x(p,w)>w'

Slutzky compensated law of demand
(p'p)[x(p',w')x(p,w)]<=0

total effects = wealth effects + substitute effects

Slutzky equation
 dx
 =D_px(p,w)dp+D_wx(p,w)dw
 =D_p x(p,w)dp + D_w x(p,w)[dp*x(p,w)]
 =[D_p x(p,w)+D_w x(p,w)x(p,w)']dp
 =S_s(p,w)dp

If WARP=> Law of Demand => dp*S_s(p,w)*dp <=0
==> S_s = substitute matrix is negative semidefinite

Substitute matrix is negative semidefinite
 1. diagonal<=0: ownprice effect
 2. ownprice effect > crossprice effect

S_s * P=0
because of Homo. degree of 0

Indifference properties
 monotonicity
 local nonsatiation (rule out thick I.C.)


Indirect utility fn.
Utility max. => x*(p,w) => U(x*(p,w))=V(p,w)

Properties of indirect u. fn.
 h.d. 0
 inc. in w
 noninc. in p
 quasiconvex (lower contour set is convex)
 preferent strictly convex & u(.) conti => V(.) conti.

Expenditure minimization
 min px
 s.t. u(x)>=u0
 => x*=h(p,u): Hicksian demand

Expenditure fn.
h(p,u)=> p*h(p,u) = e(p,u)

Properties of e(p,u)
 h.d. 1 in p
 strictly inc in u
 nondec. in p
 convex in p

Duality b/w UMP & EMP
 1. If x* is opt. in UMP s.t. px=w, then x* is optimal in the EMP s.t. u>=U(x*)
 2. If x* is opt. in EMP s.t. u>=u0, then x* is opt. in the UMP s.t. px*=w
 3. x(p,w)=h(p,V(p,w))
 4. h(p,u)=x(p,e(p,u))
5. e(p,u) is inverse of v(p,w)

Sheperd's lemma
d e(p,u)/ d p = h(p,u)

Roy's lemma
x=[dV(p,w)/dp]/[dV(p,w)/dw]

Slutzky thm.
dh/dp = dx/dp + dx/dw*x

CV vs. EV
 CV => u0 base
 EV => u1 base

Netput vector
y={q1,..,qn; z1,...,zm}

Input requirement set
Y(q0)={z: f(z)>=q0}

Output producible set
Y(z0)= {y: y<=f(z0) or (z0,y) included in Y} where Y is a production set

Possible Properties of production set
 1. nonempty
 2. closedness
 3. no free lunch (no input=> no output)
 4. possibility of inaction
 5. (strong) Free disposability (y included Y, and y'<=y, then y' included Y)
 6. irreversibility: y included Y, y cannot be included Y
 7. Additivity
 8. convexity

Returns to scale
 increasing return to scale
 constant "
 dec.

Non jointness & seperability

Profit fn. >> Hotelling's lemma

Cost fn. >> Shepherd's lemma

