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Utility fn.
x preferred y <=> u(x)>=u(y)
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If U. fn. exists, then preference is rational
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WARP
Weak Axim of Revealed Preference = Consistency among choices
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Rational choice structure satisfies WARP
Converse may not be true
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Walasian demand
- X is convex
- Competitive budget => Price takers
denoted x(p,w)
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Properties of Walasian Demands
- 1. homogeneous of degree 0: x(p,w)=x(ap,aw)
- 2. Walas' law: px=w
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Walasian demand: wealth effects
- Wealth expansion path=Engel path
- ={x(p,w):w>0}
- Normal good: income increases, then consumption inc.
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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=own-price effects
- off-diagoal=cross-price effects
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Walasian demand 3 characteristics
- Homogeneity condition (home. degree of 0)
- Cournot Aggregation (Walas' law)
- Engel Aggregation (Walas' law)
cf. WART, rationality assumption not needed
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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'
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Slutzky compensated law of demand
(p'-p)[x(p',w')-x(p,w)]<=0
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total effects = wealth effects + substitute effects
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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
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If WARP=> Law of Demand => dp*S_s(p,w)*dp <=0
==> S_s = substitute matrix is negative semidefinite
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Substitute matrix is negative semidefinite
- 1. diagonal<=0: own-price effect
- 2. |own-price effect| > cross-price effect
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S_s * P=0
because of Homo. degree of 0
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Indifference properties
- monotonicity
- local non-satiation (rule out thick I.C.)
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Indirect utility fn.
Utility max. => x*(p,w) => U(x*(p,w))=V(p,w)
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Properties of indirect u. fn.
- h.d. 0
- inc. in w
- non-inc. in p
- quasi-convex (lower contour set is convex)
- preferent strictly convex & u(.) conti => V(.) conti.
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Expenditure minimization
- min px
- s.t. u(x)>=u0
- => x*=h(p,u): Hicksian demand
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Expenditure fn.
h(p,u)=> p*h(p,u) = e(p,u)
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Properties of e(p,u)
- h.d. 1 in p
- strictly inc in u
- non-dec. in p
- convex in p
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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)
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Sheperd's lemma
d e(p,u)/ d p = h(p,u)
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Roy's lemma
x=-[dV(p,w)/dp]/[dV(p,w)/dw]
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Slutzky thm.
dh/dp = dx/dp + dx/dw*x
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CV vs. EV
- CV => u0 base
- EV => u1 base
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Netput vector
y={q1,..,qn; -z1,...,zm}
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Input requirement set
Y(q0)={z: f(z)>=q0}
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Output producible set
Y(z0)= {y: y<=f(z0) or (z0,y) included in Y} where Y is a production set
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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
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Returns to scale
- increasing return to scale
- constant "
- dec.
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Non jointness & seperability
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Profit fn. >> Hotelling's lemma
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Cost fn. >> Shepherd's lemma
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