Micro_Steve

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lucia831124
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80289
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Micro_Steve
Updated:
2011-04-18 02:36:59
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microeconomics
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Micro_Steve
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  1. Rational preference
    • complete
    • transitive
  2. Utility fn.
    x preferred y <=> u(x)>=u(y)
  3. If U. fn. exists, then preference is rational
  4. WARP
    Weak Axim of Revealed Preference = Consistency among choices
  5. Rational choice structure satisfies WARP
    Converse may not be true
  6. Walasian demand
    • X is convex
    • Competitive budget => Price takers

    denoted x(p,w)
  7. Properties of Walasian Demands
    • 1. homogeneous of degree 0: x(p,w)=x(ap,aw)
    • 2. Walas' law: px=w
  8. Walasian demand: wealth effects
    • Wealth expansion path=Engel path
    • ={x(p,w):w>0}
    • Normal good: income increases, then consumption inc.
  9. 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
  10. Walasian demand 3 characteristics
    • Homogeneity condition (home. degree of 0)
    • Cournot Aggregation (Walas' law)
    • Engel Aggregation (Walas' law)

    cf. WART, rationality assumption not needed
  11. 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'
  12. Slutzky compensated law of demand
    (p'-p)[x(p',w')-x(p,w)]<=0
  13. total effects = wealth effects + substitute effects
  14. 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
  15. If WARP=> Law of Demand => dp*S_s(p,w)*dp <=0
    ==> S_s = substitute matrix is negative semidefinite
  16. Substitute matrix is negative semidefinite
    • 1. diagonal<=0: own-price effect
    • 2. |own-price effect| > cross-price effect
  17. S_s * P=0
    because of Homo. degree of 0
  18. Indifference properties
    • monotonicity
    • local non-satiation (rule out thick I.C.)
  19. Utilily Maximization
    • max U(x)
    • s.t. xp<=w
  20. Indirect utility fn.
    Utility max. => x*(p,w) => U(x*(p,w))=V(p,w)
  21. 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.
  22. Expenditure minimization
    • min px
    • s.t. u(x)>=u0
    • => x*=h(p,u): Hicksian demand
  23. Expenditure fn.
    h(p,u)=> p*h(p,u) = e(p,u)
  24. Properties of e(p,u)
    • h.d. 1 in p
    • strictly inc in u
    • non-dec. in p
    • convex in p
  25. 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)
  26. Sheperd's lemma
    d e(p,u)/ d p = h(p,u)
  27. Roy's lemma
    x=-[dV(p,w)/dp]/[dV(p,w)/dw]
  28. Slutzky thm.
    dh/dp = dx/dp + dx/dw*x
  29. CV vs. EV
    • CV => u0 base
    • EV => u1 base
  30. Netput vector
    y={q1,..,qn; -z1,...,zm}
  31. Input requirement set
    Y(q0)={z: f(z)>=q0}
  32. Output producible set
    Y(z0)= {y: y<=f(z0) or (z0,y) included in Y} where Y is a production set
  33. 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
  34. Returns to scale
    • increasing return to scale
    • constant "
    • dec.
  35. Non jointness & seperability
  36. Profit fn. >> Hotelling's lemma
  37. Cost fn. >> Shepherd's lemma

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