Preference and Utility
continuity assumption
preference preserves under limits: xn>=yn, we have x>=y
The upper contour set and the lower contour set is closed
raionality+continuity=>utility function
rationality+continuity+monotonicity=>exists a continuous utlity function
convexity and monotonicity of preference=>quasi-concave, some sort of monotonicity
form of homothetic and quasi-linear*
Utility maximization
Basic assumptions:
preference: rational, continuous, local nonsatiation
u(.): continuous
budget sets: p>>0, w>0
existence of solutions
optimal solutions: x(p,w)
optimal value: V(p,w)
Homogeneity
walras' law
convexity/uniqueness
Homogeneiry
strictly increasing in w, nonincreasing in p
quasi-convexity in (p,w)
continous in (p,w)
Expenditure Minimization
Basic assumptions:
U(.) continuous, local nonsatiable;
p>>0, u>u(0)
Optimal choice: h(u,p)
homogeneity in p
no excess utility
convexity and uniqueness
Expenditure function e(u,p)
homogeneity in p
strictly increasing in u and nondecreasing in p
concave in p
continuous in u and p
EMP and UMP
given w>0, h(p,V(p,w))=x(p,w), e(p,V(p,w))=w
given u>u(0), x(p,e(p,u))=h(p,u), V(p,e(p,u))=u
Basic assumption:
u(.) continuous, local nonsatiable;
perference strictly convex,
p>>0
x(p,w) single-valued
partial e(p,u)/partial p=h(p,u)
Dph
Dph=D^2 e(p,h)
Dph is nsd
Dph is symmetric
Dph p=0
Hicksian Demand: [p''-p']\cdot [h(p'',u)-h(p',u)]<=0
Connected to Slusky matrix