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