Micro - Winter '16

Front 1

Preference relation

Back 1

Binary relation that is complete and transitive

Front 2

Utility function existence (binary relation must be...)

Back 2

... complete, transitive, continuous, strictly monotonic

Front 3

Completeness

Back 3

for all x1 and x2, either x1 >~ x2 or x2 >~ x1

Front 4

Transitivity

Back 4

if x1 >~ x2 and x2 >~ x3, then x1 >~ x3

Front 5

Continuity

Back 5

>~(x) and <~(x) are closed

implies ~(x) is closed b/c it is the intersection of these two

Front 6

Strict monotonicity

Back 6

if x0 >= x1 then x0 >~ x1 and if x0 >> x1 then x0 > x1

Front 7

Convexity

Back 7

If x1 >~ x0 then t x1 + (1 - t) x0 >~ x0 for all t in [0,1]

Front 8

Invariance of utility to positive monotonic transformation

Back 8

if and only if!!!

Front 9

Properties of preferences and utility functions

Back 9

1. u(x) is strictly increasing iff >~ is strictly monotonic

2. u(x) is quasiconcave iff >~ convex

3. u(x) strictly quasiconcave iff >~ strictly convex

Front 10

MRS

Back 10

absolute value of indifference curve through point

indifference curve: u(x1, f(x1)) = constant

MRS = -f'(x1)

Front 11

Indirect utility function

Back 11

demand: x(p, y)

v(p, y) = u(x(p, y))

Front 12

Properties of indirect utility function

Back 12

if u(x) is continuous and strictly increasing, then v(p, y) is:

1. Continuous

2. Homogeneous degree 0 in (p, y)

3. Strictly increasing in y

4. Decreasing in p

5. Quasiconvex in (p, y)

6. Roy's identity:

x_i(p0, y0) = -(dv(p0, y0)/dp_i) / (dv(p0, y0)/dy)

Front 13

Expenditure function

Back 13

hicksian demand: x^h(p, u)

e(p, u) = p * x^h(p, u)

Front 14

Properties of the expenditure function

Back 14

If u is continuous and strictly increasing, then e(p, u):

1. e(p, 0) = 0

2. continuous

3. if p >> 0, strictly increasing and unbounded above in u

4. increasing in p

5. homogeneous degree 1 in p

6. concave in p

7. Shephard's lemma:

de(p0, u0) / dp_i = x_i^h(p0, u0)

Front 15

Relation between indirect utility and expenditure

Back 15

e(p, v(p, y)) = y

v(p, e(p, u)) = u

e(p, u) = v^-1(p; u)

v(p, y) = e^-1(p; y)

Front 16

Core of exchange economy

Back 16

set of all unblocked feasible allocations

Front 17

Unique demand

Back 17

if u is continuous, strongly increasing, and strictly quasiconcave, then demand is unique and continuous in p

existence follows because p>>0 implies bounded budget set

uniqueness follows from strict quasiconcavity of u

Front 18

Properties of aggregate excess demand

Back 18

if all u^i are continuous, strongly increasing, and strictly quasiconcave, then for all p>>0:

1. z is continuous at p

2. z(t p) = z(p) for all t > 0

3. p * z(p) = 0 (Walras' law)

Front 19

Walrasian eqm

Back 19

A price vector p* is called a Walrasian eqm if z(p*) = 0

Front 20

Aggregate excess demand and Walrasian eqm

Back 20

If:

1. z is continuous

2. p * z(p) = 0 for all p >> 0

3. If {p^m} is a sequence of price vectors converging to pbar != 0, and pbar_k = 0 for some k, then for some good k' with pbar_k' = 0, the associated sequence of excess demands in the market for good k', {z_k'(p^m)} is unbounded above

Then there is a price vector p* >> 0 such that z(p*) = 0.

Front 21

Utility and aggregate excess demand

Back 21

if all u^i are continuous, strongly increasing, and strictly quasiconcave and if aggregate endowment of each good is strictly positive, then excess demand z satisfies:

1. z is continuous

2. p * z(p) = 0 for all p >> 0

3. If {p^m} is a sequence of price vectors converging to pbar != 0, and pbar_k = 0 for some k, then for some good k' with pbar_k' = 0, the associated sequence of excess demands in the market for good k', {z_k'(p^m)} is unbounded above

Front 22

Existence of Walrasian eqm

Back 22

if all u^i are continuous, strongly increasing, and strictly quasiconcave and if aggregate endowment of each good is strictly positive, then there exists at least one price vector p* such that z(p*) = 0

Front 23

Arrow's four things

Back 23

U. Unrestricted domain

WP. Weak Pareto principle

IIA. Independence of irrelevant alternatives

D. Non-dictatorship

Front 24

Unrestricted domain

Back 24

The domain of f must include all possible combinationsof individual preference relations on X.

Front 25

Weak Pareto principle

Back 25

For any pair of alternatives x and y in X, if x Piy for alli, then xPy

Front 26

Independence of Irrelevant Alternatives

Back 26

Consider two sets of rankings R and R'. If everybody ranks x and y the same way in R and R', then the SOCIAL ranking of x and y in R and R' should be the same

Front 27

Non-dictatorship

Back 27

There is no individual i such that for all x and y in X, x Piyimplies x P y regardless of the preferences Rj of all other individuals j = i.

Front 28

Arrow's impossibility theorem

Back 28

If there are at least 3 social states, we can't come up with a social choice function that satisfies all of Arrow's four things

Front 29

First welfare theorem

Back 29

Consider an exchange economy (u^i, e^i)i \in I. If every consumer's utility function u^i is strictly increasing, then every Walrasian equilibrium allocation is Pareto efficient

Front 30

Second Welfare theorem

Back 30

Consider an exchange economy (u^i, e^i)i \in I with aggregate endowment >>0. Suppose u^i is continuous, strongly increasing , and strictly quasiconcave. Suppose xbar is a PE allocation for (u^i, e^i)i \in I and that endowments are redistributed so that the new endowment vector is xbar. Then xbar is a Walrasian equilibrium allocation of the resulting exchange economy (u^i, xbar^i)i \in I.

Front 31

Another look at the second welfare theorem

Back 31

Consider an exchange economy (u^i, e^i)i \in I with aggregate endowment >>0. Suppose u^i is continuous, strongly increasing , and strictly quasiconcave. Suppose xbar is a PE allocation for (u^i, e^i)i \in I. Then, xbar is a WEA for some WE pbar after redistribution of initial endowments to any feasible allocation e* such that pbar * e*^i = pbar * xbar^i for all i in I.

Front 32

Existence of Walrasian Eqm with production

Back 32

Consider the economy (u^i, e^i, theta^ij, Y^j)i \in I, j \in J.

If u^i satisfies:

1. continous

2. strongly increasing

3. strictly quasiconcave

And if Y^j satisifies:

1. 0 \in Y^j

2. Y^j is closed an bounded

3. Y^j is strongly convex (for all y1, y2 \in Y^j and all t \in (0, 1) there exists ybar \in Y^j such that ybar >= t y1 + (1 - t) y2 and equality does not hold)

And if y + endowment >> 0 for some production y \in sum Y^j, then there exists at least only price vector p* >> 0 such that z(p*) = 0.

Front 33

First welfare theorem with production

Back 33

Same as without production

Front 34

Second welfare theorem with production

Back 34

Suppose the conditions for existence of WE hold and (xhat, yhat) is PE. Then there are income transfers T1, ..., TI satisfying sum Ti = 0 and a price vector pbar such that:

1. xhat^i maximizes u^i(x^i) s.t. pbar * x^i <= m^i(pbar) + Ti

2. yhat^j maximizes pbar * y^j s.t. y^j \in Y^j

Front 35

Dictatorial social choice function

Back 35

A social choice function c is dictatorial if there is an individual i such that whenever c(R1, ..., RN) = x it is the case that x R^i y for every y \in X

Front 36

Strategy proof social choice function

Back 36

A social choice function c is strategy-proof when, for every individual, i, for every pair R^i and Rtilde^i of his preferences, and for ever profile R^-i of other's preferences if c(R^i, R^-i) = x and c(Rtilde^i, R^-i) = y then x R^i y

Front 37

Gibbard-Satterthwaite Theorem

Back 37

If there are at least three social states, then every strategy-proof social choice function is dictatorial

Front 38

Pareto Efficient social choice function

Back 38

A social choice function c is Pareto efficient if c(R1, ..., RN) = x whenever x P^i y for every individual i and every y \in X distinct from x

Front 39

Monotonic social choice function

Back 39

A social choice function c is monotonic if c(R1, ..., RN) = x implies c(Rtilde1, ..., RtildeN) = x whenever for each individual i and every y in X distinct from x, x R^i y implies x Ptilde^i y