Period 1

Question 1

Name the identity: h(p, u) = …

Answer 1

d(p, E(p,u))

Question 2

Name the identity: d(p, I) = …

Answer 2

h(p, V(p, I))

Question 3

Name the identity: E(…, ….) = …

Answer 3

E(p, V(p, I)) = I

Question 4

Name the identity: V(…, …) = …

Answer 4

V(p, E(p, u)) = u

Question 5

When is a set of alternatives complete?

Answer 5

If either a (pref) b, b (pref) a, or both.

Question 6

When is a set of alternatives transitive?

Answer 6

If a (pref) b, b (pref) c, then a (pref) c

Question 7

When is a set of alternatives reflexive?

Answer 7

An alternative is always as good as itself, a (pref) a, for all a in A

Question 8

What is the definition of the best element?

Answer 8

a* in A, a* (pref) a, for all a in A

Question 9

When is a set of alternatives guaranteed of a best element?

Answer 9

If a set is finite, (pref) is complete and transitive.

Question 10

When does a set of alternatives have a single best element?

Answer 10

If a set is finite, (pref) is complete, transitive, and asymmetic.

Question 11

What is the definition of a convex preference relation?

Answer 11

if x_1 (pref) x_0, then for all t in (0, 1):

t x_1 + (1-t)x_0 (pref) x_0

Question 12

What is the definition of a strictly concave utility function?

Answer 12

for all x, y in X and t in (0, 1)

u(tx + (1-t)y) > tu(x) + (1-t)u(y)

Question 13

What is the definition of a strictly quasi-concave utility function?

Answer 13

for all x, y in X, and t in (0, 1):

u(tx + (1-t)y) > min(u(x), u(y))

Question 14

How to get the Marshillian demand function?

Answer 14

1. Get MU_k(x)/MU_1(x) = p_k/p_1
2. Free up x_1 (or x_k)
3. Plug this into the budget constraint (p_1 * x_1 + p_2 * x_2 = I)
4. Free up x_1 (or x_k), get d_1(p, I)

Question 15

How to get the compensated demand?

Answer 15

1. Get MU_k(x)/MU_1(x) = p_k/p_1
2. Free up x_1 (or x_k)
3. Plug this into the utility function (U(x)), equals to u
4. Free up x_1 or (x_k) get h_1(p, u)

Question 16

What is the maximum profit for perfect competition?

Answer 16

p - MC(q) = 0

Question 17

What is the maximum profit for a monopoly?

Answer 17

P(q) + q dP(q)/dq - dC(q)/dq = 0

Question 18

How to get the contingent demand function?

Answer 18

1. Get MP_k(x) / MP_1(x) = w_k/w_1 (just the derivative of f(x))
2. Free up x_1 (or (x_k)
3. Plug this into the production function and equals to q
4. Solve for x_1 (to get d_1(w, q))

Question 19

How to show the proof with concordet winners?

Answer 19

Proof by contradiction

Start with definition of a(bar).

Suppose that b (pref)^p a(bar) exists. Start with b < a(bar)

Get n^p(b, a(bar)) = #{} = #{… pi > a(bar)), and other.

Show that n^p(a(bar), b) > n^p(b, a(bar)), and thus contradiction.

Claim also for other b. QED.

Question 20

How to find TOP(D)?

Answer 20

All elements that are either the best, or in a cycle.

Question 21

When is an alternative covered?

Answer 21

If an alternative b that is defeated by a, all the alternatives that b defeated are also defeated by a.

Question 22

How to calculate the β-score?

Answer 22

The sum 1/#pred_b(D), for all b in Succ_a(D)

Question 23

What is a successor?

Answer 23

The set of alternatives “defeated” by an alternative a

Question 24

What is a predecessor?

Answer 24

The set of alternatives that defeat an alternative

Question 25

How to proof a unique maximizing utility?

Answer 25

_Proof by contradiction_ Suppose **at least** two of the best elements (e.g. x\*, x\*\* in X). If convex, then definition. Then definition of strictly quasi-concave utility, (or asymmetic, strictly convex (pref)). Show contradiction. QED.

Question 26

What is the demand function?

Answer 26

d(p, I)

Question 27

What is the compensated demand function?

Answer 27

h(p, u)

Question 28

What is the indirect utility?

Answer 28

V(p, I) = U(d(p, I))

Question 29

What is the expenditure function?

Answer 29

E(p, u) = p \* h(p, u)