Period 1 Flashcards
Name the identity: h(p, u) = …
d(p, E(p,u))
Name the identity: d(p, I) = …
h(p, V(p, I))
Name the identity: E(…, ….) = …
E(p, V(p, I)) = I
Name the identity: V(…, …) = …
V(p, E(p, u)) = u
When is a set of alternatives complete?
If either a (pref) b, b (pref) a, or both.
When is a set of alternatives transitive?
If a (pref) b, b (pref) c, then a (pref) c
When is a set of alternatives reflexive?
An alternative is always as good as itself, a (pref) a, for all a in A
What is the definition of the best element?
a* in A, a* (pref) a, for all a in A
When is a set of alternatives guaranteed of a best element?
If a set is finite, (pref) is complete and transitive.
When does a set of alternatives have a single best element?
If a set is finite, (pref) is complete, transitive, and asymmetic.
What is the definition of a convex preference relation?
if x_1 (pref) x_0, then for all t in (0, 1):
t x_1 + (1-t)x_0 (pref) x_0
What is the definition of a strictly concave utility function?
for all x, y in X and t in (0, 1)
u(tx + (1-t)y) > tu(x) + (1-t)u(y)
What is the definition of a strictly quasi-concave utility function?
for all x, y in X, and t in (0, 1):
u(tx + (1-t)y) > min(u(x), u(y))
How to get the Marshillian demand function?
- Get MU_k(x)/MU_1(x) = p_k/p_1
- Free up x_1 (or x_k)
- Plug this into the budget constraint (p_1 * x_1 + p_2 * x_2 = I)
- Free up x_1 (or x_k), get d_1(p, I)
How to get the compensated demand?
- Get MU_k(x)/MU_1(x) = p_k/p_1
- Free up x_1 (or x_k)
- Plug this into the utility function (U(x)), equals to u
- Free up x_1 or (x_k) get h_1(p, u)
What is the maximum profit for perfect competition?
p - MC(q) = 0
What is the maximum profit for a monopoly?
P(q) + q dP(q)/dq - dC(q)/dq = 0
How to get the contingent demand function?
- Get MP_k(x) / MP_1(x) = w_k/w_1 (just the derivative of f(x))
- Free up x_1 (or (x_k)
- Plug this into the production function and equals to q
- Solve for x_1 (to get d_1(w, q))
How to show the proof with concordet winners?
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.
How to find TOP(D)?
All elements that are either the best, or in a cycle.
When is an alternative covered?
If an alternative b that is defeated by a, all the alternatives that b defeated are also defeated by a.
How to calculate the β-score?
The sum 1/#pred_b(D), for all b in Succ_a(D)
What is a successor?
The set of alternatives “defeated” by an alternative a
What is a predecessor?
The set of alternatives that defeat an alternative
How to proof a unique maximizing utility?
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.
What is the demand function?
d(p, I)
What is the compensated demand function?
h(p, u)
What is the indirect utility?
V(p, I) = U(d(p, I))
What is the expenditure function?
E(p, u) = p * h(p, u)
