I have a question about Topkis. In order to apply Topkis, do we have to have maximizaton problem or minimization is fine, too?

Our statement of Topkis has been for maximization of a supermodular function. You can write other (more general) forms that involve minimization and/or submodularity/decreasing differences. However, I always find it easiest to just work with the basic statement of the theorem, and map your problem into “Topkis form.”

Let’s say, my original problem is Min f(x,t) wrt x and i want to figure out whether my optimizer is increasing in x or not. Can I directly find the answer by just saying f function is supermodular in (x,t) ?

Moreover is it true that if I have increasing differences in (x,t) for minimization problem, I would have increasing differences in (-x,t) for corresponding maximization problem?

Whether or not that works depends on whether x and t are single or multidimensional. To apply Topkis, you need the objective function of a maximization problem (here, for example, g(x,t) = -f(x,t) ) to be supermodular in its arguments.

Suppose x and t are single-dimensional. If f is supermodular in (x,t) as you stated, then g is supermodular in (x, -t). So by Topkis, x*(t) is nonincreasing in t.

Now suppose that x or t are multidimensional. If f is supermodular in (x,t), that means it has ID in (x_i, x_j), (t_k, t_l), and (x_i, t_k) for all i ~= j and k ~= l. You can**not** generally get supermodularity of g in anything.

In general, applying Topkis to a minimization problem requires that the objective function be *sub*modular. If it’s supermodular, you’re in luck as long as its a function of only two variables.

In the revealed preferences arguments that we used in solving Question 1 in Consumer Theory Problem Set, even after we assume local nonsatiation, the question is still tricky because we are never sure if we can observe the entire choice set. We had to argue through many results relying on the fact that the optimal bundle on the budget line is strictly preferred to any interior bundle within the budget set.

What if we also assume strict convexity of preferences? Then there would be a unique optimal bundle? Under strictly convex preferences, there is only one unique optimal bundle, and we observe only one chosen bundle. So they must be the same thing. Therefore the strict convexity assumption is equivalent to the assumption that we observe all elements (only one) of the choice function C(B). And the comparison would be easier because we are sure the unique chosen bundle is strictly preferred to anything else on or within the budget line. Under a new set of prices, then, the new optimal choice must not be any of the feasible bundles under the old set of prices. So assuming that preferences are strictly convex would save us a lot of trouble thinking whether the bundle chosen by the consumer is one of the many bundles that are equally preferred or is the unique most preferred bundle.

I think this is also related to a Choice Theory Problem Set question 4 where we are asked to show that preference based models are not falsifiable, i.e. any observation could be rationalized by the preference that everything is equally preferred. If we can somehow assume away this situation so that we observe the entire choice set, life would be so much easier.

Would you say my thought is correct?

]]>I have some questions concerning the problem set and the notes of Choice under Uncertainty.

First, for problem #3, I was confused by the question “state necessary and sufficient conditions on u(.), ensuring the consumer will underinsure/fully insure/overinsure”. In fact, I have solved the FOC, and find that the property of u'(.) rather than u(.) do matter in this problem; However, the problem doesn’t provide any information about u'(.). Can I make any additional assumption about continuity, convexity/concavity, differentiability… of u'(.)?

When you solve through the FOC, you should get that the result in question (over vs. perfect vs. underinsurance) depends on an expression that looks like E[u'(x)] <=> u'(E[x]). The problem doesn’t actually give you enough info to really state a necessary condition, I agree. But try for a pretty loose sufficient condition (i.e., u(x) = x^(5/4) may be sufficient for overinsurance, but that’s not what I’m looking for).

Doesn’t that E[u'(x)] <=> u'(E[x]) look familiar? (Perhaps something to do with Jensen’s inequality…)

Second, a question about the definition of “continuous preference over lotteries P”: if p>p’>p”, there exists some lamda, such that lamda*p+(1-lamda)*p”~p’, then whether such lamda is unique? I also found in the proof of Theorem 1 (Representation Theorem, in step 3), the author assumes that such lamda is unique. But how can we prove the uniqueness from the definition?

A good question… how can we prove that the lambda in question is unique. I won’t lay out a complete proof, but here’s a sketch. Suppose as you do that p > p’ > p” (where those are strict preference, not greater than signs). Further suppose that there were more than one lambda (i.e., L1 != L2) such that Li p + (1-Li) p” ~ p’ for both i=1 and i=2.

- By transitivity of indifference, we get L1 p + (1-L1) p” ~ L2 p + (1-L2) p”
- Since p > p”, I know p != p”. Thus since L1 != L2, I know that L1 p + (1-L1) p” != L2 p + (1-L2) p”.
- Notice that I now have four points that are colinear: p, p”, L1 p + (1-L1) p”, and L2 p + (1-L2) p”. Further, I know the first two are nonequal and the second two are nonequal. The second two are not equal.
- Using a slight extension of our that independence gives linear indifference curves, we can show that since I am indifferent between the second two points above, I must be indifferent among all four (since they are colinear). This is the slightly tricky step, but give it a think.
- Contradiction: p ~ p” but p > p”

Finally, a small question: which one is correct: Shepard’s Lemma, or Shephard’s Lemma?…

Wikipedia likes “Shephard” http://en.wikipedia.org/wiki/Shephard%27s_lemma

Google search count’s also way outnumber for this spelling compared to others.

I wonder if you will be marking our final exam papers. If yes, I wonder if I can use the trick for Cobb Douglas type preferences/production technology, that we always get interior solutions, and that at the optimal solution the fraction of income spent on each good is determined by the exponent parameters. So that we do not have to do Kuhn Tucker.

I will be one of several people marking the final exams, but it doesn’t matter… you can definitely cite without justification the results on Cobb-Douglas that you mention.

]]>On page 212 of your latest updated slides, on the “sufficiency” of Kuhn-Tucker conditions, can the conclusion be true if we do not assume local non-satiation? If we do not assume monotonicity? strict monotonicity? It’s interesting if it is, because the would mean even if one of the two goods is a bad, or whatever pathological preferences we can think of, the Kuhn Tucker conditions are still necessary and sufficient.

Given a concave, differentiable utility function, the Kuhn-Tucker conditions are indeed sufficient to describe a solution to the UMP. Even if preferences are strange. Try a few examples with strange preferences and you’ll see!

]]>As I reviewed producer theory, I found a couple of things that I should have asked you a month ago:

1 On page 60 of the latest slides, I found the first bullet used to be “Closure ensures that \pi(p) is actually achieved.” Is it because the unboundedness or the asymptotic case might still prevent the non-emptiness of the correspondence even though Y is closed?

Yes.

2 On page 50, the proof for homogeneity of degree one for the profit function is assumes the existence of maximum. I wonder why we would do this here.

The “max” should be a “sup.” Otherwise, the proof goes through precisely. Good catch.

3 On page 51, in the proof for homogeneity of y, can I argue that

\argmax_{y \in Y} \lambda p \cdot y = \lambda \argmax_{y \in Y} p \cdot y = \lambda y(p) ?

I will have to assume argmax exists, though.

No… and the conclusion is untrue! (Note your proof is that y(.) is homogeneous of degree one, when it’s actually homogeneous of degree zero.)

When you “take the lambda out of the argmax,” it doesn’t appear in front, it just disappears. Think of argmax (-2 x^2) = argmax (-x^2). (In this manner, you can cancel any strictly increasing monotone transformation from an argmax or argmin.)

FollowupSorry, some questions are not well formulated.

For 3, can I write

\argmax_{y \in Y} \lambda p \cdot y = \argmax_{y \in Y} p \cdot y ?

Because the objective function is just a strictly monotone transformation.

Yes

4. On page 49, can I use the same proof to demonstrate that if f(\cdot) is homogeneous of degree k, then its gradient is homogeneous of degree k-1?

Yes.

5. On page 77, the substitution matrix. I know the diagonal elements must be non-negative. However, I put down in my notes that the derivatives of the output with respect to input prices must be negative. I wonder if that’s true and if so, why.

We would generally expect this to be the case (and I may have said as much in lecture), but it need not be true. Consider the following (non-differentiable) example:

Y = {(0,0,0) , (2,-1,0) , (3,0,-3) }

Then y(3,2,2) = {(2,-1,0)} and pi(3,1,2) = 4

while y(3,4,2) = {(3,0,-3)} and pi(3,4,2) = 3

That is, an increase in an input price lead to an increase in output.

6. On pages 120 and 121, why do we say X*(t) is non-decreasing in t in “stronger set order” rather than “strong set order”? Is it because of multidimensionality of X?

Yes. Strong set order means that a in A and b in B implies min{a,b} in A and max{a,b} in B. Stronger set order means that a in A and b in B implies meet{a,b} in A and join{a,b} in B. “min” and “max” are not well-defined operations on partially ordered sets such as R^n for n > 1.

If X is lattice and partially ordered, can we say that each variable x_i is non-decreasing in t? I think so because of the positive feedback loop among all variables.

I don’t understand the premise of the question.

In my notes I put down something “F is supermodular in (x_1, x_2, …, x_n, t_1, t_2, …, t_m”, rather than the stated conditions in the theorem. Does it matter?

This is the stated condition in the first version I stated of multivariate Topkis’ Theorem (“F is supermodular”). As I state on the next slide, this is a stronger than is required… we can dispense with ID between t_i and t_j.

Also, it seems the notes only provided the Multivariate Topkis Thm when t is fully ordered, i.e., only the version on page120. In that case, do we still need to check if F has increasing differences in x_i and t ? Intuitively I definitely should, but I wonder why it’s not in the theorem.

I am confused. You always need to consider ID in x_i and t_j, as well as in x_i and x_j. Not sure what you mean that “it’s not in the theorem.”

FollowupFor 6, I meant to say, for example, if X has 3 variables x_1, x_2, x_3, and we have two parameters t_1 and t_2. If F is supermodular in all x_i, and has ID in all (x_i , t_j), then by Multivariate Topkis X*(t) is non-decreasing in t in the stronger set order.

So can we conclude x_1 increases in strong set order, x_2 increases in strong set order, and x_3 increases in strong set order, in t= (t_1 and t_2) ?

Intuitively I think we can, at least weakly so. Like in the beer, wine, wealth example. Otherwise many previous analysis wouldn’t go through. But I don’t know how to think about proving it even heuristically. If X increases in t in Ser SO, how can I conclude each element of X, x_i, is increasing in t in SSO?

I think we can. Suppose t’ > t; then by Topkis x*(t’) >= x*(t) in the stronger SO. That is, for any (x1 x2) in x*(t) and (x1′ x2′) in x*(t’), we have (max{x1, x1′} max{x2, x2′}) in x*(t’) and (min{x1, x1′} min{x2, x2′}) in x*(t’). Thus max{x1 x1′} in x1*(t’) and min{x1 x1′} in x1*(t). Thus x1*() is increasing in t in the strong SO.

]]>Sorry to bother but I wonder if I could ask you a couple of stupid questions regarding CV and EV:

If the new price p’ < p (for at least one) with fixed w>0, does the consumer have to be better off? Do CV and EV have to be positive?

Yes. If p’ < p, then the budget set has strictly increased, so the consumer must have become weakly better off.

The converse: If I am told there’s been a price change from p to p’ for only one good. I am told the consumer is better off. Then can I conclude that p’ < p?

Yes.

In the integration calculation for CV and EV, p26 of the notes says we must assume the Hicksian demand for only one good responds to the single price change — i.e., the own price good change. This assumption essentially rules out all cross price effects on Hicksian demand.

However, do we really need this assumption? It is not in the slides. I checked the proof for rationalizability of profit function/choice correspondence and found out that the proof uses the homogeneity of degree zero of y (Hicksian demand in our case), Euler’s Law, and the symmetry of the Hessian (Slutzky matrix in our case). In the proof, all cross-price effects are zero because of the symmetry of the Hessian we exchange i and j. Then applying Euler’s law on own-price change and everything goes to zero. So I infer that the final result of the derivative of e(p,u) w.r.t. p_i should be h_i no matter what. There’s no cross price effects at the optimal solution because they are all zero by Euler’s Law.

Am I correct in the above reasoning?

I’m still thinking about this.

In policy evaluation, can CV<0 tell us not to enact a policy, and EV>0 can tell us to enact a policy? What difference does it make to use EV<0 and CV>0 respectively?

I can’t think of why either one would be preferred over the other on *a priori* grounds. Basically the question is whether you want to think of the status quo or the world under the policy as your baseline; I suppose that using the CV criterion could be thought of as conservative (we evaluate relative to status quo utility), while using the EV criterion could be thought of as liberal (we evaluate relative to utility under the policy).

The fact that the 2 criteria CV>0 and EV<0 are destined to cycle in certain economic environments definitely means CV and EV do not always have the same sign? However, p24 of the notes says “both CV and EV are constructed to be +ve for welfare increase and -ve for welfare decrease”. I thought CV and EV must always be either both +ve or both -ve. I can’t seem to find a counter example. How can the criteria cycle ? If the consumer is better off, she is better off, and both criteria should have the same sign? (This is sort of related to Question 1)

The CV and EV for an individual consumer will have the same sign for a given policy change, and the CV associated with doing the policy will equal the EV for the same consumer associated with undoing the policy (i.e., moving from p’ to p).

The cycling doesn’t happen for an individual consumer, it happens in the the sign of the **sum** of the CV and EV across many consumers.

If demand for a good is indep’t of wealth, like that of Quasilinear, then should we call it normal or inferior? Both or neither?

Both, I’d say (since we generally define comparative statics properties that don’t use words like “strict” as including the unchanging case). Most precise would be to say “demand doesn’t depend on wealth,” just as you did.

]]>I think there is a typo in the last midterm answers of question 2, letter b, iii…

You’re absolutely right. The EV given in the solution is negative of what it should be. Sorry!

Since prices have fallen, the consumer has become better off. Thus EV should be positive.

]]>I wonder if the \alpha in Question 3(a) of Problem Set 4 is required to be between 0 and 1. If not, would it still be Cobb Douglas?

Also, could there be a condition on the \beta in 3(c) that \beta > 0 ? Otherwise it might not be a monotone increasing transformation of (a).

Both seem like eminently reasonable assumptions to me!

]]>I just want to make sure i got the intuition behind the crazy utility function: u = max(ax, ay) + min(x,y).

Because utility = max(ax, ay) + min(x,y), this means the consumer is able to substitute between two goods, and he is able to maximize one of them, consuming the less quantity possible of the other. because this is a leontieff, you also know that ax=ay, or x=y. For this homotetic function, we need to consider several cases.

One of them is px/py = a. If the consumer is indifferent among all choices such that x > y, then you need to consider the ranges such that y=0 (best scenario until the case y=x (worst scenario). when y=0 then he only consumes x, the maximum level of utility he can get is u=ax. To get the Hickisian demand, you know that at the optimum x=h. Then, h=u/a . When x=y, he will consume the minimum amount of y, and the maximum of x. That is u= ax +y. But you know that y=x then u= x(a+1), rearranging a little bit, h= u/ (a+1).

Am I in the right track? thanks!

I think you’re on the right track. One way to think of the utility function is as follows:

u(x,y) =

ax + y if x >= y

x + ay if y >= x

(related to Leontieff, but not quite Leontieff)

we know that 0<a<1. I would recommend drawing what the indifference curves look like for this thing. If you look at the shape of the indifference curves, and think of where utility would be maximized for different budget sets, you will see that there are three cases (plus the two boundary cases between the three):

0 < p_x / p_y < a

a < p_x / p_y < 1/a

1/a < p_x / p_y

Overall, I would reccommend a graphical argument on this problem, rather than trying to solve Lagrangians or anything. You can check your answers against intuition by noting that a=0 corresponds to Leontieff [u(x,y) = min(x,y)] and a = 1 corresponds to perfect substitutes [u(x,y) = x+y]

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