CV and EV questions

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.

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