# Does an "optimal" MRS exist?

I was reading a case study in Hal Varian, where the author talks about essentially a surge pricing mechanism for incentivizing households to consume less electricity during peak hours (so as to not exceed total capacity). Thus, the electricity company basically increases prices during peak hours.

This leads to the following indifference curves and budget lines. (pic attached. Reference : Hal R. Varian, Intermediate Microeconomics) Now its clear that under surge pricing (RTP budget constraint), the optimal bundle has changed but infact the consumer is better off since the previous bundle is affordable but they choose a different point because it leads to a higher indifference curve, giving more utility.

This leads to my actual question : I can technically pivot the line around the original baseline optimal point in any of the infinite ways, and at each pivot, the original bundle will be affordable but not necessary be optimal, and thus may lead to infinite different optimal points, depending on how it is pivoted.

In this sense, is there an optimal "pivot" or slope of the budget line or MRS ratio with the same income (so that it passes through the original baseline optimal point) that gives the maximum utility? If yes, how does it make sense practically? I was forced to consume less but i am more happy?

• Was the consumer also given more money or “other goods” became cheaper aside from the price increase in electricity? If not, the point on the “other goods” axis should remain unchanged. May 15 at 12:16

If yes, how does it make sense practically? I was forced to consume less but i am more happy?

You are not consuming less in general though, you are consuming less electricity but spending more money on other things.

Also, you are not "forced" to consume less electricity, you choose to consume less electricity because you are incentivized to do so. It is still very much possible to consume your original bundle.

is there an optimal "pivot" or slope of the budget line or MRS ratio with the same income (so that it passes through the original baseline optimal point) that gives the maximum utility?

If the preference is strongly monotonic, and at least one pivot results in a bundle better than the original, no optimal pivot will exist.

Let us denote the original bundle by $$(\bar{e},\bar{x})$$.

Compared to any bundle $$(\hat{e},\hat{x})$$ where $$\hat{e} < \bar{e}$$, $$\hat{x} \geq \bar{x}$$ and $$(\hat{e},\hat{x}) \succeq (\bar{e},\bar{x})$$, a better bundle attainable by some other pivot will always exist.

$$(\hat{e},\hat{x} + 1) \succ (\hat{e},\hat{x})$$ due to strong monotonicity, and there is a non-vertical line connecting the points $$(\bar{e},\bar{x})$$, $$(\hat{e},\hat{x} +1)$$ because $$\hat{e} < \bar{e}$$, $$\hat{x} + 1 > \bar{x}$$, thus a pivot that makes $$(\hat{e},\hat{x} + 1)$$ feasible exists.

Since there is always a better bundle, no bundle with maximum utility exists.

If we do not assume strong monotonicity then counterexamples exist, e.g.; with Leontief utilities or similar preferences with kinked indifference curves.