# Finding Optimal Bundle and Change in Satisfaction after Changes in Budget Constraint

I am looking at the following exercise and struggling with the solution proposed by my microeconomics book.

A consumer spends all his income on two goods, X and Y. The prices he paid and the quantities he consumed last year are as follows: PX = 15, X = 20, PY = 25, and Y = 30. We assume the consumer's preferences follow the 5 properties for preference ordering (complete, transitive, more-is-better, convex, continuous).

If the prices next year are PX = 6 and PY = 30, and the consumer's income is 1,020, will he be better or worse off than he was in the previous year? (Assume that his tastes do not change.)

For the Solution we first find the consumer's budget constraint of last year:

M = PX * X + PY * Y = 15 * 20 + 25 * 30 = 1050


We need to ask can he buy last year’s bundle given the new prices?

PXnew * X + PYnew * Y = 6 * 20 + 30 * 30 = 1020


Given that the consumer can still afford last year's bundle, he must be (in my opinion) at least as well off. I am working with a economics book and the solution proposed in the book claims, that the consumer must be better off using the following figure as explanation:

I am wondering, couldn't the indifference curve that is tangent to the bundle at X = 20, Y = 30 for the last year, also be tangent to a point on the new budget constraint and equal the highest indifference curve that can be reached?

Is there an explanation, given the 5 properties of preference ordering that the consumer MUST be better off.

EDIT:

I have added the image below to clarify my question.

I am wondering whether the indifference curve in red in the image could look like this and imply that the satisfaction level is not increasing as the consumer will stay on the same indifference curve but consume a different combination of X and Y.

• Can indifference curves intersect, as they do in your last image? – Giskard Mar 24 '19 at 10:21
• No they cannot. The red indifference curve is supposed to present a different indifference map, so different preferences of the consumer. The red and black indifference curves cannot occur for the same consumer. But I am wondering whether the red curves could be shaped like that and lead to the consumer not being better off. – Tom Mar 24 '19 at 11:05
• Ah, I see. Sure, you can easily make such preferences, e.g.: $U(X,Y) = \min(3X;2Y)$. – Giskard Mar 24 '19 at 11:30
• That is correct but these preferences would violate the preference properties I have assumed. Most notably, the more-is-better property. – Tom Mar 24 '19 at 11:40
• You don't actually need the IC curve to consist of horizontal and vertical parts, all you need is the 'kink'. For example the indifference curve of $U(X,Y) = \min(3X;2Y) + (X+Y)$ would also have a kink at (20,30), but the slope would always be negative. – Giskard Mar 24 '19 at 13:24

I understand what you though, the old indifference curve sure can touch but cannot conclude to be the optimal one.

As the evidence of the money he can pay 1050 of the last year budget but for this year he could only use just 1020 to buy the same combination of those goods with the same utility so that means he still can buy and consume more with the relative purchasing power.

I will just use 2 assumptions that are "more is better", "no bliss point". It is said that the budget which has shown or given here is not significant to the level that the consumer will keep and not spend so that as he can spend he will and with "no bliss point" that means this consumption is too small and yet not to the level that he will have negative marginal utility.

Therefore, as he can buy the old combination with the lesser price so he can allocate the leftover budget for either of two goods.

• There is no leftover budget. The new budget is 1,020 and therefore just enough to buy the same bundle as last year. I have added another image to the question to make my thoughts more clear. – Tom Mar 24 '19 at 10:10

I think I have figured the answer out myself.

Given the rule to find the optimal bundle, we know that the consumer will pick the bundle for which the slope of the budget constraint is equal to the marginal rate of substitution (MRS):

PX/PY = MRS


We know that that the old budget constraint is steeper than the new budget constraint. That means that the indifference curve needs to enter the space underneath the new budget constraint as the slope at the point where the budget constraints intersect is steeper than of the new budget constraint. This also means there needs to be a parallel indifference curve that allows the consumer to reach a higher level of satisfaction.