# How to calculate income and substitution effect when equal marginal principle is violated

I am trying to calculate substitution and income effect for 2 goods, $$x$$ and $$y$$.

Given that marginal utility $$\mathrm{MU}_x = 1$$, marginal utility $$\mathrm{MU}_y = -a$$ (unknown number), price of $$p_x = 16$$, price of $$p_y = 20$$, how should I proceed? I am unable to use the equal marginal principle, since I cannot have a negative price. But without using the marginal principle, I am also unable to find the optimal basket.

I was also considering that the marginal rate of substitution of $$x$$ will always be greater than that of $$y$$. Hence, the optimal basket would just be to spend all the income on $$x$$. Could that count as an optimal basket?

• Is there any information which says that a has to be positive? – erik Sep 10 '18 at 16:28
• Yes, it is given in the question that a > 0 – statsguy21 Sep 11 '18 at 1:35
• Then you should mention that in the post I think. Thanks. – erik Sep 11 '18 at 6:46
• This question might be helpful, although note that this is an example where both goods yield positive utility; economics.stackexchange.com/questions/25384/… – Ubiquitous Nov 10 '18 at 7:56

## 1 Answer

Given that $$a>0$$, one can see that the utility function is, $$U =x-ay$$. Which gives us linear indifference curves. Which also tells us that $$x$$ is a “good” and $$y$$ is a “bad”.

Thus the consumer spends all income on $$x$$ and purchases no units of $$y$$. You are correct in what you mention in your question.

You can not apply the equimarginal principle here since all you get is a corner solution.

• Ok thanks for the clarification. Does this mean that I shouldn't be worrying about calculations (equimarginal principles / tangency solutions) in such a question and revert back to the basics and go by the definition instead? – statsguy21 Sep 11 '18 at 6:53
• You can’t use equimarginal principle here. So yes, go by applying logic and you find that the only good bought will be x. The IE becomes really easy. And there is no SE in the sense that no money is ever spent on Y. – erik Sep 11 '18 at 6:59