# 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

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.