# Log Utiliy Function Trick

I am watching Lecture 3 of Yale's Financial Theory Lecture (by John). At about minute 50 he explains something along this line (with reference to log utility functions).

MUx/Px=MUy/Py

And simplifies down to: 0.75/Px(x)= 0.25/Py(y)

Given that Px(x)+Py(y)=Total Money

He concludes that "The total spent on x relative to 3/4 is equal to the total spent on y relative to 1/4" and therefore a consumer would spend 3/4 of their income on x and 1/4 on y.

How is he drawing that conclusion from the above equations?

Start here $$0.75/p_xx= 0.25/p_yy$$

Multiply both sides by $$p_xx p_yy$$, you get: $$0.75p_yy=0.25p_xx$$

Add $$0.75p_xx$$ to both sides, you get:

$$0.75(p_xx+p_yy)=p_xx$$

So: $$\frac{p_xx}{p_xx+p_yy}=0.75=\frac{3}{4}$$

And you spend the rest of your budget, so $$1/4$$ of it, on $$y$$.

• Thanks so much! I assumed that it could be solved algebraically; but is there an intuitive reasoning for why we could make that conclusion? He didn't do any algebraic manipulation in the lecture, and purely just deduced it from the fact that "total spent on x relative to 3/4 is equal to total spent on y relative to 1/4" – Bob Oct 13 '19 at 20:24
• Well, if you note that $0.75 \times 1/x$ is just the marginal utility of consuming $x$ and $0.25 \times 1/y$ is the marginal utility of consuming $y$ then you see that the marginal utility of consuming $x$ is 3 times higher than the marginal utlity of consuming $y$ (when you are consuming equal amounts of $x$ and $y$). Then it seems intuitive that you spend 3 times as much money on $x$. – user18214 Oct 13 '19 at 20:53
• This kind of reasoning however, is too sloppy. For example consider the following preferences for perfect substitutes: $U(x,y)=3x+y$ , here too, $MU_x$ is 3 times as large as $MU_y$. the consumer however, either only consumes $x$, only consumes $y$ or is indifferent between any budget depleting mix of the two. Which of the three we will see depends on prices. – user18214 Oct 13 '19 at 20:57