Question: So I have this utility function where I need to find t expenditure minimizing value of x. Normally, I would get the marginal utility of goods x and y, but doing that in this case doesn't leave any variables, so setting that equal to the prices would give me 2/5 = 3/3, which doesn't make sense. How do I solve this? I feel like I know how to do it but I'm just missing a major step or two.

[![](https://i.stack.imgur.com/TvuOE.jpg)](https://i.stack.imgur.com/TvuOE.jpg)

Answer: 
Since in this case, both utility function and the budget line are perfect substitutes and thus you can't equate the **Marginal Rate of Substitution** to **the price ratio**.

This problem can be solved this way,

$$\min\_{x,y} \quad 3x+3y\\
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{s.t.} \quad 2x+5y=10\\$$

or

$$\min\_{x,y} \quad x+y\\
 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \textrm{s.t.} \quad 2x+5y=10\\$$

Here,  

$$MRS=\frac{MU\_{x}}{MU\_{y}}=\frac{2}{5}<1=\frac{P\_{x}}{P\_{y}}$$ 
or, $$\frac{MU\_{x}}{P\_{x}}=2 < 5 = \frac{MU\_{y}}{P\_{y}}$$ 

Therefore the expenditure is minimum when the consumer spends his entire income on good y, that is $(x,y)=(0,2)$ is the bundle that minimizes the expenditure given the utility $u(x,y)=2x+5y=10$. This is depicted in the figure below:

[![enter image description here](https://i.stack.imgur.com/prYTl.png)](https://i.stack.imgur.com/prYTl.png)
The utility function shows that x and y are perfect complements with, y giving a MU of 5 and x giving an MU of 2. Both have the same price. Hence in equilibrium the consumer will consume only y. So quantity of x is 0.