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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.

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  • $\begingroup$ Try drawing the indifference curve (for $u(x,y)=10$) and budget line. $\endgroup$ – Herr K. Feb 24 at 4:55
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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

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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.

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Here, Mux/Muy is less than Px/Py and this utility function is perfect substitute type So we are free to choose only x or y. Here the optimal solution will be(0, 2)

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