finding optimal values of quantities through utility function when MRS is 1 and price ratio is 1:2 [closed]

Question

so this is a somewhat simple question but im stumped, here's why:

given the utility U=2$X_1$+2$X_2$ where $P_1$$X_1$+$P_2$$X_2$=20 and $P_1$=0.5$P_2$

the question asks to find optimal values of $X_1$ and $X_2$, i tried using lagrangian method but the FOCs would only leave me with $\lambda$ and $P$ values. i also tried finding the MRS, which resulted in $1$, while the price ratio is $1:2$ , in both methods the quantities of goods 1 and 2 would disappear and I'd be trying to figure it out with what feels like incomplete information.

give it a try and tell me where i went wrong if possible.

Answer 1

Linear utility functions like the one you have ($U = 2X_1+2X_2$) commonly lead to corner solutions where you only buy one of the goods. You can tell that this will occur here because utility is maximized with an internal solution by setting MRS equal to the price ratio, but MRS is always 1 and the price ratio is always .5, so they'll never be equal.

In this utility function, adding another unit of $X_1$ or $X_2$ will always bring you the same amount of marginal utility ($2$). Notice that you can rewrite the utility function as $U = 2(X_1+X_2)$. All that matters is maximizing the total amount of units of either good you have, subject to constraints.

Given that, and given that $X_1$ is cheaper than $X_2$, you have no reason to ever buy any $X_2$. So set $X_2 = 0$ and use the budget constraint to solve for $X_1$.