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To preface this, I appreciate any help you offer.

In reference to 1.(d) I'm trying to understand the MRS of perfect complements. Given: utility function U(x,y), commodity pair (2,8): y/x is 4 ...> a/b which is 3, so we say the MRS at this point is undefined (Δy/Δx = Δy/0)? Given commodity pair (5,1): y/x is 1/5...< a/b which is 3, so we say the MRS at this point is 0(0/Δx)? I'm not exactly sure why this is or how to best phrase those statements if they're correct, I've looked everywhere and have not really caught any explanation.

In reference to 2.(f) Is "the expression" requested maybe MRS = (dUx/dx)/(dU/dy) = Px/Py? I'm confused by this phrase "positive values of consumption"... is this positive x and y values? How do I find this minimal combination? I'm only familiar with the approach to finding a maximal utility producing combination.

Thanks so much if you've read this far.

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Your instructor seems to have a rather sloppy approach to the mathematics of microeconomics.

Let's begin with the easiest case, the bundle $(5,1)$. We have $\min\{3\cdot 5,1\}=1$. For $\delta<14$, we have $\min\{3\cdot 5, 1+\delta\}=1+\delta$ so the marginal utility of $y$ is $1$ at the bundle $(5,1)$. Moreover, for any $\delta>-14/3$, we have $\min\{3\cdot(5+\delta),1\}=1$, so the marginal utility of $x$ is $0$ at $(5,1)$.

A similar argument applies to the bundle $(2,8)$, but here the marginal utility of $y$ is $0$, and dividing by zero is not allowed. Maybe your instructor wants to hear that the MRS is infinite.

Now at the bundle $(3,9)$, increases in any commodity have zero effects, but decreases do. There is a kink and you cannot smoothly move along the curve. By any semi-reasonable standards, the MRS is undefined. But to answer the verbal question, the consumer is not willing to give up any amount of one commodity ( effect negative) to receive more of the other good (effect zero).

Now the idea that you get the optimal consumption bundle by setting the MRS equal to relative prices is complete nonsense. If prices $p_x$ and $p_y$ and income $m$ are all positive, we can still, however, still find the optimal bundle. First note that the optimal bundle $(x,y)$ wil satisfy $3x=y$. If $3x>y$, one could reduce the amount $x$ without decreasing the utility (the minimum is unchanged), but use the freed money to increase $y$, which does have a positive effect. In an optimal bundle, this is not possible. by a similar argument, we can rule out $3x<y$. So we must have $3x=y$, which means that at any optimal solution, the MRS is undefined!!! To solve for an optimal bundle, all that remains to do is plug the condition $3x=y$ into the budget constraint $p_x x+p_y y=m$. We can substitute $3x$ for $y$ to get $$p_x x+p_y 3x=m$$ and solve for $x$, which gives us $$x=\frac{m}{p_x+3p_y}.$$ Similarly, we get $$y=\frac{m}{p_x/3+p_y}.$$

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