Economics Asked on September 2, 2021
Question: Consider a consumer with utility function $U(x,y,z)=ymin{x,z}$. The prices of all three goods are the same. The consumer has $100 to spend on these three goods.The demands will be such that:
(a) $y<x=z$
(b) $y>x=z$
(c) $x=y=z$
(d) None of the above
My attempt: The consumer will consume equal amounts of $x$ and $z$ because otherwise the allocation would be inefficient, that is, he can obtain the same level of utility by spending less. So $x=z$. I cannot figure out how is $y$ related to $x$ and $z$. I think the answer would be (d) None of the above because it does not matter if $y$ is less than or greater than or equal to $x$ and $z$.
Let $min{x,z}=Omega$, where $P_Omega=P_x+P_z$. Now the problem becomes $U(y,Omega)=yOmega$, which is a standard Cobb-Douglas with degree 2 of homogeneity. Now in this case the choice for each good is:
$y^*=frac{alpha_y100}{P_y(alpha_y+alpha_Omega)}implies y^*=frac{100}{2P_y};;;;;;;$in this case $alpha_y=alpha_Omega=1$
For $Omega$: $;;;;;Omega^*=frac{alpha_Omega100}{P_Omega(alpha_y+alpha_Omega)}implies Omega^*=frac{100}{2P_Omega} implies Omega^*=frac{100}{2(P_x+P_z)}$
Now, since $P_x=P_y=P_z$, let $P_x=P_y=P_z=P$ a general price, therefore substituting in our optimums:
$y^*=frac{100}{2P};;;;;Omega^*=frac{100}{4P}$
Now it's straightforward (since we already know $x^*=z^*$ and as this is the optimum for $min{x,z}$ which is $x$ OR $z$) that $y^*=frac{100}{2P}>Omega^*=frac{100}{4P}$, so this implies that:
$y^*>x^*=z^*;;$****
Also I found this document, where this question is number 13.
Hope this helps.
Disclaimer: It would be helpful if other people could assess this approximation, since I hadn't never seen this problem before.
Correct answer by nrivera on September 2, 2021
Hint: Suppose the price of the goods is $P$ so that $N=100/P$ goods can be afforded in total. Now consider which of the following yields more utility:
a) $x=y=z=N/3$.
b) $x=z=N/4$ and $y=N/2$.
Answered by Adam Bailey on September 2, 2021
I have not seen this in any textbook of mine, but here's my attempt:
Since the utility function (1) is the product of the quantity of y and the minimum quantity of either x or z (so, $min {x,z}$ is a singular value, say 15 units or 27 units, etc.) and (2) $x=z$ for every value of x or z, the utility function turns into:
Taking the first case (and the same works for the second), maximization gives the solution for $$ MRS_{XY} = frac{p_X + p_Z}{p_Y} $$ where $ MRS = dy/dx = MU_x/MU_y = {y}/{x}$, and since $p_X = p_Z = p_Y$ we get $$ y/x = 2 $$ so in the end: $$ y = 2x =2z$$ and overall $y>x=z$.
Answered by S. Iason Koutsoulis on September 2, 2021
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