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I would like an example of an utility function with 2 goods where the proportion of one goes to zero (and the other goes to one). I am thinking of a problem where the household receive an endowment y and has to divide it in two goods $(x_1,x_2)$. So what I want is:

$U(x_1,x_2)$

such that the (real) bugdet constraint is $x_1+x_2=y$

and $\lim_{y\rightarrow \infty} \frac{x_1}{y} \rightarrow 0$ and $\lim_{y\rightarrow \infty} \frac{x_2}{y} \rightarrow 1$ Where both $x_1$ and $x_2$ are normal goods and $x_1>0$ and $x_2>0$ in the optimal allocation.

Would somebody help me, please?

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2 Answers 2

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Consider quasi-linear utility function: $u(x_1,x_2)=2\sqrt{x_1}+x_2$

Here demand for $x_1$ is $x_1^d = \min(y, 1)$, and demand for $x_2^d = \max(0, y-1)$. Clearly, $\lim_{y\rightarrow\infty} \frac{x_1^d}{y} = \lim_{y\rightarrow\infty} \frac{1}{y} = 0$, and $\lim_{y\rightarrow\infty} \frac{x_2^d}{y} = \lim_{y\rightarrow\infty} \frac{y-1}{y} = 1$. Also, both the demand functions are non-decreasing functions of $y$. Though it does not satisfy $x_2^d > 0$ for all $y>0$, but I thought it is an example worth mentioning.

To see how to find the demand for quasi-linear utility, please refer to this answer: https://qr.ae/pGJuvH

Another "non-standard Cobb Douglas" example of a utility function is: $u(x_1,x_2;y) = x_1x_2^y$. Please note that this is a non-standard consumer model where income directly enters the utility function of an individual. Here demand for $x_1$ is $x_1^d = \frac{y}{1+y}$, and demand for $x_2^d = \frac{y^2}{1+y}$. Clearly, $\lim_{y\rightarrow\infty} \frac{x_1^d}{y} = \lim_{y\rightarrow\infty} \frac{1}{1+y} = 0$, and $\lim_{y\rightarrow\infty} \frac{x_2^d}{y} = \lim_{y\rightarrow\infty} \frac{y}{1+y} = 1$. Also, both the demand functions are increasing functions of $y$.

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    $\begingroup$ If you don't want $y$ in the utility function (in the second example), you can change $u$ to $u(x_1,x_2)=x_1x_2^{x_1+x_2}$. Although it represents different preferences compared to the one given in the answer, but optimal solution stays the same and therefore, it satisfy all the required properties. $\endgroup$
    – Amit
    Mar 1, 2023 at 3:28
  • $\begingroup$ Thank you. The thing that most deviated from what I want is that $x_1$ is non-decreasing instead of strictly increasing function of y. $\endgroup$ Mar 1, 2023 at 23:54
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    $\begingroup$ In the second example i.e. with $u=x_1x_2^{x_1+x_2}$, both $x_1$ and $x_2$ are increasing functions of $y$. $\endgroup$
    – Amit
    Mar 2, 2023 at 0:14
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$U(x_1,x_2)=\min\{x_1^2,x_2\}$.

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  • $\begingroup$ That's a nice one :) $\endgroup$
    – Amit
    Mar 1, 2023 at 3:07
  • $\begingroup$ I think it solves what I asked. Thank you. May I add one more thing? I intend to create a dynamic problem beginning with that. The household should be able to transfer the endowment over time time at a fix rate R. I am having some trouble to write the lagrangian of the problem. What should be the shadow price of budget constraint that include a one-period bond? $\endgroup$ Mar 1, 2023 at 23:54
  • $\begingroup$ I'm not sure what exactly you are trying to do. Note also that this utility function is not differentiable, so the Lagrangian might not be that useful. $\endgroup$ Mar 2, 2023 at 11:04
  • $\begingroup$ Let's say that the household receive a aleatory endownment each period with mean 1. If the utility function is what you describe, it is possible that the household try to take part of the income (endownment) and transfer to another period. $ x_1+x_2 +\frac{B_{t+1}}{R_t} \leq y_t + B_t$ $\endgroup$ Mar 2, 2023 at 19:41
  • $\begingroup$ I think the most straightforward approach would be to calculate the (per period) indirect utility function. This gives you the value of having a certain amount of wealth in each period. Then you have the one-dimensional optimization problem of allocating wealth over the two periods. $\endgroup$ Mar 3, 2023 at 8:36

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