<p><strong>Question:</strong>
Suppose an economy consists of two price-taking people, “a” and “b.” Person a has available 1
unit of “time” which he divides between rest <span class="math-container">$R(a)$</span> and labour <span class="math-container">$L(a)$</span>. Person b has available 1 unit of “time” which he divides between rest <span class="math-container">$R(b)$</span> and labour <span class="math-container">$L(b)$</span>. Good “<span class="math-container">$x$</span>” is produced by many competitive firms according to a linear production function <span class="math-container">$F(L)=L$</span> where <span class="math-container">$L$</span> is the labour. Let the amount of good <span class="math-container">$x$</span> consumed by Person a be <span class="math-container">$x(a)$</span> and the amount of good <span class="math-container">$x$</span> consumed by Person b be <span class="math-container">$x(b)$</span>. Let <span class="math-container">$X = x(a) + x(b)$</span>.</p>
<p>Suppose production of <span class="math-container">$x$</span> causes air pollution which decreases the utility of Person b but not of Person a (perhaps because Person a lives far away from the source of the air pollution). The
utility functions of the two individuals are</p>
<p><span class="math-container">$ua = x(a)*R(a)$</span></p>
<p><span class="math-container">$ub = x(b)*R(b) - X$</span>.</p>
<p>Suppose that Person b always sees “<span class="math-container">$X$</span>” as exogenous; he never thinks of X as being <span class="math-container">$x(a) +x(b)$</span> (even though it is), and therefore he never considers the effect of his own consumption of x on the amount of air pollution. (You may think about “person a” and “person b” as “representative
consumers”, i.e., a collection of many small consumers whose individual decisions have little
effect on aggregate variables). Take the price of labour to be $1.</p>
<p>Find <span class="math-container">$x(a)$</span>, <span class="math-container">$R(a)$</span>, <span class="math-container">$x(b)$</span>, and <span class="math-container">$R(b)$</span>.</p>
<p><strong>My Attempt:</strong>
As Person b is exogenous, we can consider its utility function without X because he ignores it and won't affect how he thinks will maximize his utility. From this step, however, I was stuck. I suppose that they consume the exact amount that they produce, but I cannot confirm the validity (they may not be able to afford all of the products they produce). Additionally, I am not sure about the price of the good, so it becomes challenging for me to solve it. My primitive guess is that each person would split their time half-half, but I feel that this is incorrect.</p>
<p>Thank you in advance and I really appreciate your help!</p>

CC BY-SA 4.0
2024-04-26T06:31:15.020Z
If you want to find equilibrium you start by asking: What equations must hold in equilibrium? Then you most likely get a system of non-linear equation you can solve for the endogenous variables.
Jesper Hybel
CC BY-SA 4.0
2024-04-26T13:11:32.127Z
Hi Jesper, to determine the price of the good, I am thinking of determining the demand and supply. The supply equation should be L(a)+L(b) without doubt, but what would be the demand function though? If I am able to find that, I think I can tackle the rest of the question more easily. Thanks!
ZZZ
<mathematical-economics><utility><labor-economics><general-equilibrium>
How to find the amount of leisure time and the good consumed in a two-person and one-good economy?
ZZZ
# How to find the amount of leisure time and the good consumed in a two-person and one-good economy?

(Asked by: username_0 on 2024-04-26)

**Question:**
Suppose an economy consists of two price-taking people, “a” and “b.” Person a has available 1
unit of “time” which he divides between rest $R(a)$ and labour $L(a)$. Person b has available 1 unit of “time” which he divides between rest $R(b)$ and labour $L(b)$. Good “$x$” is produced by many competitive firms according to a linear production function $F(L)=L$ where $L$ is the labour. Let the amount of good $x$ consumed by Person a be $x(a)$ and the amount of good $x$ consumed by Person b be $x(b)$. Let $X = x(a) + x(b)$.

Suppose production of $x$ causes air pollution which decreases the utility of Person b but not of Person a (perhaps because Person a lives far away from the source of the air pollution). The
utility functions of the two individuals are

$ua = x(a)\*R(a)$

$ub = x(b)\*R(b) - X$.

Suppose that Person b always sees “$X$” as exogenous; he never thinks of X as being $x(a) +x(b)$ (even though it is), and therefore he never considers the effect of his own consumption of x on the amount of air pollution. (You may think about “person a” and “person b” as “representative
consumers”, i.e., a collection of many small consumers whose individual decisions have little
effect on aggregate variables). Take the price of labour to be $1.

Find $x(a)$, $R(a)$, $x(b)$, and $R(b)$.

**My Attempt:**
As Person b is exogenous, we can consider its utility function without X because he ignores it and won't affect how he thinks will maximize his utility. From this step, however, I was stuck. I suppose that they consume the exact amount that they produce, but I cannot confirm the validity (they may not be able to afford all of the products they produce). Additionally, I am not sure about the price of the good, so it becomes challenging for me to solve it. My primitive guess is that each person would split their time half-half, but I feel that this is incorrect.

Thank you in advance and I really appreciate your help!

**Tags:** mathematical-economics, utility, labor-economics, general-equilibrium

> **Comments:**
> **username_1**: If you want to find equilibrium you start by asking: What equations must hold in equilibrium? Then you most likely get a system of non-linear equation you can solve for the endogenous variables.  
> **username_0**: Hi Jesper, to determine the price of the good, I am thinking of determining the demand and supply. The supply equation should be L(a)+L(b) without doubt, but what would be the demand function though? If I am able to find that, I think I can tackle the rest of the question more easily. Thanks!  

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## 0 answers

