Introducing productive sector into an exchange economy where only one agent is endowed with input
Introducing productive sector into an exchange economy where only one agent is endowed with input
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RobinsonWM · External communityPost link
External question — Economics Stack Exchange
Author: RobinsonWM
Original post: https://economics.stackexchange.com/questions/57688
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
I'm trying to find a competitive equilibrium for an economy with consumers and some outside productive sector.
Consider an economy with two consumption goods
$x_1, x_2$
and two individuals
$A,B$
.
Endowments are given
$\omega^A = (3,0), \omega^B = (0,3)$
.
There is a productive sector not owned by any individual that takes the first consumption good as an input and produces the second consumption good via the function
$y_2 = f(y_1) = a y_1$
.
My question is this: How do we setup the budget constraint for individual
$A$
?
He is the only agent who can supply the firm with its inputs, so it could be
$$p_1 x_1^A + p_2 x_2^A \le (3 + y_1) p_1,$$
but in the new equilibrium, won't the prices and demand functions reflect the information that
$A$
is the only supplier of
$y_1$
?
So then the budget constraint should be
$$ p_1 x_1^A + p_2 x_2^A \le 3 p_1.$$
I'm happy to provide anymore detail.
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tdm · External communityPost link
External answer — Economics Stack Exchange
Author: tdm
Original post: https://economics.stackexchange.com/a/57708
License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/
Adaptation: HTML converted to plain text; contact email addresses removed.
Individual 1 has 3 units of the first good.
Assume that a part
$z \in [0,3]$
of this good is used to produce good 2. Then she will have in total
$(3 - z)$
units of good 1 and
$a z$
units of good 2.
If she sells those goods on the market, she will receive:
$$
p_1 (3 - z) + p_2 az.
$$
This income will then be used to buy her final consumption bundle.
Note that the best she can do is to pick
$z \in [0,3]$
in order to optimize this budget. In other words, she will first solve.
$$
\max_{z \in [0,3]} 3 p_1 + (a p_2 - p_1)z.
$$
If
$p_1 < a p_2$
, she will pick
$z = 3$
, which will give her a total income of:
$$
3 p_1 + (a p_2 - p_1)3 = 3 a p_2.
$$
If
$p_1 > a p_2$
, she will pick
$z = 0$
, which will give her a total income of:
$$
3 p_1.
$$
In other words, her final income will be given by:
$\max\{3 a p_2, 3p_1\} = 3\max\{a p_2, p_1\}$
.
This means that her budget constraint can be written as:
$$
p_1 x_1^A + p_2 x_2^A \le 3 \max\{a p_2, p_1\}.
$$
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Quoted from Forex.com.bd-Editorial External answer — Economics Stack Exchange Author: tdm Source score (net votes, not local likes): 1 Original post: https://economics.stackexchange.com/a/57708 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Individual 1 has 3 units of the first good. Assume that a part $z \in [0,3]$ of this good is used to produce good 2. Then she will have in total $(3 - z)$ units of good 1 and $a z$ units of good 2. If she sells those goods on the market, she will receive: $$ p_1 (3 - z) + p_2 az. $$ This income will then be used to buy her final consumption bundle. Note that the best she can do is to pick $z \in [0,3]$ in order to optimize this budget. In other words, she will first solve. $$ \max_{z \in [0,3]} 3 p_1 + (a p_2 - p_1)z. $$ If $p_1 < a p_2$ , she will pick $z = 3$ , which will give her a total income of: $$ 3 p_1 + (a p_2 - p_1)3 = 3 a p_2. $$ If $p_1 > a p_2$ , she will pick $z = 0$ , which will give her a total income of: $$ 3 p_1. $$ In other words, her final income will be given by: $\max\{3 a p_2, 3p_1\} = 3\max\{a p_2, p_1\}$ . This means that her budget constraint can be written as: $$ p_1 x_1^A + p_2 x_2^A \le 3 \max\{a p_2, p_1\}. $$
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