3
$\begingroup$

Consider a continuum of firms indexed by $a\in [0,1]$ and that $a$ is uniformaly distributed. a firm of type $a$ has a production set $Y^n$ as depicted in the figure below. note that good 1 is used as an input to produce good 2. suppose that the price of the input (good 1) is fixed at $w=1$ while the price of output is $p>0$.

How should I get the input demand and output supply of a firm of type $a$ when the price of output is $p$? And what is the aggregate or average input demand and output supply when the output price is $p$? And what is the aggregate (or average) production set?

enter image description here

$\endgroup$

2 Answers 2

4
$\begingroup$

Observing the highest possible iso-profit curve passing through the feasible set, we obtain the supply of $y_2$ and demand for $y_1$ as follows. Here iso-profit curve is set of $(y_1,y_2)$ pairs that yields the same level of profit given $p$.

enter image description here

So, \begin{eqnarray*} (y_1^d,y_2^s)(p)\in\begin{cases}\{(0,0),(1+a,1)\} & \text{if } p = 1+a \\ \{(1+a,1)\} & \text{if } p > 1+a \\ \{(0,0)\} & \text{if } p < 1+a \end{cases} \end{eqnarray*}

Corresponding optimal profit is

\begin{eqnarray*} \pi(p)=\max(0,p-(1+a)) \end{eqnarray*}

Given that $a\sim\text{Unif}(0,1)$, aggregate input demand is \begin{eqnarray*}Y_1^d(p) = \int_0^1y_1^d(p)da = \begin{cases} \int_0^{1}(1+a)da=\frac{3}{2} & \text{if } p \geq 2 \\ 0 & \text{if } p \leq 1 \\ \int_0^{p-1}(1+a)da=\dfrac{(p-1)(p+1)}{2} & \text{if } 1 < p < 2\end{cases} \end{eqnarray*}

Aggregate output supply is \begin{eqnarray*}Y_2^s(p) = \int_0^1y_2^s(p)da = \begin{cases} \int_0^{1}da=1 & \text{if } p \geq 2 \\ 0 & \text{if } p \leq 1 \\ \int_0^{p-1}da=p-1 & \text{if } 1 < p < 2\end{cases} \end{eqnarray*}

$\endgroup$
2
  • $\begingroup$ Hi - What is the average production set in this case? $\endgroup$ Commented Dec 8, 2023 at 15:27
  • $\begingroup$ Boundary of the aggregate Production Set is given by $g(y_1)=\begin{cases} 1 & \text{for } y_1<-2\\ \int_0^1 \min(-y_1-a,1)da & \text{for } -2\leq y_1\leq -1 \\ \int_0^1 \max(-y_1-a,0)da & \text{for } -1<y_1\leq 0 \end{cases}$ and aggregate production set is simply $\{(y_1,y_2)|0\leq y_2\leq g(y_1)\}$ $\endgroup$
    – Amit
    Commented Dec 11, 2023 at 2:34
4
$\begingroup$

Let's first look at the profit of a single firm with parameter $a$. Let us denote by $x = -y_1$ the input and $f(x) = y_2$ the output.

The production function is the following: $$ f(x) = \begin{cases} 0 &\text{ if } x < a,\\ x - a &\text{ if } a \le x \le 1+a,\\ 1 &\text{ if } x > 1+a\end{cases} $$

The firm optimizes profits: $$ x^\ast = \arg\max_x p f(x) - x. $$

You can solve this by looking at various scenarios and taking the best (profit maximising) one.

  1. if $x^\ast < a$ then profits are $p 0 - x^\ast$, so here it is best to put $x^\ast = 0$.
  2. if $a \le x^\ast \le 1+a$ then profits are $p(x^\ast - a) - x^\ast= (p-1)x^\ast - pa$. If $p > 1$ then profits increase in $x^\ast$ so it is best to put $x^\ast = 1+a$ and profits will be $p - 1 - a$. if $p < 1$, then it is best to set $x^\ast = a$ and profits will be $-a$ (which is negative). In the latter case, it will be best to produce nothing.
  3. if $x^\ast \ge 1+a$ then profits are $p - x^\ast$ so here it is optimal to put $x = 1+a$ and profits are given by $p - 1 - a$.

As such, we see that profits are either equal to $p-1-a$ with optimal demand $x^\ast = 1+a$ or equal to zero with optimal demand $x^\ast = 0$.

The first will be optimal if $a \le p-1$, while the second is optimal if $a > p-1$. This gives the following optimal demand for a firm with parameter $a$: $$ x^\ast = \begin{cases} 1+a &\text{ if } a\le p-1 \\ 0 &\text{ if } a > p-1\end{cases} $$

Now let us derive optimal aggregate demand. If $p < 1$ then $a \ge 0 > p-1$ so no firm will demand anything: aggregate demand equals zero.

If $p > 2$ then $a \le 1 \le p - 1$ so the firm of type $a$ will demand $1+a$. Aggregate demand is then given by: $$ \int_0^1 (1+a) da = [a + a^2/2]^1_0 = \frac{3}{2}. $$

If $1 \le p \le 2$ then firms with $a \le p-1$ will demand $1+a$ and firms with $a > p-1$ will not demand anything. As such, aggreagate demand is: $$ \begin{align*} \int_0^{p-1} (1+a) da &= [a + a^2/2]^{(p-1)}_0 \\ &= (p-1) + \frac{(p-1)^2}{2} \\ &= p-1 + \frac{p^2}{2} - \frac{2p}{2} + \frac{1}{2},\\ & = \frac{p^2}{2} - \frac{1}{2}. \end{align*} $$

We get $$ \text{aggregate demand} = \begin{cases} 0 &\text{ if } p < 1\\ \frac{p^2}{2} - \frac{1}{2} &\text{ if } 1 \le p \le 2\\ \frac{3}{2} &\text{ if } p > 2.\end{cases} $$

Aggregate demand is given below. aggregate supply

$\endgroup$
3
  • $\begingroup$ According to the "production function" (<<a production set Yn as depicted in the figure below>>), the firm cannot produce more than 1 unit of output. $\endgroup$ Commented Nov 3, 2023 at 9:37
  • $\begingroup$ @escaiguolquer thanks. I made some corrections (supply->demand) $\endgroup$
    – tdm
    Commented Nov 3, 2023 at 12:48
  • $\begingroup$ Hi - What is the average production set in this case? $\endgroup$ Commented Dec 8, 2023 at 15:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.