<p><strong>Q1</strong></p>
<p>This is not a straightforward problem. Two things to note:<br />
<strong>First,</strong> the production function is linear in the inputs. This means that their marginal products are constant, and so are their marginal revenue products (presumably the firm is treated as a price taker in the output market). This implies that if price is sufficiently low, then production will be zero.<br />
<strong>Second:</strong> the production inputs are not complementary, production can happen using both or either one of them.</p>
<p>The profit function is</p>
<p><span class="math-container">$$pQ - C = p(2x_1+ x_2) - 4x_1 - 3x_2 = (2p-4)x_1 + (p-3)x_2.$$</span></p>
<p>From this we see that</p>
<p><span class="math-container">$$p &lt; 2 \implies Q = 0$$</span></p>
<p>because with any choice of inputs the firms will incur losses.</p>
<p>At exactly <span class="math-container">$p=2$</span>, the firm can have zero-profits if it uses only input <span class="math-container">$x_1$</span> for any output level, so here</p>
<p><span class="math-container">$$p=2 \implies Q = {\rm indeterminate.}$$</span></p>
<p>&quot;Indeterminate&quot; in the sense that it could be zero, or any positive quantity, or even &quot;infinity&quot; (see below) - the firm is indifferent between these choices.</p>
<p>Suppose now that <span class="math-container">$p&gt;2$</span>.</p>
<p>If price is <span class="math-container">$2 &lt; p \leq 3$</span>, the firm has an incentive to use only input <span class="math-container">$x_1$</span>, because using also input <span class="math-container">$x_2$</span> will incur some loss, while it can use only input <span class="math-container">$x_1$</span> and have profit for every output unit it produces. And then, the more it produces using just <span class="math-container">$x_1$</span> the more profit it has. So here</p>
<p><span class="math-container">$$2 &lt; p \leq 3 \implies Q = \infty$$</span></p>
<p>Certainly, one could wonder whether this is meaningful, since<br />
a) there is nothing infinite in this world and<br />
b) as the firm produces more and more output wouldn't that eventually push the price down? How much of the good do consumers are willing to buy after all?</p>
<p>These are valid remarks, but the solution that the supply function will be &quot;infinite&quot; indicates the <em>profit-maximizing tendency</em> of the firm, not what will actually happen in the real world.</p>
<p>If <span class="math-container">$p&gt;3$</span> the conclusion as regards the supply function does not change. In theory, the firm can here use a combination of both inputs, but because the marginal profit of <span class="math-container">$x_1$</span>, <span class="math-container">$2p-4$</span> is always greater than the marginal profit of <span class="math-container">$x_2$</span>, <span class="math-container">$p-3$</span>, for these range of prices, the firm will here too want to use only input <span class="math-container">$x_1$</span> for its &quot;infinite&quot; supply.</p>

CC BY-SA 4.0
2020-12-10T12:52:44.160Z
2020-12-10T14:22:45.713Z
2020-12-10T14:22:45.713Z
Alecos Papadopoulos
<p>Q1: A company has the following production function:
<span class="math-container">$$f(x_1,x_2) = 2x_1 + x_2$$</span>.</p>
<p>The factor prices are <span class="math-container">$w_1=4$</span> and <span class="math-container">$w_2=3$</span>. Calculate the company's supply function.</p>
<p>Q2: A company's cost function is
<span class="math-container">$$C(y) = y^2 + 1$$</span>
Calculate the company's supply function.</p>
<p>Unfortunately, I don't understand how the supply function can be calculated based on the production or cost function. I would be very grateful for solutions.</p>
<p>Thank you very much!</p>

CC BY-SA 4.0
2020-12-10T10:56:43.567Z
Hint for Q2: Assume the firm faces price $p$ in the market. At profit maximization point: $MR=MC$. MR is marginal revenue MC is marginal cost. Hint for Q1: Use MRTS to calculate per unit cost of production (maximizing output under budget constraint). This will help get you the cost function, then it is similar to Q2.
Dayne
CC BY-SA 4.0
2020-12-10T11:11:26.363Z
Thank you! My idea for Q2 was: MC = Supply function. That would mean supply(p) = p/2. Is that also true?
jeet31
CC BY-SA 4.0
2020-12-10T11:20:16.283Z
Your supply function is correct but I don't know about the expression`MC=supply`. Supply function is a function of $p$. MR here is $p$ so you will get $2Q=p$
Dayne
CC BY-SA 4.0
2020-12-10T11:42:25.643Z
Alright, I understand it now. Thank you
jeet31
<macroeconomics><microeconomics><supply-and-demand><production-function><cost-functions>
Calculate supply function based on production or cost function
jeet31
# Calculate supply function based on production or cost function

(Asked by: username_0 on 2020-12-10)

Q1: A company has the following production function:
$$f(x\_1,x\_2) = 2x\_1 + x\_2$$.

The factor prices are $w\_1=4$ and $w\_2=3$. Calculate the company's supply function.

Q2: A company's cost function is
$$C(y) = y^2 + 1$$
Calculate the company's supply function.

Unfortunately, I don't understand how the supply function can be calculated based on the production or cost function. I would be very grateful for solutions.

Thank you very much!

**Tags:** macroeconomics, microeconomics, supply-and-demand, production-function, cost-functions

> **Comments:**
> **username_1**: Hint for Q2: Assume the firm faces price $p$ in the market. At profit maximization point: $MR=MC$. MR is marginal revenue MC is marginal cost. Hint for Q1: Use MRTS to calculate per unit cost of production (maximizing output under budget constraint). This will help get you the cost function, then it is similar to Q2.  
> **username_0**: Thank you! My idea for Q2 was: MC = Supply function. That would mean supply(p) = p/2. Is that also true?  
> **username_1**: Your supply function is correct but I don't know about the expression`MC=supply`. Supply function is a function of $p$. MR here is $p$ so you will get $2Q=p$  
> **username_0**: Alright, I understand it now. Thank you  

---

## 1 answers

### Answer by username_2 on 2020-12-10. Score: 1 | Accepted answer

**Q1**

This is not a straightforward problem. Two things to note:  
**First,** the production function is linear in the inputs. This means that their marginal products are constant, and so are their marginal revenue products (presumably the firm is treated as a price taker in the output market). This implies that if price is sufficiently low, then production will be zero.  
**Second:** the production inputs are not complementary, production can happen using both or either one of them.

The profit function is

$$pQ - C = p(2x\_1+ x\_2) - 4x\_1 - 3x\_2 = (2p-4)x\_1 + (p-3)x\_2.$$

From this we see that

$$p < 2 \implies Q = 0$$

because with any choice of inputs the firms will incur losses.

At exactly $p=2$, the firm can have zero-profits if it uses only input $x\_1$ for any output level, so here

$$p=2 \implies Q = {\rm indeterminate.}$$

"Indeterminate" in the sense that it could be zero, or any positive quantity, or even "infinity" (see below) - the firm is indifferent between these choices.

Suppose now that $p>2$.

If price is $2 < p \leq 3$, the firm has an incentive to use only input $x\_1$, because using also input $x\_2$ will incur some loss, while it can use only input $x\_1$ and have profit for every output unit it produces. And then, the more it produces using just $x\_1$ the more profit it has. So here

$$2 < p \leq 3 \implies Q = \infty$$

Certainly, one could wonder whether this is meaningful, since  
a) there is nothing infinite in this world and  
b) as the firm produces more and more output wouldn't that eventually push the price down? How much of the good do consumers are willing to buy after all?

These are valid remarks, but the solution that the supply function will be "infinite" indicates the *profit-maximizing tendency* of the firm, not what will actually happen in the real world.

If $p>3$ the conclusion as regards the supply function does not change. In theory, the firm can here use a combination of both inputs, but because the marginal profit of $x\_1$, $2p-4$ is always greater than the marginal profit of $x\_2$, $p-3$, for these range of prices, the firm will here too want to use only input $x\_1$ for its "infinite" supply.


---

