Calculate supply function based on production or cost function

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!

• 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. Dec 10 '20 at 10:56
• Thank you! My idea for Q2 was: MC = Supply function. That would mean supply(p) = p/2. Is that also true? Dec 10 '20 at 11:11
• Your supply function is correct but I don't know about the expressionMC=supply. Supply function is a function of $p$. MR here is $p$ so you will get $2Q=p$ Dec 10 '20 at 11:20
• Alright, I understand it now. Thank you Dec 10 '20 at 11:42

1 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.