I have a given product $y$ that is produced by the input $x$ in the following relation: $2x=y$. In our example, we are given the unit price of $x$ is $16$. Find the unit cost of $y$. The answer is $8$. But I strongly disagree: surely it should be $32$ because to create another unit of $y$ I need $2$ units of $x$ each at cost $16$, and therefore $32$ is the cost. Why is my answer wrong? I can’t see a fault in my logic?


To be formal, the production function is (under strict positiivity constraints)

$$y = F(x) = 2x \implies x(y) = \frac{y}{2}$$

and the Total cost function is

$$TC = p_x\cdot x(y) = p_x \cdot \frac{y}{2}$$

Then the Average Cost function is

$$AC = \frac {TC}{y} = \frac{p_x}{2}$$

So the exercise gives the correct answer. Where did the OP's logic went wrong?

The OP wrote

...because to create another unit of $y$ I need $2$ units of $x$

Is the OP sure about the correctness of that statement? It seems to me that if I have one unit of $x$ I can produce two units of $y$ since $y=2x\implies y = 2\cdot 1 = 2$, therefore to one unit of $y$ corresponds half a unit of $x$.

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