# Production technology of $y=2x$

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 $$x$$.