Total cost

Input prices and total cost

How do firms determine which technology to use? They seek to maximize profit, so they use the technology that makes producing goods least expensive.

We assumed that each technology only uses two inputs, labor and electricity. This means we can write an equation expressing how total costs are calculated based on the input prices (denoting \(p\) for the price of electricity and \(w\) for the hourly wage) and quantities (denoting \(E\) for the units of electricity and \(L\) for the hours of labor) needed to produce one unit of output:

\[C = pE + wL\]

Simply put, this equation is calculating total costs by adding the price of electricty multiplied by the quantity of electricity to the price of labor multiplied by the cost of labor.

Notice that in our graph to visualize technologies, we had the units of electricity (\(E\)) in the \(y\)-axis and the hours of labor (\(L\)) in the \(x\)-axis. Therefore, we can rearrange the total cost equation into the formula of a line:

\[E = \frac{C}{p} - \frac{w}{p}L\]

Visualizing an isocost line

The total cost (\(C\)) is the total amount of money a firm spends on production. Holding input prices and total cost constant, we can draw a line using the total cost equation that we derived above. This line is called an isocost line. Isocost lines map out all possible combinations of our inputs, labor and electricity, that have the same total cost. These lines are downward sloping because if you use less of one input but maintain the same total cost, you must be using more of the other input.

To draw an isocost line, you need three things: price of Input \(X\), price of Input \(Y\), and the total cost. We place Input \(X\) on the \(x\)-axis and Input \(Y\) on the \(y\)-axis. If Input \(X\) (hours of labor) is priced at $10, Input \(Y\) (units of electricity) at $15, and total cost at $300, then the equation of the isocost line is:

\[Y = \frac{300}{15} - \frac{10}{15}X\]

The intercepts are the maximum amounts of Input \(X\) or Input \(Y\) that can be purchased given the total cost if none of the other input is used. Notice, the \(y\)-intercept is 300 (total cost) divided by 15 (price of Input \(Y\)). This gives you the maximum units of Input \(Y\) that lies on this isocost line, which is 20 units.

The slope of the isocost line is the hours of labor that have the same cost as one unit of electricity. Mathematically, the slope is \(\frac{-w}{p}\). Therefore, in the example above, the slope is -10 (the hourly wage) divided by 15 (cost of one unit of electricity).

The following tool allows you to enter input prices and the total cost to graph an isocost line. Try changing the input prices and total cost. Observe how these changes affect the slope and the intercepts of the isocost line.

As total cost changes, the line shifts up (with a higher total cost) or down (with a lower total cost). This means that both intercepts will change but the slope stays the same. Changes in the price of individual inputs will affect the intercepts and the slope.