3.2 Firm's Costs
In order to choose optimally its level of production, a firm must first compute how much it costs to produce (in an efficient way) each possible quantity of output. Naturally, this cost in turn depends on the firm’s productivity. In this section we will learn how to compute the firm’s costs, and how they depend on the firm’s productivity.
From Production to Costs
A firm must pay two types of costs. The fixed cost, which we will denote by $FC$, is the cost of the fixed input, i.e. the money the firm needs to pay to keep its unit running (e.g. rent, machinery, ecc.). As the name says, the fixed cost is independent of the chosen level of output. The variable cost of the firm, which instead depends on the desired output level, is the money that the firm needs to pay for labor.
How can we compute, for each possible quantity of output $Q$, the labor cost that the firm needs to pay in order to obtain that quantity of output? To answer this question, it suffices to recall that the inverse production function specifies, for each possible quantity of output $Q$, the number of units of labor needed to obtain that quantity of output: $L=Q^2/A^2$. Now it should be clear that, by multiplying this number by the price of one unit of labor, i.e. the daily wage, which we denote by $W$, we get the answer to our question. We have thus defined the variable cost function of the firm:
\(\begin{gathered} VC(Q) = \frac{W}{A^2} Q^2 \end{gathered}\)
Adding the fixed cost to the variable cost gives the total cost of the firm:
\(\begin{gathered} C(Q) = FC + VC(Q) = FC + \frac{W}{A^2} Q^2 \end{gathered}\)
In the figure below, based on the example discussed in Section 3.1, we illustrate the calculation of the firm’s costs, starting from its production function $Q=80\sqrt{L}$. The firm’s fixed cost is assumed to be $FC=100$ and the daily wage $W=64$.
The next figure shows how costs vary with firm’s productivity, fixed cost, and daily wage (labor is assumed to be finely divisible).
From Costs to Unit Costs: Average Costs
How much does it cost the firm to produce one unit of output? How much of this cost is due to the fixed cost, and how much is due to labor? We will see in Section 3.3 that answering these questions is important for determining the optimal choice of the firm.
If we divide the fixed cost by the number of units of output produced by the firm, we obtain the average fixed cost function. It measures the fixed cost that the firm needs to pay, on average, in order to produce one unit of output:
\(\begin{gathered} AFC(Q) = \dfrac{FC}{Q} \end{gathered}\)
Analogously, dividing the variable cost by the number of units of output produced by the firm, gives the average variable cost, measuring the labor cost that the firm needs to pay, on average, in order to produce one unit of output:
\(\begin{gathered} AVC(Q) = \dfrac{VC(Q)}{Q} = \dfrac{W}{A^2} Q \end{gathered}\)
Intuitively, the latter cost should be the higher, the less productive the firm. Recall that the average product measures the quantity of output obtained, on average, with one unit of labor. Thus, its reciprocal, $1/AP_L$, measures the number of units of labor needed, on average, to produce one unit of output. Moltiplying this number by the unit price of labor therefore gives the labor cost that the firm needs to pay, on average, to produce one unit of output, i.e. the average variable cost. Indeed, it is easy to see that
\(\begin{gathered} AVC = \frac{W}{AP_L} \end{gathered}\)
Finally, adding average fixed cost and average variable cost gives the average cost, i.e. the cost of fixed and variable inputs that the firm needs to pay, on average, to produce one unit of output:
\(\begin{gathered} AC(Q) = \dfrac{C(Q)}{Q} = \dfrac{FC+VC(Q)}{Q} = \frac{FC}{Q} + \frac{W}{A^2} Q \end{gathered}\)
In the following figure we illustrate the computation of the average cost functions starting from the firm’s costs, and discuss their shape. As before, we are assuming that the production function is $Q=80\sqrt{L}$, the fixed cost $FC=100$, and the daily wage $W=64$.
From Costs to Unit Costs: Marginal Cost
As we shall see in Section 3.3, the average cost function is useful for understanding whether the firm is able to generate positive profits. The notion of unit cost that we introduce here, marginal cost, will play a crucial role in determining the firm’s optimal choice. The marginal cost is the cost that the firm must pay, on average, to produce one of the last $\Delta Q$ units of output. Like average costs, marginal cost is a function of the quantity $Q$ of output produced by the firm. Its formula is
\(\begin{gathered} MC(Q) = \frac{\Delta C}{\Delta Q} = \frac{C(Q)-C(Q-\Delta Q)}{\Delta Q} \end{gathered}\)
When labor and output are finely divisible, the marginal cost function is the derivative of the total cost function (equivalently the derivative of the variable cost function, given that $AC$ and $AVC$ only differ by a constant, namely the fixed cost):
\(\begin{gathered} MC(Q) = \frac{2W}{A^2} Q \end{gathered}\)
Here, too, it is worth discussing the relationship between costs and productivity. Recall that the marginal product of labor measures the quantity of output obtained, on average, by one of the last $\Delta L$ units of labor hired by the firm. Its reciprocal $1/MP_L$ therefore measures the number of units of labor needed, on average, to produce each of the last $\Delta Q$ units of output. Multiplying this number by the price of labor therefore gives the number of euros the firms must spend, on average, to produce one of those last units, i.e. the marginal cost. Indeed, it is easy to see that
\(\begin{gathered} MC = \frac{W}{MP_L} \end{gathered}\)
The next figure illustrates the calculation of marginal cost from total cost.
The figure below summarizes the concepts introduced in this section, and shows how costs and unit costs depend on the firm’s technology (parameter $A$), fixed cost and price of labor (parameters $FC$ and $W$).
Minimum Average Cost and Efficient Scale of Production
Following up on our discussion about the U-shape of the average cost function (with $\alpha=1/2$ and, more generally, whenever $\alpha<1$), let us observe the following: for low levels of output, marginal cost is below average cost, while for high levels of output, marginal cost is above average cost. In the first case, average cost must therefore be decreasing. Think again about your school grade average: if the last grade received is lower than the average grade, the average grade will go down. In the second case, average cost must instead be increasing: if the last grade received is higher than average, the average will go up.
When marginal cost equals average cost, i.e. at the point where $MC$ intersects $AC$, average cost stops decreasing and starts increasing. That is, average cost reaches its minimum, which we denote by $AC_\text{min}$ in the figure below. The quantity of output at which this happens, denoted by $Q^\text{eff}$, is the efficient scale of production of the firm. As we will see in the next chapter, the notions of minimum average cost and efficient scale of production play an important role in the firm’s long-run decisions.