This is “Myriad Costs”, section 9.6 from the book Beginning Economic Analysis (v. 1.0).
For more information on the source of this book, or why it is available for free, please see the project's home page. You can browse or download additional books there. You may also download a PDF copy of this book (7 MB) or just this chapter (524 KB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline).
How much does it cost to produce a given quantity q? We already have a detailed answer to this question, but now we need to focus less on the details and more on the “big picture.” First, let’s focus on the short run and suppose L is adjustable in the short run but K is not. Then the short-run total costThe total cost of output with only short-run factors varying. of producing q,—that is, the total cost of output with only short-term factors varying—given the capital level, is
$$\text{SRTC(q|K)=}\underset{L}{\mathrm{min}}\text{}rK+wL\text{,overall}L\text{satisfying}\mathrm{F(K,L)}\ge q\text{.}$$In words, this equation says that the short-run total cost of the quantity q, given the existing level K, is the minimum cost, where L gets to vary (which is denoted by “min over L”) and where the L considered is large enough to produce q. The vertical line | is used to indicate a condition or conditional requirement; here |K indicates that K is fixed. The minimum lets L vary but not K. Finally, there is a constraint F(K, L) ≥ q, which indicates that one must be able to produce q with the mix of inputs because we are considering the short-run cost of q.
The short-run total cost of q, given K, has a simple form. First, since we are minimizing cost, the constraint F(K, L) ≥ q will be satisfied with equality, F(K, L) = q. This equation determines L, since K is fixed; that is, $F(K,{L}_{S}(q,K))=q$ gives the short-run value of L, LS(q, K). Finally, the cost is then rK + wL = rK + wLS(q, K).
The short-run marginal costThe derivative of short-run total cost with respect to output., given K, is just the derivative of short-run total cost with respect to output, q. To establish the short-run marginal cost, note that the equation F(K, L) = q implies
$$\frac{\partial F}{\partial L}(K,{L}_{S}(q,K))\text{\hspace{0.17em}}dL=dq,$$or
$${\frac{dL}{dq}|}_{F=q}=\frac{1}{\frac{\partial F}{\partial L}(K,{L}_{S}(q,K))}.$$The tall vertical line, subscripted with F = q, is used to denote the constraint F(K, L) = q that is being differentiated. Thus, the short-run marginal cost is
SRMC(q|K) = $SRT{C}^{\prime}(q)=\frac{d}{dq}(rK+wL)=w{\frac{dL}{dq}|}_{F=q}=\frac{w}{\frac{\partial F}{\partial L}(K,{L}_{S}(q,K))}.$There are three other short-run costs we require to complete the analysis. First, there is the short-run average cost of production that we obtain by dividing the total cost by the quantity produced:
$SRAC(q|K)=\frac{SRTC(q|K)}{q}\text{.}$Second, there is the short-run variable costThe total cost minus the cost of producing zero units. that is the total cost minus the cost of producing zero units—that is, minus the fixed cost—which in this case is rK. Finally, we need one more short-run cost: the short-run average variable cost. The short-run average variable cost is the short-run variable cost divided by quantity, which is given,
$$SRAVC(q|K)=\frac{SRTC(q|K)-SRTC(0|K)}{q}=\frac{w{L}_{S}(q|K)}{q}.$$The short-run average variable costThe average cost ignoring the investment in capital equipment. is the average cost ignoring the investment in capital equipment.
The short-run average costThe average of the total cost per unit of output. could also be called the short-run average total cost, since it is the average of the total cost per unit of output, but “average total” is a bit of an oxymoron.An oxymoron is a word or phrase that is self-contradictory, like “jumbo shrimp,” “stationary orbit,” “virtual reality,” “modern tradition,” or “pretty ugly.” Oxymoron comes from the Greek “oxy,” meaning sharp, and “moros,” meaning dull. Thus, oxymoron is itself an oxymoron, so an oxymoron is self-descriptive. Another word that is self-descriptive is “pentasyllabic.” Consequently, when total, fixed, or variable is not specified, the convention is to mean total. Note that the marginal variable cost is the same as the marginal total costs, because the difference between variable cost and total cost is a constant—the cost of zero production, also known as the short-run fixed costThe difference between variable cost and total cost. of production.
At this point, we have identified four distinct costs, all of which are relevant to the short run. These are the total cost, the marginal cost, the average cost, and the average variable cost. In addition, all of these costs may be considered in the long run as well. There are three differences in the long run. First, the long run lets all inputs vary, so the long-run total costThe total cost of output with all factors varying. is the total cost of output with all factors varying. In this case, $\text{LRTC}(q)\text{=}\underset{L,K}{\mathrm{min}}rK+wL$ over all L and K combinations satisfying F(K, L) ≥ q.
Second, since all inputs can vary, the long-run cost isn’t conditioned on K. Finally, the long-run average variable costThe long-run total cost divided by output. is the long-run total cost divided by output; it is also known as the long-run average total cost. Since a firm could use no inputs in the long run and thus incur no costs, the cost of producing zero is zero. Therefore, in the long run, all costs are variable, and the long-run average variable cost is the long-run average total cost divided by quantity.
Note that the easiest way to find the long-run total cost is to minimize the short-run total cost over K. Since this is a function of one variable, it is straightforward to identify the K that minimizes cost, and then plug that K into the expression for total cost.
One might want to distinguish the very short run from the short run, the medium run, the long run, and the very long run.. But a better approach is to view adjustment as a continuous process, with a gradual easing of the constraints. Faster adjustment costs more. Continuous adjustment is a more advanced topic, requiring an Euler equation approach.
For the Cobb-Douglas production function $F(K,L)=A{K}^{\alpha}{L}^{\beta},$ with α + β < 1, K fixed in the short run but not in the long run, and cost r of capital and w for labor, show
SRTC(q|K) = $rK+w{\left(\frac{q}{A{K}^{\alpha}}\right)}^{\frac{1}{\beta}},$
SRAVC(q|K) = $w\frac{{q}^{\frac{1-\beta}{\beta}}}{{\left(A{K}^{\alpha}\right)}^{\frac{1}{\beta}}},$
SRMC(q|K) = $w\frac{\text{\hspace{0.17em}}{q}^{\frac{1-\beta}{\beta}}}{\beta {\left(A{K}^{\alpha}\right)}^{\frac{1}{\beta}}},$
LRTC(q|K) = $\left({\left(\frac{\alpha}{\beta}\right)}^{\frac{\beta}{\alpha +\beta}}+{\left(\frac{\beta}{\alpha}\right)}^{\frac{\alpha}{\alpha +\beta}}\right){w}^{\frac{\beta}{\alpha +\beta}}{r}^{\frac{\alpha}{\alpha +\beta}}{\left(\frac{q}{A}\right)}^{\frac{1}{\alpha +\beta}}.$
Consider a cost function of producing an output q of the form c(q) = q^{2} + 2q + 16. Determine the following:
Graph the long-run supply curve, assuming the cost function is for a single plant and can be replicated without change.