11.1 Overview of the "Theory of the Firm"
In the first part of this course, we imagined a single agent, Chuck, who was both a consumer and a producer. He consumed what he produced, and no money changed hands. He could transform resources like capital and labor into goods via production functions, and those goods in turn were transformed into utility via a utility function.
We then “decentralized” these two functions of production and consumption: rather than a single agent consuming what he produces, we had an agent (a “consumer”) who buys goods that someone else produced. We now turn our attention to the other side of that transaction: the producer, or firm, who uses resources to produce goods and sells them to consumers.
Just as we derived the demand for a good from consumers’ utility functions, we will now derive the supply of a good from firms’ production functions. In the theory of the firm, we will assume that the firm takes resources (labor and capital) and uses a production technology to produce goods which it sells to consumers. We will assume that the firm does not own the resources, but hires labor at a “wage rate” of $w$, and hires capital at a “rental rate” of $r$; it then sells its goods for some price $p$.
In the consumer model, we analyzed how a consumer would spend a given budget on two goods to maximize a utility function. This was a constrained optimization problem: the agent was trying maximize an objective function (in that case, a utility function) subject to an exogenously given constraint (a budget line). The consumer took their income $m$, as well as the prices $p_1$ and $p_2$, as given.
The firm faces a different kind of constrained optimization problem: it must simultaneously choose the amount of output to produce ($q$) and the way in which to produce it ($L$ and $K$), subject to the constraint that $q \le f(L,K)$. In other words, the firm is trying to maximize its profits $\pi$, which we define as its total revenue (price of output times quantity of output sold, or $pq$) minus the total cost of inputs (cost of labor $wL$, plus cost of capital $rK$): \(\pi(L,K,q) = pq  (wL + rK)\) subject to the constraint \(q \le f(L,K)\)
Our approach will be to split the firm’s profitmaximization problem into two steps:

Step 1: Find the lowestcost way of producing $q$ units of output by solving the cost minimization problem \(\min_{L,K}\ wL + rK\) \(\text{s.t. }f(L,K) = q\) The solution functions for this problem are the conditional demands $L(q)$ and $K(q)$. The cost of the costminimizing combination of inputs: \(c(q) = wL(q) + rK(q)\)

Step 2: Embed this cost function $c(q)$ into the firm’s profit maximization problem: that is, if the firm sells each unit of output at some price $p$, their profit $\pi$ as a function of $q$ is \(\pi(q) = p \times q  c(q)\)
We can think of step 1 as the costminimization problem of an operations manager within a firm, who receives an order from “management” to produce $q$ units of output and has the job of choosing the lowestcost way of fulfilling that order; and step 2 as the profitmaximization problem of the CEO of the firm, whose job it is to determine the amount of output to produce.
For this entire unit, we’ll look in particular at a firm with the CobbDouglas production function $f(L,K) = \sqrt{LK}$. This function is constant returns to scale, has diminishing $MP_L$ and $MP_K$, and has isoquants of the usual shape:
While you should expect to see different types of production functions, using this canonical example can be helpful since it will let us keep the production function constant while solving a variety of different problems.