13.1: Initial Solution
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Recall that the firm’s backbone is the production function. Inputs, or factors of production, typically labor (L) and capital (K) are used to make output, or product (q).
In previous chapters, we explored the firm’s input cost minimization and output profit maximization problems. This chapter returns to the input side and works on the firm’s third optimization problem: input profit maximization.
We continue working with a perfectly competitive (PC) firm, but we extend the assumption of perfect competition to input markets. Thus, not only is the firm one of many sellers of a perfectly homogeneous product with free entry and exit, it is also one of many buyers of labor and capital. Our firm is an output and input price taker.
This means that our PC firm only chooses the amount of input to hire, not how much to pay for it. If it has market power, then the firm not only determines how much to hire, but also gets to choose the input price. In this case, we say the firm has monopsony power.
While you have surely heard of monopoly, monopsony may be new to you. They are similar in that one is selling (monopoly) and the other buying (monopsony) and that means price (output or input) is no longer exogenous. A classic example is the only hospital in a small town hiring nurses. Another example is a big box retailer. Walmart is such a big buyer that they have monopsony power. They can negotiate with suppliers and extract cheaper prices from them. Notice that a firm can have both monopoly and monopsony power.
In a Labor Economics course, you study how firms can take advantage of the ability to set input prices to make greater profits. We assume this possibility away and stay with a PC firm that takes the wage rate (w) and rental rate of capital (r) as given. Our PC firm is such a small buyer that it can hire as much L and K as it wants at the going w and r.
Setting Up the Problem
There are three parts to every optimization problem. Here is the framework for a PC firm.

Goal: Maximize profits (\(\pi\)), which equal total revenues minus total costs. To distinguish the input from the output side, we use the terms total revenue product (TRP) and total factor cost (TFacC). The idea is that labor and capital are used to make product that is sold so price times the number of units produced is the TRP.

Endogenous variables: labor and capital, in the long run; only L in the short run.

Exogenous variables: price (of the product, P), input prices (the wage rate and the rental rate of capital), and technology (parameters in the production function).
As usual, we will work with a CobbDouglas production function, with \(\alpha >0\), \(\beta >0\), and \(\alpha + \beta < 1\). \[q=AK^\alpha L^\beta\] Revenues are the output price multiplied by the output produced, \(TR=Pq\). We substitute the production function for q in TR to get total revenue product: \[TRP=PAK^\alpha L^\beta\] The units of TRP are dollars (just like total revenue). The "revenue product" language indicates that we are considering the amount of revenue ($) produced by the inputs.
The costs are simply the amounts spent on labor and capital, \(wL + rK\). These are called total factor costs.
The firm chooses L and K to max profits. \[\begin{gathered} %star suppresses line # \max\limits_{L,K} \pi = PAK^\alpha L^\beta  (wL+rK)\end{gathered}\]
Finding the Initial Solution
First the problem is solved using numerical methods, and then the analytical approach is used.
STEP Open the Excel workbook InputProfitMax.xls and read the Intro sheet,then go to the TwoVar sheet to see the problem implemented in Excel.
The sheet is named TwoVar because both inputs are choice variables, which means this is a long run profit maximization problem. As usual, the sheet is organized into the colorcoded components of an optimization problem, with goal, endogenous, and exogenous cells.
STEP Read the description of the firm, a bakery, and scroll down to the endogenous variables.
On opening, the sheet has 500 hours of labor hired and 100 units of capital rented, yielding a profit of $936. Is this the best this firm can do? Cells B48 and B49 show the marginal revenue product of labor and marginal factor cost. By hiring one more hour of labor, revenues would rise by more than costs, so profits would increase. Clearly, therefore, this bakery is not optimizing.
STEP Run Solver to find the initial solution. Your screen should look like Figure 13.1.
The firm hires roughly 1,431 hours of labor and rents 153 machines (but click on cells B34 and B35 to see more decimal places). This yields a maximum possible profit of just over $1,900.
Notice that the marginal revenue product and marginal factor cost cells are now exactly equal at $20/hour. This is no coincidence. The equimarginal condition for input profit maximization is that \(MRP=MFC\). Since the firm is an input price taker, \(MFC=w\) (just like \(P=MR\) for a PC firm) so it is also true that \(MRP=w\) at the optimal solution.
Finally, notice the breakdown of the firms revenues in rows 44 to 46. Labor’s share (wL), capital’s share (rK), and profits (whatever is left) add up to 100%. K and L’s shares, 75% and 20% equal \(\alpha\) and \(\beta\). Is that a coincidence? No, that’s a property of the CobbDouglas functional form. The exponent tells you the share of revenues that factor will receive.
We can also solve this problem via the analytical approach. We know the objective function and can substitute in each of the parameter values. \[\begin{gathered} %star suppresses line # \max\limits_{L,K} \pi = PAK^\alpha L^\beta  (wL+rK)\\ \max\limits_{L,K} \pi = 2*30*K^{0.2} L^{0.75}  (2L+3K)\end{gathered}\] Next, we take derivatives with respect to L and K, set them equal to zero, and use algebra to solve the two equation system of firstorder conditions.
We can move the 20 and 50 to the right hand side and this immediately reveals the equimarginal conditions: \(MRP_L = w\) and \(MRP_K = r\).
We solve the first equation for L and substitute it into the second equation to solve for optimal K. We use the rule that \((x^a)^b = x^{ab}\) to solve for L.
Substitute the expression for L into the second firstorder condition.
Compute optimal L from the expression for L. \[L \mbox{*}=2.25^4K^{0.8}=2.25^4[152.6842]^{0.8}=1431.414\] Compute maximum profits. \[\pi \mbox{*}=2*30*[152.6842]^{0.2}*[1431.414]^{0.75}2*[1431.414]3*[152.6842]=\$1908.55\] This analytical solution is extremely close to Excel’s solution. Practically speaking, as we would expect, the two solutions are the same.
The Short Run
A slightly different version of the firm’s input profit maximization problem involves the short run when capital is not variable. By putting a bar over K, we highlight that capital is fixed. \[\max\limits_{L} \pi = PA\bar{K}^\alpha L^\beta  wLr\bar{K})\] We do the analytical solution first this time and in general form. There is only one derivative (since there is only one choice variable) and one firstorder condition.
STEP To see the numerical version of this problem, proceed to the OneVar sheet.
Notice that there is only one endogenous variable, L. Capital has been moved to the exogenous list because we are in the short run.
Notice also that there are two graphs. Each one can be used to represent the initial solution.
Below the graphs, you can see that the marginal revenue product of labor does not equal the wage. As you know, this means you need to run Solver because the firm is not optimizing.
STEP Run Solver to find the initial solution. Your screen should look like Figure 13.2.
The bottom graph shows that the optimal labor use can be found where the marginal revenue product of labor (the curve) equals the wage (at $20/hr). This is the canonical graph for the input side profit maximization problem. Like \(MR=MC\) on the output side, the intersection of the two marginal relationships instantly reveals the optimal solution.
The top graph is a different way of viewing the exact same problem. It is using the production function as a constraint (the TRP curve) and three representative isoprofit lines are displayed. Each isoprofit line shows the combination of L and q that gives the same profit. The firm is trying to get on the highest isoprofit (to the northwest) while meeting the constraint. It can roll on the TRP curve (like it rolled on the isoquant) until it hits an isoprofit line that is tangent to the TRP.
The constrained optimization problem can be written like this: \[\begin{gathered} %star suppresses line # \max\limits_{L,q} \pi = Pq  wLr\bar{K}\\ \textrm{s.t. } q=A\bar{K}^\alpha L^\beta\end{gathered}\] The Lagrangean method could be applied to solve this problem. Naturally, the exact same solution is obtained if we use the Lagrangean or the more common approach of directly substituting the constraint (the production function) into the revenue function.
Suppose we wanted to check if the analytical and numerical results are the same. We need to evaluate the expression for optimal L at the parameter values in the OneVar sheet.
The expression is complicated enough that entering it in a cell as you would write it is a bad idea. The parentheses are likely to cause confusion. It is better to create houses for each part then fill them in. Here’s how.
STEP Watch this short video on how to enter a complicated formula in Excel: vimeo.com/415967747.
Entering parentheses as pairs, is a good habit to develop when working in a spreadsheet. It is easy to make an order of operations mistake or get mismatching parentheses if you try to enter the formula like you would on a piece of paper.
STEP Enter the formula in cell M28 (just like in the video) to practice building houses in formulas in Excel.
In so doing, you confirm that the analytical and numerical methods yield substantially the same answer.
Another Short Run Production Function
A CobbDouglas production function has many advantages, including that the sum of exponents reveals whether returns to scale are increasing, constant, or decreasing if they are greater, equal, or less than one. However, once the exponents are set, the function can only exhibit those returns to scale.
Likewise, in the short run, with K fixed, our CobbDouglas functional form showed the Law of Diminishing Returns because \(\beta = 0.75\). A more flexible functional form would enable production to have increasing and diminishing returns as more labor is added.
Like the cubic polynomial we used for the total cost function, a cubic functional form can give us an Sshaped TRP curve. \[TRP=aL^3+bL^2+cL\]
STEP Proceed to the Graphs sheet to see this functional form implemented in a set of four graphs that can be used to represent the firm’s input profit maximization problem (Figure 13.3).
It is striking that these graphs mirror the four graphs we used to describe the firm’s output side profit maximization problem. The two top graphs show total revenue and total cost on the top left, along with total profits on the top right. The bottom graphs display a series of marginal and average curves on the bottom left and marginal profit on the bottom right.
If you look carefully, you will notice that things are switched around a bit. Instead of total cost being a curve (as it is on the output side), it is a straight line because total factor cost on the input side in the short run is \(wL+ r\bar{K}\). On the other hand, total revenue product (so named to distinguish it from total revenue on the output side) is a curve (instead of a straight line).
Unlike the canonical output side profit maximization graph with Ushaped MC, ATC, and AVC curves and a horizontal \(P = MR\) line, the bottom left graph has a horizontal MFC line and the MRP and ARP functions are curves and they are upside down.
But there are also key similarities. The equimarginal rule is in play: \(MFC=MRP\) reveals the labor use that maximizes profits. Also, a rectangle of \((ARPAFC)L\) gives an area that is equal to profits. The length of the profit rectangle ranges from zero to the chosen amount of labor hired. The height is the difference between average revenue product, ARP, and average factor cost, AFC. The area of this rectangle is profit because \(ARP  AFC\) is profit per hour so multiplying by L, measured in hours, yields profits. Another way to think about this is that multiplying L by ARP yields total revenues (since \(L*TRP/L=TRP\)) and multiplying L by AFC gives total costs (since \(L*TFacC/L=TFacC\)). Subtracting the total cost rectangle from the total revenue rectangle leaves the profit rectangle.
Another similarity between output and input profit maximization is that the firm has the same four profit positions.
STEP In the Graphs sheet, click on the pull down menu (near cell P4) and cycle through all of the profit positions.
As with the output side, the shock is output price. As it falls, so do maximum profits.
The Neg Profits, Cont Prod and Neg Profits, Shutdown options show that the firm will shut down when the \(w>ARP\). This is analogous to the \(P < AVC\) Shutdown Rule. Keep your eye on the total profits in the top right graph to see that the story is the samethe firm is deciding whether the negative profit at best of the positive levels of L is better than hiring no L at all.
The connection between input and output is simple. The firm shuts down when \(w > ARP\) which we can multiply by L to give \(wL > TRP\). But wL and TRP are TVC and TR on the output side. Divide both by q and we get \(AVC>P\), which is the same as \(P<AVC\), the usual output side Shutdown Rule. In addition, the \(wL > TRP\) version of the Shutdown Rule supports the claim that revenues must cover variable costs for a firm to produce.
Input Profit Maximization Highlights
At this point, you might be suffering from repetitive stress syndromewe seem to be going over and over the same ideas. That is an important level to attain in mastering the economic way of thinking. The body of knowledge in economics is grounded in a core methodology of optimization and comparative statics. The framework is used over and over and over again.
Like every optimization problem, the input side profit maximization problem can be organized into a goal, endogenous, and exogenous variables. This problem has a canonical graph (with MFC and MRP as the key elements) and an equimarginal rule \(MFC = MRP\).
Because the firm is an input price taker, \(MFC=w\). This means that every additional hour of labor adds w to total cost. If the firm was a monopsony, this would not be true and the optimization problem would be more complicated.
Finally, because the input profit maximization problem is the flip side of the output side profit maximization problem, it should not be surprising that we can represent the initial solution with a set of four graphs. The parallelism carries through all the way to the Shutdown Rule, where \(w>ARP\) is equivalent to \(P < AVC\). We will stress the connections between input and output side again in the next chapter.
Exercises

Use the TwoVar sheet to compute the long run beta elasticity of \(L \mbox{*}\) from beta = 0.75 to beta = 0.74. Show your work.

In the Q&A sheet, question 4 asks you to find short run beta elasticity of \(L \mbox{*}\) from beta = 0.75 to beta 0.74. The InputProfitMaxA.doc file in the Answers folder shows that the answer is about 28. Explain why the short run elasticity (which is admittedly quite large) is much smaller than the long run elasticity that you computed in the previous question.

Use Excel to set up and solve (with Solver, of course) the constrained version of the input profit maximization problem in the OneVar sheet. Take a screenshot of your solution (including the constraint cell) and paste it in your Word document.

In the Graphs sheet, select the Neg Profits, Shutdown case. Does the top, right graph support the \(w>ARP\) Shutdown Rule? Explain.
References
The epigraph, from John Palmer at thesportseconomist.com/whatisthemarginalrevenueproductofbarrybonds, points to two avenues for further reading: sports economics and blogs.
The worlds of economics and sports are increasingly intertwined. There are courses, conferences, and journals dedicated to the economics of sports. For a classic paper on baseball, see Simon Rottenberg’s "The Baseball Players’ Labor Market," The Journal of Political Economy, Vol. 64, No. 3 (June, 1956), pp. 242–258, www.jstor.org/stable/1825886.
There are, of course, many blogs dedicated to economics. The marginalrevolution.com and cafehayek.com are often informative and entertaining. For macroeconomics, see Greg Mankiw at gregmankiw.blogspot.com and Brad DeLong at delong.typepad.com. John Cochrane will give you a free market perspective at johnhcochrane.blogspot.complus, The Grumpy Economist is a great name for a blog.