Input Markets and Optimal Choice

How many workers should you hire?  How much should you pay for that loan?  What is the most that you could pay for that land?  How much should you spend on a new capital investment project?  These are very different from the questions we have looked at up to this point - questions that dealt with the pricing and output decisions of firms.  These are related questions, however, since the production and pricing decisions are closely connected to the decisions regarding the use of inputs. 

The focus of the current unit is the input markets.  In a later unit we will include this in an analysis of the means by which input prices are established.

The major inputs and their prices are enumerated in the table below.  The inputs we will examine here are capital and land, holding off to later a more detailed look at the labor market.  Before we discuss the inputs and their prices in detail, we will examine a generic framework used to describe the pricing of these resources.

Inputs and their Prices

Not surprisingly, at the heart of the traditional theory of input pricing is a supply-demand model. Input prices, just like output prices, are set in markets which may differ in the details, but possess a common denominator. As a result, the approach will be the same as that we used before. In each case you will need to follow the cookbook approach and identify the following:

Price
Buyers
Sellers
Determinants of behavior
Peculiarities of input

In all the markets, demand for the input is based on firms' demand for those inputs.  What guides the firms in their choices?  The same thing that guided them in their output decisions.  As you will recall from our earlier work, firms are assumed to be maximizing their profit when their output and pricing decisions are guided by the MR=MC condition.  A firm would expand production if MR > MC, and cut production if MR < MC.

There is no reason to assume firms change their behavior when they hire inputs. They will hire inputs as long as the MR of the input is greater than the MC of the input, and they will cut back on their use of the input if MR < MC. But what is the MR of an input? To answer this question we must return to our earlier discussion of marginal productivity, which we can illustrate with two examples from RIU.   In the first we will examine the tutoring problem of Chris and in the second we will determine how many faculty members will be demanded at various salary levels. 

Optimal Choice of Inputs: How many hours of tutoring?

The question we want to examine here is: how much of the input should be used?  Just as we found in the output market, the answer to the question depends upon the firm's objective.  In this case we will follow our initial approach and assume the goal is maximization of profit.  To demonstrate the situation we will return to the tutoring problem we first saw in our discussion of production and cost.  At that time we examined the relationship between the grades in ECN and hours of tutoring help.  Here we will add two pieces of information  to the earlier work, the same information we used in the earlier unit on output choice, the prices of inputs and output.  To make our life easy we will assume each grade point is valued at $.5 and the value per grade is independent of the level of grades - we are talking about perfect competition.  Things would differ a bit if we did not assume perfect competition, but the essence of the problem would remain unchanged.  

The complete information set appears in the table below.  The column headings look similar to what we have already seen, but you should note one important difference.  In this table we are looking at the relationship between the hours spent with the tutor and costs and revenues (inputs with revenue and cost), while in the previous table we looked at the relationship between grades and costs and revenues (outputs with revenue and cost). Our attention here is focused on the relationship between input use and profit rather than output levels and profit.  [To see the differences you should return to the comparable table in the output choice section.  In that table, for example,  MC = $.50 was the additional cost of a grade when output increased from a grade of 36 to a grade of 46.  Here a MC of $5 equals the additional cost of using one more hour of tutoring input which in this example is assumed to be $5 per hour.

Cost and Revenue Relationships

Price per grade is simply $.50 regardless of the hours studied since Chris will always value the grades at $.5 per point. The Total Revenue column is output times price per unit of output.  When Chris studies 4 hours with the tutor total revenue generated is $23 [46*$.5].   Marginal Revenue Product is the ratio of the change in revenue to the change in hours studied (input).  This provides a measure of the additional revenue created by adding one more hour of study time.  The marginal revenue product generated by adding the 2nd hour of studying is $6 [ ($11-$5)/(2-1)].  This is very much like the MR concept from the last unit - the difference being this one is measuring the revenue generated by the input and not the output.

Actually the MRP is determined by two factors - the marginal revenue of the output and the marginal product. Let's look at the MRP figure of $5 when we use the 4th hour of the tutor.  The additional tutoring time produces additional grades - the MP of tutoring.  In this case the MP = 10 - the additional hour of work increases the grade by 10 points.  The additional grades create additional revenue.  In this case, because the price remains constant at $.5, the additional 10 grades create additional revenue of $5. 

In the last column we have Profit which is simply the difference between total revenue and total cost.  In the case of Chris's studying, we see the profit generated by the studying peaks at 3-4 hours of studying.  The question that has not been answered, but is certainly of interest, is: are there are any generalizations concerning input use we might make based on our analysis of Chris's studying? 

The answer is yes, and we can find it in the table if we compare the Marginal Revenue Product and the Marginal Cost columns.  It is no accident profit is maximized when the two are equal. In fact we can make the generalization that profit will be maximized when the additional revenue generated by employing the last unit of input (MRP) equals the additional cost of employing that unit (MC).  If the addition to costs were less than the addition to revenue (MC < MRP), then input use should be expanded since profit would increase.  Similarly, if the additions to cost of employing additional inputs were greater than the additions to revenue (MC > MR), then the optimal decision would be to cut back on your use of the input. In this example the optimal use of tutoring would be 4 hours.  By moving to the fifth hour the additional cost (MC) equaled $5, more than the additional revenue (MR) generated which equaled $4. 

To make sure you understand the situation, determine how many hours you would want if the tutor's cost per hour was $3.75. In that situation you would hire 5 hours of tutoring time since the fifth hour now costs you $3.75 and it creates $4 of revenue.  Actually what we are beginning to see is a demand curve for tutoring.  At a cost (price) of $4 per hour you will buy 5 hours of tutoring, while at $5 you will buy 4 hours.  

One of the things that is obvious here is that the demand for the input is a derived demand - it is derived from demand for the output.  If there is an increase in demand for the output, this will increase the price of the output which would increase the marginal revenue product. To convince yourself of this you should redo the analysis under the assumption that the price of the grade has risen to $1.  The Total Revenue and Marginal Revenue columns have now doubled as you can see in the table below.  The graph that follows includes the original MRP curve and the new one.  As you can see, if the price of labor is $6, then when grades are priced at $.5 there will be a demand for 3 hours, but once the price has risen to $1, then demand will be a demand for 6 hours. This is reflected in the outward shift in the MRP curve.

Cost and Revenue Relationships:
Price of a Grade Rises to $1

Optimal Choice of Inputs: Demand for faculty at RIU

What is the optimal, profit maximizing size of the faculty at RIU?  This is a question which we have already looked at from the perspective of the number of students, but we could also look at it from the perspective of the number of faculty.  We could turn our attention from outputs to inputs.  The basis for this example can be found in the unit on output and pricing in the output market where we looked at RIU's choice regarding its optimal size.  Now we want to extend that analysis and ask the question: If the average  faculty salary is $200,000, how many faculty would the school want to hire? The relevant data appears in the table below.   

The headings, with the possible exception of MRP, should be familiar to you. As the university considered the move from 650 to 700 faculty, the number of students would increase from 8,775 to 9,800. With the 50 additional faculty the student population rose 1,025 - an increase of 20.5 students per faculty. This is the marginal product of the faculty being hired. But what is the revenue being generated by these faculty. To answer this we need to know the tuition that is being paid by the students. If we assume that the students all pay a tuition of $10,100, then the income generated by each of these additional faculty, what we refer to as the marginal revenue product, will be the number of students times the tuition [20.5*$10,100 = $207,050]. This figure is the maximum the university could pay the additional faculty without losing money on the expansion.  If we considered the move to 750 faculty, each of these faculty would generate $65,650 which would be the maximum one could pay for their services.

RIU

So how many faculty should RIU hire?  If we graphed the relationship between marginal revenue product and the number of faculty we have a curve representing the maximum amount RIU would pay faculty at each salary - what may sound very much like the demand curve that it is. In fact, if you rename this you will have the demand curve for faculty that is derived from the demand for seats at the university. The university will pay faculty over $200,000 for a 700 faculty university and only $65,650 for a 750 faculty university. To test your understanding of the concept, think of what would happen to the demand curve if the tuition rate fell as a result of decreased demand for seats.  Because the demand for faculty is derived from the demand for seats, you would expect it to decrease (shift inward).

Definitions of  concepts

Marginal (revenue) product: additional revenue obtained by selling output produced by 1 more unit of input

Optimal input choice rules for profit maximizer

Short run

Marginal Revenue Product of input (MRP_A) = Marginal Cost of input (P_A).  A firm will maximize profit if the marginal cost of the input (P_A) equals the marginal revenues (MRP_A).  The formula is given below.

MRP_A =  MR_X * MP_A  = P_A

Long run

In the long-run the firm will follow the same process, except here the firm will have the choice of a mix of inputs.  In the short-run, the optimal choice decision was based on the assumption that only one input could be varied.  In the long-run firms have more degrees of flexibility and they will choose inputs in a way that is related to their prices.  If there are two inputs, they will choose inputs such that one $ spent on input A will generate the same revenue as one $ spent on input B.  

Marginal Revenue Product from input divided by the price of A (/P _A ) =  Marginal Revenue Product from input divided by the price of B (P _B ).

MRP_A /P_A = MRP_B /P_B

Imperfect Competition in the Output Market

There are times that the firm will have market power in the output market and this will have an impact on input demand.  When a firm has market power in the output market where it sells X, then MR_X <P_X - as the use of the input expands and output increases, then this drives down the price of the output.  As a result  the marginal revenue product of A where the firm has market power in the output market is less than the marginal revenue product of A where the firm has no market power in the output market.  Stated somewhat differently, the firm's market power in the output market reduces the firm's demand for the input.  

 MRP_A = MR_X * MP_A=  P_A <  P_A* MRP_A

The Bottom Line: A profit maximizing firm will always continue to hire additional inputs (labor) as long as the contribution of that resource (Marginal revenue) is greater than the increase in cost of the extra resource (marginal cost).

Market for Input

We now have one half of our market for faculty - the demand curve.  We now need to look at the supply side of the market. We can once again use the RIU example and attempt to put ourselves in the positions of suppliers. What do you think would happen if the price (wage) the university paid its faculty would increase? If the wage increased, you would expect more people would be willing to work for the university. The result is what you expect, the supply curve to be positively sloped.  At a higher price there will be a greater supply of the input. If we combine this supply information with our information on demand we get the following diagram for the input market. As you would expect, the analysis will be the same as what we saw earlier with our supply-demand analysis. The market, to the extent that it works, will tend toward the equilibrium where supply = demand.  At any other price the shortage (price too low) or surplus (price too high) there will be adjustments in the price to bring us back to the equilibrium.

The Input Market

Before we leave our discussion of input markets, let's look a bit more carefully at two markets - the market for land and the market for capital.  The specifics of the markets are different, but the underlying theme is the same.  Demand is derived from profit maximizing firms that use the MR = MC decision rule as a guide to their behavior.  In the input market this led to the development of a new concept, Marginal Revenue Product (MRP), and a new variation on the equilibrium condition which equates Marginal revenue Product of the input and the price of the input [MRP_A = P_A].