Intermediate Microeconomics Module 6 Firms and their Production Decisions

FIRMS AND THEIR PRODUCTION DECISIONS 99 CHAPTER Firms and Their Production Decisions THE POLICY QUESTION IS INTRODUCING TECHNOLOGY IN THE CLASSROOM THE BEST WAY TO IMPROVE EDUCATION ?

In chapter , we covered individual and market demand . We now turn to the supply side of the the goods and services that are offered for sale in markets come from . Agood is a physical product like a candy bar or a car , while a service is a task , like an accountant doing your taxes or a dresser cutting your hair . In economics , we say that all goods and services come from firms . Firm is a term that describes any entity that produces a good or service for sale in a market . Thus , a firm can be a mammoth conglomerate like General Electric that makes everything from kitchen appliances to advanced medical equipment engines . Or a firm can be an child selling lemonade on the street corner . Our goal in this section is to understand where supply curves come from and what influences their shape and movement . We can then make reasonable predictions about how changes in production affect the price and sales of goods and services . When we understand and are able to describe supply curves , we can match them up to the demand curves and talk about markets and market outcomes in section . EXPLORING THE POLICY QUESTION Reform efforts are common in the history of public education in the United States . The Bush tion No Child Left Behind Act and the Obama administration Race to the Top initiative are examples of broad efforts to reform education . Yet despite these and many other efforts in recent decades , the United States still does relatively poorly in rankings of educational outcomes . This suggests that policy makers need to better understand how knowledge is acquired and ted in a group setting . The following are important questions that need to be answered in order to make an effective policy How important is the training of teachers , and what types of training are most effective ?

How important is caring for the health and of students , and what are the best ways of doing so ?

How important are good textbooks , and what makes a textbook good ?

Does the quality of school infrastructure matter ?

Is it important to have computers integrated into learning in the century ?

99 100 PATRICK EMERSON These are all part of what an economist would describe as the education production function the set of inputs ( teachers , students , supplies , and infrastructure ) that are combined by education systems to an output , educated children . This chapter explains production functions in economics and will give us some guidance in thinking about the education production function that we might use to guide and shape education policy . LEARNING OBJECTIVES Inputs Learning Objective Identify the four basic categories of inputs in production and give examples of each . Production Functions , Inputs , and Short and Long Runs Learning Objective Explain the concept of production functions , the difference between fixed and variable inputs , and the difference between the economic short run and long run . Production Functions and Characteristics in the Short Run Learning Objective Explain the concepts of the marginal product of labor , the total product of labor , the average product of labor , and the law of diminishing marginal returns . Production Functions in the Long Run Learning Objective Describe and illustrate generally and for , perfect complements ( fixed ) and perfect substitutes . Returns to Scale Learning Objective Define the three categories of returns to scale and describe how to identify them . Technological Change and Productivity Growth Learning Objective Discuss technological change and productivity increases and how they affect production functions . The Policy Question Is Introducing Technology in the Classroom the Best Way to Improve Education ?

Learning Objective Use a production function to model the process of learning in education . 61 INPUTS Learning Objective Identify the four basic categories of inputs in production , and give examples of each . Previously , we distilled the essence of consumers to a utility function that describes their preferences and is used to choose a consumption bundle among many possible alternatives . In this chapter , we are

FIRMS AND THEIR PRODUCTION DECISIONS 101 going to do something very similar with producers , boiling down their complexity to the very essence of their function . Another term for a producer is firm . The most basic definition of a firm is an entity that combines a set of inputs to produce a good or service for sale in a market . There are four basic categories of inputs that describe most of the potential inputs used in any production process . Labor ( This category of input encompasses physical labor as well as intellectual labor . It includes less skilled or manual labor , managerial labor , and skilled labor ( engineers , scientists , lawyers , etc . human element that goes into the production of a good or service . Capital ( This input category describes all the machines that are used in production , such as conveyor belts , robots , and computers . It also describes the buildings , such as factories , stores , and offices , and other elements of duction , such as delivery trucks . Land ( Some goods , most notably agricultural goods , need land to produce . Fields that grow crops and forests that grow trees for lumber and pulp for paper are examples ofthe land input in production . Materials ( This input category describes all the raw materials ( trees , ore , wheat , oil , etc . or intermediate products ( lumber , rolled aluminum , flour , plastic , etc . used in the production of the final good . Note that one firm final good , like , is often another firm input . Not all firms use all the inputs . For example , a child who runs a lemonade stand uses labor ( the time ting , squeezing , mixing , and selling ) capital ( the knife to cut , the juicer to juice , and the pitcher to hold the lemonade ) and materials ( lemons , water , and sugar ) This young entrepreneur does not use land , though the producer of the lemons does . The providers of services use inputs as well . An accountant , for example , might use labor , a computer ( capital ) office supplies ( materials ) and so on . PRODUCTION FUNCTIONS , INPUTS , AND SHORT AND LONG RUNS Learning Objective Explain the concept of production functions , the difference between fixed and variable inputs , and the difference between the economic short run and long run . The way inputs are combined to produce an output is called the firm technology or production process . We describe the production process with a production function a mathematical expression of the maximum output that results from a specific amount of each input . Let start with the very simple example of our lemonade stand . Suppose for each cup of lemonade the child can sell , it takes exactly one lemon , two cups of water , one tablespoon of sugar , and ten minutes

102 PATRICK EMERSON of labor , including the making and the selling of the beverage . A production function that describes this process would look something like this Cups of Lemonade ( lemons , sugar , water , labor time ) where stands for some functional form . For the moment , we are not describing the actual function but that these inputs are what go into the production of lemonade . A more generic description of a duction function would look like this ( MaN ) Similar to the way we simplified the consumer choice problem , we will generally use two input tion functions to keep the problem simple and tractable . By convention , we typically use labor ( and capital ( as the two inputs , and so the generic production function is ( When we put in actual amounts for labor and capital , this function tells us the maximum output that can be achieved with the two amounts . Producing less than that output is possible , but we will soon find that firms that are trying to maximize profits or minimize costs would never produce less than the full amount of something that they can sell for a positive price . Said another way , If a firm has a specific put in mind , it has no interest in using more than the minimum amounts of inputs . Producing a good or service for the market requires firms to make choices about the type and number of inputs to use . As we will see , this choice depends critically on the contribution the input makes to the final output relative to its cost . But a firm can only make choices about inputs that are variable inputs whose quantities can be adjusted by the firm . In shorter time periods , some inputs are not variable but fixed the firm can not adjust these quantities . Consider a firm that produces tablet computers for use in classrooms using an assembly line . A belt ( capital ) carries the computers in various stages of construction past stations where the next part is added by a worker ( labor ) The firm might decide that it wants to expand production and add a second conveyor belt , but it takes time to expand the factory , build the belt , and get it operational . In this case , capital ( the conveyor belt ) is a fixed input than can not be adjusted by the firm in a given time period . Ifthe firm wants to increase production for the time being , it will have to adjust labor , a variable input that can be adjusted by the firm in a given time period . Given enough time , this firm could add the second conveyor belt and increase production on its new level of capital . This concept is how we define short run and long run in economics the short run is a period of time In which some inputs are fixed , and the long run is a period of time long enough that all inputs can be adjusted . It is important to note that these are not defined by any objective period of time ( day , month , year , etc . but are specific to the particular firm and its particular inputs . Suppose our tablet computer manufacturer needs three months to install a second assembly is , its short run is less than three months , and its long run is three months or more . For our lemonade salesperson , it might take an trip to the store to buy a or pitcher . In this case , the short run is less than an hour , and the long run is an hour or more . PRODUCTION FUNCTIONS AND CHARACTERISTICS IN THE SHORT RUN Learning Objective Explain the concepts of the marginal product of labor , the total product of labor , the average product of labor , and the law of diminishing marginal returns . The short run , as wasjust described , is a period short enough that at least one input is fixed . We will low the convention that the capital ( input is fixed in the short run and labor is variable . This tion is not always true , but labor hours often can be adjusted example , by having workers

FIRMS AND THEIR PRODUCTION DECISIONS 103 put in a little capital inputs generally take some time to adjust . To describe the short run using our production function , we write it like this ( where is the quantity of the output , is the quantity of the labor ( generally measured in labor hours ) and is the quantity of capital ( generally measured in capital hours ) The bar over the capital variable indicates that it is fixed Let go back to our computer manufacturer . In the short run , the firm can not add capital in the form of a new assembly line . But it can add workers . It could , for example , add a second shift to production , using the same assembly line through the night . Or if already running hours , the firm could add more workers to the same assembly line and squeeze more production out of it . The relationship between the amount of the variable input used and the amount of output produced ( given a level of the fixed input ) is the total product , or in this case the total product of labor ( as labor is the variable input . Table summarizes this information for our tablet computer manufacturer along with some additional information , which we will discuss . Table Product and Product Output , total . 112 ' i ' the tablets tablets assembled with number of tablets assembly line ) assembled ) addition ofone worker to the line ) assembled per worker ) 20 14 10 44 24 50 15 15 70 10 14 78 13 12 11 86 10 so As the number of workers ( labor ) increases incrementally by one in table , the output also increases . Note that at first , increases in production are pretty big . This is due to the efficiencies rom increased made possible by more workers . After a while , the increases in output from increased labor get smaller and smaller as the assembly line gets crowded and the additional workers have difficulty being as productive . The maximum number of workers is reached at ten due to the firm being unable to fit more workers on the assembly line . The extra contribution to output of each worker is critically important to the firm decision about how many workers to employ . The extra output achieved from the addition of a single unit of labor is the marginal product of labor ( In our case , the extra unit is an additional worker , but in general , we might measure units as worker hours . is a key measure listed in the fifth column of table . Formally , it is

104 PATRICK EMERSON Table shows that this marginal product increases at first as we add more workers , peaks at five workers , but then starts to decline and even goes negative as we reach ten workers . Also important to keep track of is the average product of labor ( or how Calculus much output per worker is being at each level of employment Note that we use the partial derivative here even though it is a I function inthe short run , because in general , bivariate function . In table , we see that the average product of labor increases at first , peaks at five workers , but then starts to decline . These measures of labor productivity can also be represented as curves . We will draw them as smooth continuous curves because we can hire fractional workers in the sense that we could hire one employee and another who works 60 percent of the time , and so on . In figure , we represent the total product of labor curve in the upper panel and the marginal and average product of labor curves in the lower panel .

FIRMS AND THEIR PRODUCTION DECISIONS 105 ( Li ) Total Product 88 ' of Labor 60 44 24 Labor , 15 , Labor , Figure curve and average and curves The shape of the total product of labor curve confirms that , at first , adding workers allows more and more output per worker . This can be seen in the initial part of the curve where the slope is getting steeper . Then given the fact that the capital is fixed , adding more workers continues to increase output but at a decreasing rate . We see this past point A on the graph , where the slope is still positive but decreasing . Finally , more workers actually cause output to fall . From the total product of labor graph , we can actually measure the average and marginal product of labor . The average product of labor is the same as the slope of the ray from the origin that intersects the total product curve ( note that the slopes is ) This reaches its highest slope right at point , where the ray is just touching the total product curve . The marginal product of labor curve is the change in output over the change in labor , which is the same as the slope of the total product curve itself . Notice that the maximum slope is right at point A so that the average and the marginal are exactly the same at point and that the slope turns negative at point point of maximum output . These observations can help us draw the average and marginal product curves in the lower panel . We know marginal product reaches a maximum at A , crosses the average product curve at , and turns negative

106 PATRICK EMERSON at We also know that when the marginal product is above the average , the average is increasing , and when the marginal product is below the average , the average is decreasing . Thus the two sets of curves relate directly to each other . There is one more important characteristic of production to note before we move on the law of diminishing marginal returns . This law states that ifa firm increases one input while holding all others constant , the marginal product ofthe input will start to get smaller , just like in our example at three labor units . It is important to remember that we are talking about the marginal returns the output will be positive but smaller and smaller . We can also talk about diminishing happens to total output as labor is increased , which in our case does eventually diminish ( at nine labor units ) but this is not a general rule . 64 PRODUCTION FUNCTIONS IN THE LONG RUN Learning Objective Describe and illustrate generally and for , perfect ( fixed proportions ) and perfect substitutes . In the long run , all production inputs are variable . Because of this , firms have more flexibility in how they choose to produce , expand , and contract their output in the long run than in the short run . But this situation also complicates firms . They have to weigh the marginal contribution to put an extra unit of each input would provide and the cost of that extra unit . We will now focus on this decision process . Since we have simplified our firm to one that uses only two inputs , capital ( and labor ( we have a choice problem that shares many similarities with the consumers problem we studied in section . It should come as no surprise then that the way we solve the problem is similar as well . Let start with the production function . Note that there is no bar above the , meaning that both labor and capital are now variable . This makes the function mathematically similar to the utility function from chapter . The graph of these two variable functions is . Table Output from the use of two variable inputs , capital ( and labor ( Labor ( From table , we can see the diminishing marginal productivity of one variable input . Holding one input example , holding capital at can see that the increase in the total output falls as we increase labor . The extra output from going from one unit of labor to two is 130 ( 380 250 ) from two to three is 120 , from three to four is 60 , and from four to five is 40 . Like indifference curves for utility functions , we can visualize this output function in two dimensions by drawing contour lines lines that connect all combinations of labor and capital that lead to an identical output . To draw one , we can start by arbitrarily choosing an output level and then finding all the nations of labor and capital the firm could possibly use to produce this quantity

FIRMS AND THEIR PRODUCTION DECISIONS 107 ( Drawing this on a graph with capital on the vertical axis and labor on the horizontal axis produces an a curve that shows all the possible combinations of inputs that produce the same output . means the same and means quantity . So an is a curve that for every point on it , the output quantity is the same . Figure shows three for table , corresponding to the outputs 180 , 300 , and 500 . In the figure , the combinations of inputs that yield the same output are connected by a smooth curve . We draw them as smooth curves to illustrate that firms can use any fraction of a unit of input they desire . Capital , Labor , Figure Selected from table You can , of course , do this for any output level you choose so the entire space is filled with potential figure simply sketches three . Note that we do not have a specific functional form for the production function , so we are drawing hypothetical . We will derive an from a specific production function momentarily . Properties of Just like indifference curves , have properties that we can characterize . that output are further from the origin . Quite simply , if you add more of both inputs , you get more output . Stated the other way , if you want more output , you have to increase the inputs . do not cross each other .

108 PATRICK EMERSON To understand this , try a thought experiment . If two crossed , it would mean that at the crossing point , the same inputs would yield two ent and distinct output levels . Under the assumption that firms make efficient decisions , they would never produce a lower output . are downward sloping . This is the same as saying there are with inputs you can replace some capital with labor and produce the same amount of output and vice versa . Really this requirement is only a prohibition on quants , which would imply that you could cut back on both inputs and produce the same output . do not bow out . This is a similar property to the one for indifference curves , which are bowed in . Note that we do not say that can only be bowed in because complements and substitutes play a big role in production functions . But that curve in represent the idea that it is often more efficient to have a mix of labor and capital working together than too much of one or the other ( this is quite similar to the motive in consumption ) Three Types of Production Functions and Their Related As with preferences , there are three basic types of production functions . Though they are essentially the same as the three indifference curves that represent the three basic types of preferences , it is good to keep in mind that the context is very different . Depending on the production process , inputs into the duction process could be easily substitutable , might need to be used in fixed proportions , or might be mixed together but have flexibility in terms of proportions . Perfect Substitute Production Functions Some production processes can substitute one input for another in a fixed ratio . For example , our child might have an electric juicer that will transform whole lemons into lemonade without any human input . This child might also be able to make lemonade entirely by hand . Suppose the juicer can always make one gallon of lemonade in half an hour and the child can always make one gallon in one hour . Thus the production function for the quantity of lemonade ( given labor ( and capital ( looks like this To see this , think in terms ofan producing exactly one gallon of lemonade ( requires either one hour of labor ( or hour of capital , the juicer ( You can also mix example , you could use hour of labor and hour of capital . To produce two gallons ( you could use

FIRMS AND THEIR PRODUCTION DECISIONS 109 two hours of labor or one hour of capital . You could also mix and use one hour of labor and hour of capital or any linear combination of the two inputs . for a perfect substitute production function are therefore straight lines . Figure presents three for a production function , 234 Figure for a production function with inputs that are perfect substitutes Perfect substitute production functions generally have the form aL , where and are positive constants . A more general form of this production function that rates a measure of overall productivity is this A ( aL ) where A is also a positive constant . Careful observers will notice that the A is unnecessary because it can be incorporated into a and but expressing the production function this way provides an easy

110 PATRICK EMERSON method of adjusting for ( and empirically estimating ) productivity . We will examine A and productivity in more detail in section . Perfect Complement ( Production Functions If inputs have to be used in fixed proportions , then we have a production function where the inputs are perfect complements , also known as . Imagine a business where the business takes metal sheets from its clients and stamps them into shapes , such as auto body parts . Suppose the stamping machines require exactly one operator . Without a operator , the machine will not work . Without a machine , a worker can not stamp the metal sheet . Thus the inputs are perfect have to be applied in fixed proportions to get any extra output . Suppose also that one machine and one worker can produce one hundred stamped panels in a day . A production function that describes the daily output ofthe business looks like this 100 , where the min function simply takes the smaller of the two values of the arguments and For example , if the business has ten workers ( and eight machines ( the value of the function , would be eight times one hundred , or eight hundred . Similarly , if the business has twelve machines but only seven workers , the daily output would be seven hundred . We can draw the for this firm starting with equal inputs . At point A , there are five machines and five workers , and the daily output is five hundred . Now suppose we move to point , where there are still five machines and seven workers . Does this yield any more output ?

No . So this point is on the same indifference curve . This is similar to point , where there are seven machines and five workers . Perfect complement production functions have the general functional form of A min , where A , oz , and are positive constants . As in the case of perfect substitutes , the A is unnecessary because it can be incorporated into at and but we will keep it for simplicity and clarity . Figure presents three for the production function min ,

FIRMS AND THEIR PRODUCTION DECISIONS 111 , I I I I I I I I I I Figure for a production function with inputs that are perfect complements Production Functions Perfect substitute and production functions are special cases ofa more general tion function that describes inputs as imperfect substitutes for each other . In other words , we can get rid of some machines ( capital ) in exchange for more workers ( labor ) but at a ratio that changes depending on the current mix of workers and machines . The function that we used to describe choice with the preference for variety assumption is used forjust such a production process . An example ofa production function is , where A , 04 , and are positive constants . Notice that A represents overall productivity ( as A increases , the same number of inputs yields more output ) and the and parameters represent the contribution to the output of each input . In other words , the higher the level of , the more one unit of capital will produce in extra output , and likewise for labor . The essential aspect of this production function is always having the ability to substitute one input for the other and maintain the same level of output .

112 PATRICK EMERSON From this production function , we can study the of inputs into production . In order to do this , we need to define the marginal rate substitution , which is similar to the marginal rate of substitution from consumer theory . The marginal rate of technical substitution ( describes how much you must increase one input if you decrease the other input by one unit in order to produce the same output . The is simply the ratio of the marginal product of labor to the marginal product of capital To see where this ratio comes from , rememberthat the marginal product ofan input tells us how much extra output will result from a increase in the input . Earlier we defined the marginal product of labor as Similarly for capital , we have Note that we can rearrange the marginal product of labor and express it as So ifthe is four and the firm increases labor by one unit , then the extra output is four . Similarly , if the is two and the firm decreases capital by one unit , then the decline in output is two , or AK . Notice that in order to stay on the same , the net change in output has to be zero , or ( AK ) Rearranging the steps , we get the following ( AK ) which we can simplify to give us AK AL Note that the describes the slope ofthe many units of capital you would need to add ( rise ) ifthe labor decreases by one unit ( run ) to maintain the same output ( stay on the )

FIRMS AND THEIR PRODUCTION DECISIONS 113 Capital , 16 ' 16 Labor , Figure with rate of technical substitution ( Now let return to the utility function , and Calculus derive the . Since the parameters or and are positive constants that Since i and ( and describe the contribution of each input to the final output , they are related to the marginal products . It turns out that the , marginal product of labor is just times the ave ) then , And the marginal product of capital is just times the marginal product of Thus for the , we have a

114 PATRICK EMERSON Notice the absence ofA in this ratio . Since we are talking about marginal returns to inputs and A only affects total return , it does not appear in this calculation . Let look at an example ofa specific duction function . Suppose a firm Calculus duction function is estimated to be , ac , 37 and In this case , and . Thus the for this production Similarly , AK ( On ' 57 AK ( Li a Thus , Bi A ( aKa ) ax . So the slope of this firm depends on the specific mix of capital ( and labor ( but is ( Note that when is relatively large and is relatively small ( as in point , 16 in figure ) the slope is negative and steep . Alternatively , when is relatively small and is relatively large , the slope is negative and shallow . The slope is also always changing along the , and thus we have the shape ( concave to the origin ) typical to the production function . RETURNS TO SCALE Learning Objective Define the three categories of returns to scale , and describe how to identify them . Up to now , our focus on marginal returns has centered on the change of one input , holding the other input constant . In this section , we consider what happens to output when we increase or decrease both inputs . Increasing or decreasing both inputs can be described as scaling the firm up or down , and we want to know how it affects output . In particular , we would like to know about proportional changes to output . For example , if the firm doubles both inputs , does output double as well ?

Does it increase more or less than double the previous amount ?

The answer to these questions determines the returns to scale of the firm the rate at which the output increases when all inputs are increased proportionally . Constant Returns to Scale ( If a firm output changes in exact proportion to changes in the inputs , we say the firm exhibits constant returns to scale ( Let consider this situation using the doubling example . Suppose a firm has a production function of ( Now the firm doubles its inputs . We can show this by multiplying the current inputs by two ( If this yields exactly double the output , we can write this expression as (

FIRMS AND THEIR PRODUCTION DECISIONS 115 This expression says that if both inputs double , output doubles . It is an example of a production function . The condition can be expressed more generally as ( I , where ) is a strictly positive constant , like two in our example . In the case of scaling down , would be a fraction , such as . Increasing Returns to Scale ( IRS ) Increasing returns to scale ( IRS ) production functions are those for which the output increases or decreases by a greater percentage than the percentage increase or decrease in inputs . In the case of bling inputs , the IRS expression would look like this ( 262 Look carefully at what this says . The first term on the left is the output that results from the doubling of the two inputs . We compare this to the doubling of the output , represented by the two expressions on the right of the inequality sign . This expression says that you get more output from doubling the inputs than you get from doubling the that if you double the inputs , you get more than double the output . In general terms , again where ( I ) is a strictly positive constant . Decreasing Returns to Scale ( Finally , decreasing returns to scale ( production functions are those for which the output increases or decreases by a smaller percentage than the increase or decrease in inputs . In the case of doubling the inputs , the expression would look like this ( This expression says that you get less output from doubling the inputs than you get from doubling the that ifyou double the inputs , you get less than double the output . In general terms , again where ( I ) is a strictly positive constant . Varying Returns to Scale Some production processes have varying returns to scale . The most common is where there are ing returns to scale for small output levels , constant returns for medium output levels , and decreasing for very large output levels . Returns to Scale and Production Functions Let think about the three types of production functions we have substitutes , perfect complements , and their returns to scale . The question we want to answer can be expressed as , If the amounts of inputs are doubled , by how much does the output increase ?

The answer will tell us if the production function exhibits constant , increasing , or decreasing returns to scale .

116 PATRICK EMERSON Perfect Substitutes Recall the expression for perfect substitutes aL Here we double both inputs , so we replace and with and By simple algebra , we can factor out the common two and come up with ( aL ) By the above equation , we know that the term in the parentheses is equal to So we know that by doubling both and , we exactly double or we have constant returns to scale . This means that production functions for perfect substitutes are . Perfect Complements The perfect complement function is A min oiL , Again , we want to begin by replacing and with and and note that since we are doubling both inputs , the minimum of the two numbers must also be doubled so we can pull the two out in front ofthe min , Since the order of multiplication does not matter , we know , which is the same as doubling output min , The first step is not immediately obvious , but think about the way the min function works itjust picks the smaller of the two numbers in the square brackets . Doubling them both does not change which one is smaller , and in the end , the value is just double the smaller one , so it is the same as doubling after we choose the min of the two numbers . From there , since order does matter in multiplication , we can move the two out in front ofthe A , and now we have two multiplied by the original production functions ( in parentheses ) which is the same as Again , we know that by doubling both and , we exactly double or we have constant returns to scale . Thus production functions for perfect complements are also . Recall the function AL We will start the same way , replacing and with and A ( You have to be careful here because the exponents and are on and , respectively , and when we replace with , for example , the is now the exponent for . By the rules of exponents , we can separate the and the and Next , we can rearrange terms at will , as the order of operations does not matter in multiplication

FIRMS AND THEIR PRODUCTION DECISIONS 117 ' We can also , by the rules of exponents , express 20 25 as 20 20 ( AL ) Now we have an expression with and can evaluate the returns to scale . We do so by considering the three possibilities for the exponent a . Any number to the power of one is itself , so 21 . So in this case , we have exactly double the output , and the production function is . In this case , we have some number less than two multiplied by and so we will have something less than double the output . That is , the production tion exhibits decreasing returns to scale ( In this case , is multiplied by a number larger than two , so the output is more than double . That is , the production function exhibits increasing returns to scale ( IRS ) So for production functions , the returns to scale depend on the sum of the exponents if they equal one , they are ifthey are less than one , they are and if they are more than one , they are IRS . Returns to Scale and We can see returns to scale in by examining how much they increase with Input increases for a particular production function . For example , in figure ( a ) when we double inputs from ten to twenty , we double output from one hundred to two hundred . Since output increases at the same rate as inputs , we know it is . The in figure ( illustrate increasing returns to scale . When we double Inputs from 10 to 20 , we Increase output by more than double the original quantity , from 100 to 235 . The in figure ( illustrate decreasing returns to scale . When we double Inputs from 10 to 20 , we Increase output by less than double the original quantity , from 100 to .

118 PATRICK EMERSON a ) Constant returns to scale ) Increasing returns to scale ) Decreasing returns to scale 150 150 150 20 10 10 20 Figure and returns to scale TECHNOLOGICAL CHANGE AND PRODUCTIVITY GROWTH Learning Objective Discuss technological change and productivity increases and how they affect production functions . A firm production function need not be static over time . Many firms adopt new technologies or new ways of organizing and doing things in order to become more productive . Technological change refers to new production technology or knowledge that changes firms production functions so that more put is produced by the same amount of inputs . Technological change is also known as growth . We can express changes in overall productivity with production functions . So far we have assumed that firms are , meaning that they do not waste use the minimum of inputs necessary to produce a particular output level . You can see this by the way we assume that certain inputs yield amounts of output . There is no waste in this scenario . Productivity , on the other hand , refers to how much overall output a firm gets from a set amount of inputs . The more a firm produces with the same inputs , the more productive it is . As was noted in section , the constant A is a measure of overall productivity in the types of duction functions we have been studying . All else equal , increasing A increases the output that a firm produces from a set amount of inputs . Graphically adjusting A in any of our three production function types our . As figure shows , when A is increased from one to three in the case ofa simple production tion , all the outputs associated with the individual increase by three as well .

FIRMS AND THEIR PRODUCTION DECISIONS 119 Capital , Labor , Figure with outputs increasing We measure productivity increases by comparing output before and after the productivity change . In the case above , our production function ( changed to ( The output of the firm thus increased by or ( or two . In percentage terms , output increased by 200 percent . THE POLICY EXAMPLE IS INTRODUCING TECHNOLOGY IN THE CLASSROOM THE BEST WAY TO IMPROVE EDUCATION ?

Learning Objective Use a production function to model the process of learning in education . Let return now to the policy question introduced earlier . One way to think of education is similar to a firm you take a mix of , teachers , books , buildings , pedagogy , and mix them together and get an output . The output in this case is the education much the students

120 PATRICK EMERSON learn . If learning is the output and we can identify the inputs , then it is possible to think of education as a type of production that is governed by a production function . Modeling education as a production function is mechanical and has limitations for sure . The true test ofthis endeavor is whether we can gain any insight into how student learning is accomplished . We would have to start by coming up with a plausible production function . By far the most perhaps the most relevant to the production function , where inputs are imperfectly substitutable . For example , you do get any learning without a teacher , but you might be able to substitute some instruction for some live teaching . This is the essence ofthe production function . We can express an education production function , apply it to data , and estimate its parameters to find out , for example , how important good teachers are and how substitutable they could be using technology . Doing so will give policy makers a better idea of the most effective ways to spend scarce resources ifthe ultimate goal of education funding is learning . For example , suppose we posit that learning is a function of teacher quality , the technology used in the classroom , the quality of the class materials , and the size of the class ( number of students per teacher ) So we imagine that the learning production function looks something like this learning Teacher Class Size With the appropriate data , the parameters alpha , beta , gamma , and delta can be measured , giving icy makers an idea about where the highest return on investment is found . In fact , economists have been doing this for quite some time . See Measuring Investment in Education . EXPLORING THE POLICY QUESTION . What do you think are the relevant inputs into the learning production function ?

Which type of production function best represents the grade school learning environment ?

If you believe that teachers are easily replaced by online videos , where one hour of online video is exactly equivalent to one hour of live teaching , what type of production function would you use to describe this ?

If there were a magic pill that students could take and that would , regardless of the current level and mix of inputs , double their learning , how would you express this in a production function ?

REVIEW TOPICS AND RELATED LEARNING OUTCOMES Inputs Learning Objective Identify the four basic categories of inputs in production and give examples of each . Production Functions , Inputs , and Short and Long Runs Learning Objective Explain the concept of production functions , the difference between fixed and variable inputs , and the difference between the economic short run and long run .

FIRMS AND THEIR PRODUCTION DECISIONS 121 Production Functions and Characteristics in the Short Run Learning Objective Explain the concepts of the marginal product of labor , the total product of labor , the average product of labor , and the law of diminishing marginal returns . 64 Production Functions in the Long Run Learning Objective Describe and illustrate generally and for , perfect complements ( and perfect substitutes . Returns to Scale Learning Objective Define the three categories of returns to scale and describe how to identify them . Technological Change and Productivity Growth Learning Objective Discuss technological change and productivity increases and how they affect production functions . 67 The Policy Question Is Introducing Technology in the Classroom the Best Way to Improve Education ?

Learning Objective Use a production function to model the process of learning in education . LEARN KEY TOPICS Terms Labor ( Input that encompasses physical labor as well as intellectual labor . Includes less skilled or manual labor , managerial labor , and skilled labor ( scientists , lawyers , etc , ie , the human element that goes into the production of a good or service . Capital ( Input category which describes all the machines that are used in production conveyor belts , robots , and computers . It also describes the ings , such as factories , stores , and other elements of production , such as delivery trucks . Land ( Input category that encompasses land used for production , that grow crops and forests that grow trees for lumber and pulp for paper . Materials ( All the raw materials ( trees , ore , wheat , oil , etc . or intermediate products ( lumber , rolled aluminum , plastic , etc . used in the production of the good . Production function A mathematical expression of the maximum output that results from a amount of each input . Variable input An input that can be adjusted by the in a given time period .

122 PATRICK EMERSON Fixed input An input that can not be adjusted by the in a given time period . Short run A period of time in which some inputs are not by any objective period of time ( day , month , year , etc . but are to the and its particular inputs . Long run A period of time long enough that all inputs can be adjusted not by any objective period of time ( day , month , year , etc . but are to the particular and its particular inputs . Total product of labor ( The relationship between the amount of the variable input used and the amount of output produced ( given a level of the input ) Average product of labor ( How much output per worker is being produced at each level of employment . Marginal product of labor ( The extra output achieved from the addition of a single unit of labor . Law of diminishing marginal returns If a increases one input while holding all others constant , the marginal product of the input will start to get smaller . It will still remain tive , just smaller and smaller . From iso meaning same and quant meaning quantity . A curve that shows all the possible combinations of inputs that produce the same output . production functions An example AL A production function that describes a ratio that changes depending on the current mix of workers ( labor ) and capital ( machines , etc ) A variant of the function that describes consumer choice with the preference for variety assumption . Perfect substitute production functions A production function used when two means of production can substitute for one another in a ratio . Generally have the form where and are positive constants Perfect complement ( production functions A production function where the inputs are perfect complements A min , where A , and are positive constants Marginal rate of technical substitution ( A ration of the marginal product of labor ( and the marginal product of capital ( Returns to scale

FIRMS AND THEIR PRODUCTION DECISIONS 123 The rate at which the output increases when all inputs are increased proportionally . Constant returns to scale ( Exhibited when a firm output changes in exact proportion to changes in the inputs ( I , where is a strictly positive constant . becomes fractional in the case of scaling down . Increasing returns to scale ( IRS ) When the output increases or decreases by a greater percentage than the percentage increase or decrease in inputs ( I , I ) I ( I where is a strictly positive constant . Decreasing returns to scale ( When the outputs increases or decreases by a smaller percentage than the percentage increase or decrease in inputs . I , I ) I ( I where is a strictly positive constant . Graphs total product curve and average and marginal product curves

124 PATRICK EMERSON slope 670 15 ( Total Product 88 of , ab ( Labor , Labor , Figure , I ( and average and marginal ( Examples of Capital , I , Labor , Figure from table , examples for a production function with inputs that are perfect complements FIRMS AND THEIR PRODUCTION DECISIONS 125 , Figure a production function with inputs that are perfect for a production function with inputs that are perfect substitutes 126 PATRICK EMERSON Figure for a that are with FIRMS AND THEIR PRODUCTION DECISIONS 127 Capital , 16 I . 16 Labor , Figure marginal rate ( and returns to scale a ) Constant returns to scale 13 ) Increasing returns to scale ) Decreasing returns to scale Figure and returns to ( a ) constant returns ( returns ) de ( returns ) with outputs increasing

128 Capital , PATRICK EMERSON Figure with outputs Equations Average product of labor ( A Marginal product of labor ( AL Labor , The partial derivative is used when working with , regardless if it is in presentation . In general , it is a bivariate function . Marginal rate of technical substitution (