Chapter 7: Production Theory Production Theory attempts to provide a framework for studying the production of goods and services within a firm. It is the first theory that will be studied within the more encompassing Theory of the Firm. Additionally, Production Theory shares a close relationship with both Cost Minimization Theory and Profit Maximization Theory, which will be presented in later chapters. One additional note to make about Production Theory is that it must be studied in both the long run and the short run. The Production Function Much like the Laws of Supply and Demand, Production Theory has its own function. It is known as the Production Function. The Production Function is simply a mathematical equation which links the inputs of production into the eventual output (Q). Following is an example of a production function where the inputs are capital (K), labor (L), raw materials (RM), and technology (T): Figure 7.1: Complex Production Function Q=fK,L,RMT This type of production function is known as a complex production function. Any production function with three or more inputs is considered complex. Conversely, any production function with two or less inputs is considered a simple production function. Complex production functions require three or more dimensions to represent graphically, so we re going to stick with analyzing a simple production function: Figure 7.2: Simple Production Function Q=fK,LRM,T Short Run vs. Long Run The production process in any firm can occur in either the short run or the long run. The short run is simply the period of time when production occurs with at least one fixed input. A fixed input refers to any input that remains constant, while the others may be either fixed or variable. In our analysis, capital (K) will be the fixed input while labor (L) will be the variable input. A variable input may change at any time in the production process. Production in the long run occurs with all variable inputs. Hence, the long run production process has no fixed (constant) inputs. In our case, both capital (K) and labor (L) will be variable. Production in the Short Run We will begin by analyzing production in the short term. The first important fact to note is that all production processes within the short run are subject to the Law of Diminishing Returns. The Law of Diminishing Returns states that when a variable input (labor) is added to a fixed input (capital), initially output (Q) will increase at an increasing rate, then output will increase at a decreasing rate, and eventually output will decrease. As an example, consider a car factory with a set amount of machinery, which will act as the fixed input (capital). If more and more workers are added (labor), they will eventually overcrowd and there won t be enough machinery for production or room for them to walk. Let us now consider an apple orchard s production in the short run where the amount of land (capital) is fixed and the amount of workers (labor) is variable. We are going to introduce three new terms in order to work with this example. The first is total production (TP), which is simply the output. The second is the marginal product of labor (MPL), which is the additional amount of output after each new worker is added and is calculated using TP L. Finally, there is the average product of labor (APL), which is the average amount of output generated by each worker and is calculated using TPL. Table 7.1: Production Process of an Apple Orchard in the Short Run Fixed Input (K) 1 acre 1 1 1 1 1 1 1 1 1 Variable Input (L) 0 workers 1 2 3 4 5 6 7 8 9 Output (TP) 0 bushels 1000 2700 5000 7000 8300 9000 9300 9300 9000 MPL -1000 bushels 1700 2300 2000 1300 700 300 0 -300 APL -1000 bushels 1350 1667 1750 1660 1500 1328 1162 1000 The first thing to note about this chart is that the marginal product of labor is first increasing then begins decreasing. When it reaches its maximum, this is the exact point at which diminishing returns begins setting in. The second thing to notice is that the marginal product of labor specifies exactly how much each additional worker contributed to the total output. Now, let s examine these values graphically: Figure 7.3: Production of Apple Orchard Graphs Total Product Curve There are two very important properties about the total product curve which relate to both the marginal product of labor and the average product of labor. First, remember that the marginal product of labor is simply the change in total product over the change in labor ( TP L). Therefore, the value of MPL at any value of TP is simply the slope of the tangent line at that point. Additionally, remember that the rate of change of the TP curve is constantly increasing until diminishing returns sets in; therefore, the maximum value of MPL is the slope of the tangent line at the point where diminishing returns sets in. This point is denoted by a square marker on the total product curve. The next important property involving the total product curve involves the average product of labor. Remember that the average product of labor is simply the total product over the labor (TPL). Graphically, this is simply a line whose slope is total product over labor, starting at the origin. Therefore, the value of APL at any point on a TP curve is simply the slope of the line both tangent to that point and intersecting the origin. Also, notice that the slope of this line would be highest at the maximum value of TP. Therefore, the maximum value of APL is simply the slope of the line which intersects both the origin and the maximum TP point. This is denoted by a triangular marker on the total product curve. Recall that curves have both an average rate of change and an instantaneous rate of change. The marginal product of labor is simply the instantaneous rate of change of total product at any point. On the other hand, the average product of labor is simply the average rate of change of the total product starting from the origin and ending at any point. Marginal Product of Labor and Average Product of Labor Curves First of all, notice that the average product of labor and the marginal product of labor equal each other when the amount of labor is equal to one. This will always be true, as there is only one value to be considered for the average product. The next important point to take note of is the top of the red marginal product curve, where it reaches its maximum. It is at this exact point that diminishing returns begins setting in. Now, notice that the average product and the marginal product equal each other once more at the point where they intersect. Also, this is the point where the average product of labor reaches its maximum and begins falling. This is because the marginal product begins dragging it down. Finally, the last important point is where the marginal product of labor reaches zero. This is also the point on the total product curve where it reaches its maximum. Recall that the marginal product of labor is simply the first derivative of the total product of labor. Therefore, when marginal product crosses the axis the total product curve reaches an extreme. One final observation to note about these curves is that the blue average product of labor curve measures the average of the red marginal product of labor curve, which explains why it behaves as it does. The Three Stages of Short Run Production With some further analysis of our marginal/average product of labor graph, we may break down our short run production into three further stages. These will be known as the first, second, and third stages of short run production respectively. We can define the First Stage of Short Run Production as the time from the vertical axis (L=0) until the time when APL reaches its maximum. The Second Stage of Short Run Production will be the time from when APL reaches its maximum until MPL reaches zero. Finally, the Third Stage of Short Run Production will be from when MPL becomes negative and beyond. Now, we must answer the question: which stage is the optimal stage of production in the short run? Right away, we should be able to rule out the third stage as each additional worker begins contributing negative amounts to the total product. In this case, we say that the variable input (L) is over-utilized. Additionally, we can rule out the first stage since it is missing out on some contribution to total product as MPL is still positive after that point. We will say that the variable input (L) is under-utilized in this stage. Therefore, we must draw the conclusion that the Second Stage of Short Run production is the optimum, since it is neither underutilized nor over-utilized. Optimum Amount of Workers Now that we know that the second stage is the optimum stage, we must find out exactly where in the second stage is total production most optimal. In the case of our apple orchard, we are asking: exactly how many workers between 4 and 8 should be hired? In order to answer this question, we need two pieces of information. The first is the price at which a bushel of apples sells in the market, which is known as the output price (PQ). Also, we need to know what the wage rate is which is paid to the apple pickers. So, to answer this question, we are going to construct the supply and demand curves for labor. Both of these curves will be derived from data that we already have involving production. <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of Labor and Demand for Labor We are going to introduce a new concept in order to proceed with our calculations. This new concept is known as the <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of labor (MRPL). The MRPL is simply the product of the output price and the marginal product of labor. It tells us the amount of additional revenue that each worker who is hired contributes. Note that we are not considering the wage rate of each worker in this calculation; we are simply considering a firm s demand for workers based on their generated revenues. We will consider the wage paid to each worker when talking about the supply curve of workers, which will be explained in the next section. Table 7.2: <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of Labor MPL 1000 1700 2300 2000 1300 700 300 0 -300 Output Price (PQ) $15 $15 $15 $15 $15 $15 $15 $15 $15 MRPL $15,000 $25,500 $34,000 $30,000 $19,500 $10,500 $4,500 $0 -$4,500 From this table, we can make several observations about the <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of labor. First of all, we know that as the marginal product of labor or the output price increases, the <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of labor increases as well, and vice versa. We can also draw another conclusion, this time involving the demand for labor (DL). As the <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of labor increases, the demand for labor increases, and vice versa. Now, let s have a look at the graph of MRPL: Figure 7.4: <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of Labor Graph Notice that this graph will always have exactly the same shape as that of the marginal product of labor. Additionally, notice that the demand for labor increases as the <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of labor increases, and decreases as the <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of labor decreases. In fact, the graph of demand for labor (DL) corresponds exactly with the <a href="/keyword/marginal-revenue-product/" >marginal revenue product</a> of labor (MRPL) graph. They are one in the same. Therefore, the above graph actually is the demand curve for labor. Wage Rate and Supply of Labor Now that we have derived the demand curve for labor, we must do the same for the supply curve. In order to do so, we will need our second piece of information: the annual wage rate. For our example, we will say that the annual wage rate of apple orchard pickers is $10,500. Like every other variable we ve worked with so far, we shall graph it: Figure 7.5: Annual Wage Rate Notice that no matter how much labor our firm has, the annual wage rate of apple pickers will always remain constant. Since there will always be people willing to work for this wage rate as apple pickers, then we can say that this is the same as the supply curve for apple pickers. Equilibrium of Apple Pickers and Optimum Short Run Production Finally, we can combine our derived supply and demand curves of labor together in order to see where our equilibrium lies: Figure 7.6: Equilibrium of Apple Pickers By looking at a graph such as this, a firm can decide the amount of workers it should hire. In this case, we can see that at six workers, the demand and supply of labor rests at equilibrium. Remember that market equilibrium is the point at which no surpluses or shortages exist. Therefore, in the second stage of short run production, the point at which an optimum production is reached is the point at which the supply and demand of labor is at equilibrium. As a result, the optimum number of workers to hire for our particular example is six. Production in the Long Run Now that production in the short run has been thoroughly examined, we must turn our attention to the other half of Production Theory: production in the long run. Remember, the difference between short and long run production is that long run production has no fixed inputs. As a result, this would make analyzing production in the long run difficult, if it were not for a useful tool developed by economists. This tool is known as a production grid, which shows the maximum amounts of output that can be produced with different combinations of capital and labor. Before we jump into the details, let s take a look at the production grid for our apple orchard: Table 7.3: Production Grid for the Apple Orchard Capital (K) X X X X X X X 5 4 3 2 1 0 X X X X 280 1 X X 530 X X 2 X 650 X 530 X 3 730 X 650 X X 4 X 730 X X X 5 X X X X X 6 X X X X X Labor (L) The first important characteristic of production grids is that each cell in the grid represents a short run production process. Additionally, you should be able to see that the same amount of output can be achieved with different amounts of capital and labor. For example, 3 capital and 2 labor produces the same amount of output as 2 capital and 3 labor: 530 bushels. This simple yet important characteristic allows us to represent a production grid on a graph. Production grids may be graphed in capital-labor space. A line in a production grid represents a specific level of output. These lines are known as isoquonts. Here is a visual representation of our production grid: Figure 7.7: Isoquonts The three isoquont series listed here are 530, 650, and 730 respectively. They conveniently show us all the different combinations of capital and labor which can be used to create different levels of output. There are three important characteristics about isoquonts to remember. The first is that the further to the topright an isoquont lies, the higher its output will be. The second is that isoquonts never intersect, as this would violate the Law of Transitivity. This simply means that any combination of capital and labor may only have one output value. Two isoquonts intersecting each other would cause there to be two different output values for a single combination. The third property of isoquonts is that there are infinitely many on any graph, even though not all of them are depicted. How Firms Use Isoquonts Now that we know all about isoquonts, we must answer the question: how does a firm actually use an isoquont to determine the level of production in the long run? The first step in determining the level of production in the long run is for a firm to analyze the market in order to determine its optimum level of production. This work is usually done by an economist. Next, the firm needs to figure out which combination of capital and labor is sufficient to achieve the chosen level of output. Firms always choose whichever combination costs the least amount of money. In order to figure out which combination of capital and labor costs the least; we must introduce a new concept: the isocost line. Figure 7.8: Isocost Line The isocost lineis simply a negatively-sloped line (as in the graph above) whose slope is PLKL. This line tells a firm how much capital and labor it can have given a budget constraint. Firms will always use the combination of capital and labor where the isocost line intersects the isoquont line in order to achieve an optimum price combination of capital and labor. Notice that there exists an infinite amount of isocost lines and they all have the same slope. Additionally, each isocost line has the same total price combination on each of its points and the total price combination increases as the position of an isocost line on a graph increases.