Production and Cost Review for AP Economics (page 3)
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Main Topics: Short-Run Production Functions, Law of Diminishing Returns, Short-Run Costs, Connecting Production and Cost, Long-Run Costs, Economies of Scale
Short-Run Production Functions
How do economic resources like labor, capital, natural resources, and entrepreneurial talent become a cup of lemonade, or a ton of copper, or a 30-second television commercial? A production function is the mechanism for combining production resources, with existing technology, into finished goods and services. In other words, a production function takes inputs and creates output. In a production function that uses only labor (L) and capital (K):
Fixed and Variable Inputs
The short run is a period of time too brief to change the plant capacity. This implies that some production inputs cannot be changed in the short run. These are fixed inputs. During the short run, firms can adjust production to meet changes in demand for their output. This implies that some inputs are variable inputs. Using only labor and capital, we assume that labor can be changed in the short run, but capital (i.e., the plant capacity) is fixed.
Short-Run Production Measures
By its very nature, production lends itself to be quantified and as a result you need to study these three production measures. To keep it simple, capital is assumed to be fixed while labor can be changed to produce more or less output.
- Total Product (TPL) of Labor is the total quantity, or total output, of a good produced at each quantity of labor employed.
- Marginal Product (MPL) of Labor is the change in total product resulting from a change in the labor input. MPL = ΔTPL/ΔL. If labor is changing one unit at time, MPL = ΔTPL.
- Average Product (APL) of Labor is also a measure of average labor productivity and is total product divided by the amount of labor employed. APL = TPL/L.
As you can see, MPL and APL are both derived from TPL. It is useful to see how these three measures are related with a numerical example.
In the production period of a month, Molly's lemonade stand combines variable inputs of her labor (and the raw materials) to the fixed inputs of her table and her license to operate. Molly adds employees to her plant and forecasts the change in production (cups per day) in Table 8.2
As Molly employs more workers to the fixed plant capacity (the table on the corner), total product increases, eventually peaks, and then begins to fall. This production function can be seen in Figure 8.1
Law of Diminishing Marginal Returns
Imagine what happens to the lemonade stand as Molly adds more and more workers. At first, tasks are divided. (For example, Josh squeezes the lemons; Molly adds the sugar; Kelli stirs.) Specialization occurs. The marginal productivity of successive workers is rising in the early stage of production, but at some point, adding more workers increases the total product by a lesser amount. Maybe the fourth worker is pouring the lemonade and stocking while the fifth is taking money and making change. Beyond the fifth worker, the table is too crowded with employees, cups are spilled, product is wasted, and total production actually falls. The marginal contribution of these workers is negative. This illustrates one of the most important production concepts in the short run, the Law of Diminishing Marginal Returns, which states that as successive units of a variable resource are added to a fixed resource, beyond some point the marginal product falls.
Marginal product is the incremental change in total product as one more unit of labor is added. Marginal product is the geometric slope of total product. In Figure 8.1, the total product curve is initially getting steeper as more labor is added. This is seen in Figure 8.2 as increasing marginal product. From the third to the fifth worker, the slope of total product is still positive, but is becoming less steep. In Figure 8.2 marginal product from workers 3 to 5 is still positive but is falling. Beyond the fifth worker, total product is falling and thus has a negative slope. This turn of events is seen below when marginal product becomes negative.
Average product, also plotted below, initially rises, reaches a peak, and then begins to fall. So long as the marginal (next) worker adds production that is above the current average, they are pulling the average up. This is why we see APL rising so long as MPL is above APL. If the marginal worker adds production that is below the current average, the worker pulls the average down. Thus when MPL is below APL, you see that APL is falling. Logically then, MPL intersects APL at the peak of APL. Average product cannot be negative.
It is important to note that we have discussed production theory without including the nagging necessity of paying for our hired inputs. For every employed input, fixed or variable, a cost is incurred.
In the short run, there is at least one input that is fixed and so these costs are also fixed. All inputs that are variable incur variable costs.
- Total Fixed Costs (TFC) are those costs that do not vary with changes in short-run output. They must be paid even when output is zero. These include rent on building or equipment, insurance or licenses.
- Total Variable Costs (TVC) are those costs that change with the level of output. If output is zero, so are total variable costs. They include payment for materials, fuel, power, transportation services, most labor, and similar costs.
- Total Cost (TC) is the sum of total fixed and total variable costs at each level of output.
TC = TVC + TFC
Table 8.3 summarizes Molly's costs of producing cups of lemonade per minute. Her total fixed costs are assumed to be $6 per minute and total variable costs increase as production increases.
Figure 8.3 illustrates the three total cost functions. Total fixed cost is a constant at all levels of output. Total variable cost quickly rises at first, briefly slows, and then proceeds to increase at an increasing rate. Total cost is simply the sum of TFC and TVC at every level of output and so it lies parallel to TVC. Thus the vertical distance between TC and TVC is equal to TFC.
Marginal and Average Costs
Similar to our discussion of production, we can derive marginal and per unit measures of cost from the total cost functions. These are in Table 8.4.
- Marginal Cost is the additional cost of producing one more unit of output MC = ΔTC/ΔQ. Since TVC are the only costs that change with the level of output, marginal cost is also calculated as MC = ΔTVC/ΔQ. If quantity is changing one unit at a time, MC = ΔTC = ΔTVC.
- Average Fixed Cost (AFC) is total fixed cost divided by output. AFC = TFC/Q. It continuously falls as output rises.
- Average Variable Cost (AVC) is total variable cost divided by output. AVC = TVC/Q.
- Average Total Cost (ATC) is total cost divided by output ATC = TC/Q. Note that ATC = AFC + AVC.
If marginal product is the slope of total product, it should be no surprise that marginal cost is the slope of total cost, or total variable cost. We can see that marginal cost initially falls due to specialization, but soon begins to rise as more output is produced. This is the Law of Increasing Costs and is a direct result of the Law of Diminishing Marginal Returns to production. Both being U-shaped curves, average variable and average total costs initially fall, hit a minimum point, and begin to rise. Average Total Cost is vertically above AVC by the amount of AFC. Figure 8.4 illustrates this.
Marginal cost and average variable and average total cost are related in much the same way as marginal product is related to average product of labor. When the marginal cost of producing another cup of lemonade exceeds the current average cost the average is rising. When the marginal cost of producing another cup of lemonade falls below the current average cost the average is falling. Therefore, marginal cost equals average total cost at the minimum of ATC and equals average variable cost at the minimum of AVC.
Bridge over (Troubling) Economic Waters
Many students think that production and cost concepts are two sets of theoretical topics. This separation creates the impression that "there's twice as much to remember." These students are surprised to find out that production and cost are closely connected.
Think about it from Molly's point of view. If the next worker employed has a high marginal product, then the marginal cost of producing that increased product must be quite low. When things are going well with production, they must be going well with cost. Try to see the concepts of production and cost not as two isolated bodies of theory, but as two related sets of concepts that just need to be bridged. Let us try to build this bridge with a little algebra.
Marginal Product and Marginal Cost
MC = ΔTVC/ΔQ and since the only variable input is labor being paid a fixed wage w,
MC = w/(ΔQ/ΔL) = w/MPL. MC and MPL are inverses of each other!
Average Product and Average Variable Cost
AVC = TVC/Q and with the only variable input being labor paid a fixed wage w,
AVC = wL/Q which can be modified as,
AVC = w/(Q/L) = w/APL. AVC and APL are inverses of each other!
If we put smoother versions of our production and cost figures together, we can see these relationships in Figures 8.5 and 8.6. The output q1 where MPL is at a maximum is the same as the output where MC is at a minimum. Likewise, the output q2 where APL is at a maximum is the same as the output where AVC is at a minimum.
Since all inputs are variable in the long run, discussion of production levels isn't so much about output per hour or day; it's more a question of plant size or capacity. In the short run, the firm asks, "With our current plant size, how much must we produce today?" The long run is long enough to adjust the plant capacity so the issue is really one of scale. The firm might ask itself, "At what scale do we want to operate?"
Long-Run Average Cost
I like to think of the firm's short-run average costs as a snapshot of the firm's ability to produce efficiently at the fixed plant size. Over time, the firm may grow and expand the plant size and begin to produce efficiently, but at the larger fixed plant size, giving us another snapshot. This process repeats itself as the firm expands or contracts and each time we receive another short-run snapshot of average cost. If we could put these short-run snapshots together into a kind of motion picture, we would see a more continuous long-run home movie of the firm's average costs. The example and Figure 8.7 illustrate the connection between short- and long-run average costs.
- In year one, Molly's firm operates at a "small" scale, producing on SRAC1.
- In year two, Molly could expand and operate at a "medium" scale, producing on SRAC2, but only if she can sell more than 100 gallons of lemonade. At quantities below 100, SRAC1 < SRAC2, so expansion would not be wise.
- In year three, Molly might expand to operate at a "large" scale and move to SRAC3, but only if she can sell more than 250 gallons.
- Beyond the "large" scale exists a "grande" scale, but very quickly SRAC4 > SRAC3 and so this plant capacity actually begins to pay rising per unit costs.
Each of these four short-run snapshots of average costs can be smoothed out into the home movie long-run average cost curve, which is composed of sections of each short-run average cost curve at each of the four plant sizes that Molly might choose for her firm. In Figure 8.7, the long-run average cost curve would lie along the segments a→b→c→d→e.
Economies of Scale
Construction of a smoother version of Figure 8.7 allows us to see more easily some important stages of the long-run average cost curves (Figure 8.8).
- Economies of scale are advantages of increased plant size and are seen on the downward part of the LRAC curve. LRAC falls as plant size rises.
- Labor and managerial specialization is one reason for this.
- Ability to purchase and use more efficient capital goods also can explain economies of scale.
- Constant returns to scale can occur when LRAC is constant over a variety of plant sizes.
- Diseconomies of scale are illustrated by the rising part of the LRAC curve and can occur if a firm becomes too large.
- Some reasons for this include distant management, worker alienation, and problems with communication and coordination.
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