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Page 1: Modern Economics.production

Ralph T. Byrns Modern Microeconomics ©2001

10/29/01 11:56AM Chapter 7 Production page 0

Chapter 7 Production

The Nature of Production

Production and Utility Inputs and Outputs

Production Functions Production in the Short Run

Total Product Curves Average Physical Product Marginal Physical Product The Law of Diminishing Marginal Returns

The Geometry of Total, Average, and Marginal Product Curves Total Product Average Physical Product Marginal Physical Product Stages of Production

Production in the Long Run Isoquants

A 3-D Production Function Geometric Derivation of Isoquants Ridge Lines and the Stages of Production The Marginal Rate of Technical Substitution Returns to Scale Appendix: Cobb-Douglas Production Functions

Page 2: Modern Economics.production

Ralph T. Byrns Modern Microeconomics ©2001

10/29/01 11:56AM Chapter 7 Production page 1

Chapter 7 Production

Life in a state of nature is nasty, brutish, and short.

Thomas Hobbes

Few of us know much detail about the operations of the automobiles, computers, appliances, telecommunications, �, that enable us to conduct our daily lives. In fact, the tremendous prosperity now enjoyed by so many people in developed economies depends crucially on complex manufacturing processes often understood only by industry specialists. Fortunately, however, many production processes have sufficiently common characteristics that we can broadly describe certain aspects of production and costs, our foci for the next two chapters.

Production and Utility

All forms of production require combining productive resources to alter inputs.

Production entails using knowledge or technology to apply energy to transform materials in ways that make goods and resources more valuable in form, place, possession, or time.

More valuable" means that the goods ultimately generate greater consumer utility. Altering form to create value entails reshaping materials; crude oil is less valuable than

gasoline. Augmenting place utilities requires movement; a lobster is worth more in a restaurant's glass tank than in the ocean. Possession utilities arise by shifting ownership from people who value goods less to people who value them more; realtors match home buyers who are moving into an area with sellers who are moving out. Time utilities are created when goods are made available when they are wanted most; speculators buy newly harvested wheat and store it to sell when wheat output is nil. Firms make goods more valuable in form, place, or time, or they transfer ownership to people who value them more highly.

Inputs and Outputs

Did you ever operate a lemonade stand (even if only for an hour or so) when you were a kid? If so, what types of resources did you use? For a lemonade stand, you would need capital (a table, sign, pitcher, glasses, etc.), land (a place to set up your stand), and labor (you have to make the lemonade and try to sell it).

Inputs are resources used in production processes, and are broadly categorized as capital (machinery, tools, etc.), land (all natural resources), or labor (human effort and entrepreneurship).

Whether inputs are either variable or fixed depends on how long it takes to fully adjust the amounts employed. Labor is usually a variable input because the number of workers or hours worked normally adapts quickly to changes in the business climate. Capital, on the other hand, is often a fixed input because considerable time and planning may be required to acquire such

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Ralph T. Byrns Modern Microeconomics ©2001

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specialized new capital as machinery or new buildings, or for complete liquidation or depreciation.

The Sort Run and the Long Run. The period during which the amount of at least one resource is fixed is known as the short run, while in the long run all resources are variable. The short run and long run are not defined by specific time intervals but instead denote periods of differing resource variability. Long run adjustments can vary considerably. For example, a lemonade stand may reach the long run in a single day by opening for business and then closing, but long runs for giant multinational corporations (e.g., Exxon or Toyota) can span decades.

The end product of a production process is output � goods and services. A firm�s output may be final (cars, haircuts, or CDs) or intermediate (steel, computer chips, or fabric). Final goods are ready for consumption or investment, while intermediate goods are components (or inputs) in the production of final goods. For example, steel is used to produce automobiles and buildings, computer chips are integrated into computers, and fabric is transformed into clothing and linen.

Production Functions

A production function (f) identifies how a specific technology transforms certain combinations of inputs into outputs.

A production function summarizes all currently possible technologies (recipes) for producing specific quantities of output from various possible combinations of inputs.

For simplicity, let�s ignore other resources for the moment and consider a simple production function requiring only capital (K) and labor (L):

Q = f(K,L) Thus, quantity (Q) is a function of capital (K) and labor (L). A production function for

solar powered ovens, for example, might be: Q = 3KL where capital and labor are measured in weekly units (40 hours of productive effort from each unit). If five units of capital and labor are employed, then 3 × 5 × 5, or 75 solar powered ovens can be produced weekly. Holding capital constant while increasing labor to 7 units a week yields 3 × 5 × 7 or 105 solar powered ovens produced in a week.

Production functions incorporate the technology embodied in the capital and labor used in a production process. Obviously, any piece of capital embodies a specific technology, but labor also reflects technologies in the forms of the skills and knowledge workers possess. Technological advances alter the production function to enable given quantities of capital and labor to produce more output than their earlier counterparts. Figure 7-1 illustrates the effects of a technological advance on a production function. Curve PF0 depicts a segment of a production function for an early time period with a cruder technology. Technological advances over time make capital and labor more productive, thereby shifting this total output curve up to PF1.

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Ralph T. Byrns Modern Microeconomics ©2001

10/29/01 11:56AM Chapter 7 Production page 3

Figure 7-1 Production Functions and Technological Advance

Curve PF0 illustrates one aspect of a production function at a given point in time. If technology advances, making capital and labor more productive, the production function shifts up to PF1. Specific numbers of workers using fixed amounts of capital are now able to produce more output than their earlier counterparts, as shown by the vertical distance between PF1 and PF0.

The total physical product curve shows the quantities of output or total product that can be produced with various quantities of labor, the variable input. Technology and all other resources are assumed fixed. Thus, if technology changes, the total product curve changes.

Production in the Short Run

Suppose that after college you start a small business that produces snowboards. You find a small workshop on the edge of town, purchase the necessary equipment (saws, planers, sanders, etc.), and place a sign outside your door: Thrasher Snowboards. Now all you need are some employees.

The short run total product curve in Figure 7-2 shows the weekly output you can expect to produce when you hire between one and eight employees (capital is assumed to be fixed).

A total product curve illustrates how output responds when a single input (e.g., labor) is varied, with all other resources and technology assumed fixed.

Data from your total product curve are listed in the total product column of Table 7-1. One employee can produce four snowboards a week, two employees can produce 14 snowboards a week, three employees can produce 30 snowboards a week, and so on.

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10/29/01 11:56AM Chapter 7 Production page 4

Figure 7-2 Total Physical Product Curve for Thrasher Snowboards

Two very useful measures of labor productivity can be calculated from total product data:

the average physical product of labor (APPL) and the marginal physical product of labor (MPPL).

The average physical product of labor (APPL) equals total product (output) divided by the quantity of labor used: APPL = TP/L = Q/L.

The average physical product of labor translates into output per worker. Table 7-1 and Figure 7-3 show the average productivity of your employees. The average product for 2 employees is 7 (14/2), the average product for 3 employees is 10 (30/3), and so on.

The marginal physical product of labor (MPPL) is the additional output produced by an additional unit of labor and is calculated by dividing the change in total product by the change in labor: MPPL = ∆TP/∆L = ∆Q/∆L.

Marginal physical product translates into output per extra worker. The marginal productivity of your employees is also shown in Table 7-1 and Figure 7-3. The marginal product of the second employee you hire is 10 ([14 - 10]/[2-1]), indicating that second employee increases Thrasher's output by ten snowboards a week.

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Table 7-1 Weekly Total, Average, and Marginal Physical Products of Labor at Thrasher Snowboards

Quantity of Labor Total Product or Output

Average Physical Product of Labor

Marginal Physical Product of Labor

0 0 -- -- 1 4 4 4 2 14 7 10 3 30 10 16 4 40 10 10 5 48 9.6 6 6 50 8.3 2 7 48 6.9 -2 8 42 5.3 -6

Figure 7-3 Average and Marginal Product Curves of Labor at Thrasher Snowboards

The average physical product of labor (APPL) is calculated by dividing total output by the quantity of labor used (TP/L). The marginal physical product of labor (MPPL) is calculated by dividing the change in total product by the change in labor employed (∆TP/∆L). These curves reflect the data in Table 7-1.

Page 7: Modern Economics.production

Ralph T. Byrns Modern Microeconomics ©2001

10/29/01 11:56AM Chapter 7 Production page 6

The Law of Diminishing Marginal Returns

Take another look at the total, average, and marginal product curves for your firm. In all three cases, labor productivity reaches a maximum and then begins to decline. Why the deadline? The adage, "Too many cooks in the kitchen spoils the broth", hints at the answer. Initially, each additional employee hired increases total product at an increasing rate (marginal product is increasing). In this stage (1-3 employees), dividing tasks up and working as a team significantly increases output. After the third employee is hired, additional employees continue to increase total product, but at a decreasing rate (marginal product begins to decline). Beyond six employees, total output actually falls (marginal product becomes negative). In this stage, your workshop is overcrowded and capital is spread too thinly across your employees. Idle workers get in the way and actually detract from the work of others, reducing total output. This eventual decline in output is an extreme example of the law of diminishing marginal returns, which helps explain the shapes of the total, average, and marginal product curves.

The law of diminishing marginal returns: All else constant (e.g., technology and at least one fixed resource) given successive increments of variable resources eventually result in declining marginal products.

The existence of at least one fixed resource means that this law is a short run concept. The law of diminishing marginal returns is unavoidable when at least one resource is fixed, and is without exception. Were it not for diminishing returns, enough food might be grown in a flower pot to feed the world.

Geometric Relationships between Total, Average, and Marginal Product Curves

The shapes of the average and marginal product curves and how they are related are tightly linked to the shape of the total product curve. This should come as no surprise -- average and marginal products are calculated from total product data.

Graphically, the average physical product of labor equals the slope of a ray from the origin to a given point on the total product curve. For example, in Figure 7-4, ray R1 has a slope of 4 and passes through the total product curve at the point where one unit of labor is employed. Following the dotted line down from this intersection shows that the APPL for one employee equals 4. Similarly, ray R2 has a slope of 7, and intersects the total product curve where 2 and just less than 7 employees are hired. Following the vertically dashed lines down to the average product curve shows that APPL equals 7 when 2 and just fewer than 7 workers are employed. The average physical product of labor reaches a maximum when a ray from the origin is just tangent to the total product curve. This is shown by ray R3, which has a slope of 10, and corresponds with the maximum value for the average physical product of labor.

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Figure 7-4 Deriving the Average Physical Product of Labor From the Total Product Curve

The average physical product of labor (APPL) is derived from the total product curve by drawing rays from the origin to points on the total product curve. The slope of the ray to a point on the total product curve equals the APPL at that point, as shown by rays R1, R2, and R3. The APPL is maximized at the point on the total product curve where a ray (R3) from the origin is just tangent to the total product curve.

The marginal physical product of labor graphically corresponds to the slope at particular points on a total product curve � the slope of a line drawn just tangent to a specific point. In Figure 7-5, ray R3 is just tangent to the total product curve at the point where 3.5 workers are employed. The slope of R3 is 10; following the dashed vertical line down from the point of tangency shows that the MPPL for 3.5 workers is 10. It is no coincidence that the MPPL and APPL are equal at this point (3.5 workers). When a ray from the origin (average product) is just tangent (marginal product) to the total product curve, MPPL = APPL.

When fewer than 3.5 workers are employed, the MPPL curve lies above the APPL curve because the slope of the total product curve is greater at any point than a ray drawn to that point (compare rays R1 and R2). The converse occurs when more than 3.5 workers are hired: the MPPL curve lies below the APPL curve because the slope of a ray to any given point is greater than the slope of the total product curve at the corresponding point.

There is one point where a line of tangency cannot be drawn without piercing the total product curve -- the inflection point. The inflection point occurs where the total product curve stops increasing at an increasing rate (MPPL is rising), continuing to increase, but only at a decreasing rate (MPPL is declining). The marginal physical product of labor attains its maximum value (16) at the inflection point, as shown in Figure 7-5. When the total product curve reaches its maximum value, the MPPL is zero. This occurs when 6 workers are employed. Beyond 6 workers in Figure 7-5, the total product curve declines because the MPPL is negative.

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10/29/01 11:56AM Chapter 7 Production page 8

Figure 7-5 Deriving the Marginal Physical Product of Labor from a Total Product Curve

The marginal physical product of labor (MPPL) equals the slope of the total product curve at any given point. Ray R3 is just tangent to the total product curve when 3.5 workers are employed. (You might employ part-timers.) The slope of R3 is 10 and equals both the MPPL and APPL at that point. When fewer than 3.5 workers are employed, the MPPL exceeds the APPL. When more than 3.5 workers are employed, the MPPL is below the APPL. The MPPL is maximized at the inflection point on the total product curve, equals zero when total product reaches a maximum (6 workers), and becomes negative when the total product curve declines (more than 6 workers).

Recall our discussion of the relationships between "averages" and marginals in Chapter 2. The relationship between the APPL and the MPPL can be summarized:

When MPPL > APPL, APPL is rising. When MPPL = APPL, APPL is at a maximum. When MPPL < APPL, APPL is falling.

As an example, think about your grade point average (GPA). Your semester GPA is a marginal value, while your overall GPA is an average value. Whenever your semester GPA exceeds your overall GPA, your overall GPA will increase. Whenever your semester GPA is less than your overall GPA, your overall GPA declines.

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Stages of Production

Now that you understand that relationship between total, average, and marginal products, we can identify three stages of production.

The first stage of production occurs when the APPL is rising. In the second stage of production, the APPL is falling but the MPPL is positive. The third stage of production occurs when the MPPL becomes negative.

All three stages of production are illustrated in Figure 7-6. In which stage of production should your snowboard company operate? Stage III is obviously out, because additional employees in this stage actually reduce total product (their MPPL is negative). You may be tempted to say Stage I, where MPPL is at a maximum, but in Stage I, the marginal product of capital (MPPK) is actually negative (i.e., the MPPK is in Stage III).

That leaves you with only one option: Stage II. In Stage II, the marginal products of both productive resources (capital and labor) are positive. Exactly where you should produce in Stage II (i.e. what is the optimal number of workers to hire) will depend on resource costs and the prices you receive for snowboards. These matters will be addressed in upcoming chapters.

Figure 7-6 The Stages of Production

Production can be divided into three stages graphically determined by the average and marginal product curves of labor. Stage I occurs from the origin up to the point where the APPL reaches a maximum. Stage II occurs where the APPL is declining and the MPPL is still positive. Stage III occurs where the MPPL is negative. Firms will want to operate in stage II because the marginal products of both factors of production (capital and labor) are positive in this stage.

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Production in the Long Run Although we've never known anyone who engaged in such a barbaric practice, the old saying, "There's more than one way to skin a cat" is especially relevant for firms in the long run. When all productive resources are variable, firms can use almost countless resource combinations to produce output. For example, you could produce your snowboards by employing (a) many laborers equipped only with hand tools, (b) a moderate number of workers using all electrical tools, or (c) one or two workers to maintain a team of robotic power tools. Obviously numerous resource combinations fall between these possibilities. The main point is that in the long run, firms can alter the amounts of all inputs to produce output in a variety of ways. By convention, however, technology is assumed constant, an issue we will deal with in a bit.

Isoquants

Isoquants graphically depict the input choices firms face in the long run to produce given quantities of output.

An isoquant (Iso = equal, quant= quantities) shows all combinations of resources that will produce a given quantity of output.

Figure 7-7 shows two isoquants for your snowboard company. The isoquant labeled Q = 30 shows all the combinations of capital and labor capable of producing 30 snowboards each week. As you move down the isoquant from point a to point c, labor is substituted for capital in the production process. Compared to point b, production at point a employs more capital and less labor, while production at point c employs relatively more labor and less capital.

With further moves along a ray away from the origin, each successive isoquant corresponds to a greater level of output (compare isoquants Q = 30 and Q = 40). Intuitively, higher output levels necessarily absorb more inputs. For example, to increase snowboard production from Q = 30 to Q = 40, you must increase at least one, if not both of the inputs.

Isoquants, as you are probably beginning to suspect, are very similar to indifference curves. Just like indifference curves, isoquants are: (1) convex to origin, (2) nonintersecting, and (3) negatively sloped (in the economically relevant region). The big difference between the two, however, is that indifference curves show relative rankings since utility is only ordinally measurable, while isoquants are cardinally measurable and denote specific and amounts of output.

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Figure 7-7 Isoquant Curves for Thrasher Snowboards

Isoquants show all combinations of two productive inputs that will produce a given quantity of output. On isoquant Q = 30, points a, b, and c show three different capital/labor combinations that are capable of producing 30 snowboards weekly. Isoquants Q = 40 and Q = 30 illustrate that isoquants further from the origin show a greater level of output.

A 3-D Production Function: Geometric Derivation of Isoquants

Isoquants are usually graphed, as in Figure 7-7, in two dimensions with resources on the axes. Quantity is left to your imagination. Let's repeat the mental experiment we used in Chapter 4 to illustrate preference functions in three dimensions. To truly incorporate quantity into an isoquant, three dimensions are needed. To visualize a three-dimensional graph, look at (or imagine) a corner in a room. Next, imagine that you are standing in a room with your back to a corner, and that a pile of sand is poured against the opposite corner of the room. Now look at Panel A of Figure 7-8, which shows the pile of sand heaped against the opposite corner and sloping down to your feet (you are standing in the corner labeled 0). The intersection of the wall and floor on your right represents the labor axis, which indicates that more labor is used the further you move away from the corner. Likewise, the capital axis is represented by the intersection of the wall and floor

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on your left, with capital inputs increasing as you move away from the corner. Quantity is represented by height, and increases from zero at floor level to a maximum at the ceiling. As capital and labor inputs are increased, the quantity of sand or output increases as shown by the increased height of the pile.

Figure 7-8 Geometric Derivation of Isoquants from a Three Dimensional Graph

Panel A illustrates a three-dimensional production mountain in which output is represented by height. A horizontal plane inserted into the mountain yields a contour line that shows all the combinations of capital and labor yielding the same level of output (e.g., Q1). Similar planes and contours can then be projected down into the two dimensional graph in Panel B, yielding isoquants Q1- Q5, which horizontally intersect the production mountain (Panel A) at varying heights, and are then projected into Panel B as two dimensional isoquants.

Imagine that you are looking straight at the sand pile, and that you insert a thin piece of wood horizontally into it. The quantity of sand (output) is constant everywhere under the board. The intersection of the sand pile by the piece of wood is shown by curve ab in Panel A of Figure 7-8. All combinations of labor and capital inputs that lie on curve ab yield the same amount of output (Q1). Curve ab can be projected to the two dimensional graph shown in Panel B, yielding isoquant Q1. Considering horizontal intersections at varying heights on the sand pile can generate an endless number of isoquants.

Ridge Lines and the Stages of Production

So far isoquants have been shown as being negatively sloped throughout their entirety because no rational firm would produce on a positively sloped portion of an isoquant. To understand why, suppose Figure 7-9 depicts some isoquants for your firm, Thrasher Snowboards. Points b-f depict input combinations capable of producing 40 snowboards in a week. As a rational manager you would not want to produce at points b or f because you could produce the same quantity of snowboards using less of both inputs (compare points b and c, and points e and f). Rational producers will never willingly produce on the positively sloped portion of an isoquant because this entails using more resources (inputs) than is necessary to produce a given level of output.

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Figure 7-9 Ridge Lines and the Stages of Production for Thrasher Snowboards

Ridge lines separate isoquants into positively and negatively sloped regions. Rational producers will never produce on the positively sloped portion of an isoquant because this entails using more inputs than is necessary to produce a given level of output (compare point b with point c). Ridge lines also define the three stages of production. Above the top ridge line, the MPPK is negative, indicating Stage I. To the right of the bottom ridge line, the MPPL is negative, indicating Stage III. Between the two ridge lines, the MPPK and the MPPL are both positive, identifying Stage II and the relevant range of production.

Isoquants can be divided into negatively and positively sloping portions by ridge lines like the ones shown in Figure 7-9. Ridge lines also divide isoquants into the three stages of production discussed earlier in this chapter. Stage III, where the MPPL is negative, occurs to the right of the bottom ridge line. Starting at point e, and holding capital constant, additional labor results in less output, so the MPPL must be negative in this region. Stage I, where the MPPK is negative, occurs above the top ridge line. Additional capital beyond point c, while holding labor constant, results in less output, so the MPPK must be negative in this region. Stage II, where the MPPL and the MPPK are positive, occurs between the two ridge lines and defines the area of relevant economic activity. Since rational firms will always produce in stage II, only the negatively sloped portion of isoquants will be shown and considered from this point on (except in special cases).

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The Marginal Rate of Technical Substitution (MRTS)

When a firm reduces its use of capital in Zone II, it must hire additional labor to hold its production constant, as shown on any isoquant. A crucial question is, how much additional labor must be employed to compensate for a given reduction in capital, or vice versa? This question can be answered by determining the slope of the appropriate isoquant along its relevant portion, which yields the marginal rate of technical substitution (MRTS).

The marginal rate of technical substitution (MRTS) is the rate at which one input must be increased to maintain a constant output when another input is reduced, and is given by the absolute value of the slope of the isoquant (|∆K/∆L|).

The MRTS is analogous to the marginal rate of substitution (MRS) discussed in chapter 3. The only difference is that the MRTS determines how much of one input must be exchanged for another input to keep output constant, whereas the MRS determines how much of one good must be exchanged for another good to maintain consumer satisfaction. In Figure 7-10, the marginal rate of technical substitution of labor for capital (MRTSLK) between points a and b equals 2 (|∆K/∆L| = 2/1 = 2). This means that you need to hire an additional unit of labor to replace the two units of capital to continue producing 30 snowboards weekly. Between points b and c, the MRTSLK equals 1/2, indicating that 2 additional units of labor are needed to replace one unit of capital to keep output constant. The MRTS can be determined at any point along an isoquant by calculating the slope of a line drawn just tangent to the point in question. For example, the MRTSLK at point b in Figure 7-10 equals 1, which is the absolute value of the slope of this tangency.

The MRTSLK can also be calculated as the ratio of the marginal products of labor and capital (MPPL/MPPK). A movement down along an isoquant requires the additional output produced by the additional labor to offset the loss of output because of reduced capital. More specifically, the change in labor multiplied by the marginal physical product of labor must equal the change in capital multiplied by the marginal product of capital, or:

(∆L)(MPPL) = (∆K)(MPPK)

Rearranging yields:

MPPL/MPPK = ∆K/∆L

which equals the MRTSLK. The MRTSLK can therefore be calculated by taking the absolute value of the slope of the isoquant or by calculating the ratio of the marginal products of labor and capital.

As a firm moves down along an isoquant, the MRTSLK decreases, indicating that it takes increasingly more labor to replace a unit of capital. This occurs because the marginal physical product of labor (MPPL) declines as more and more labor is employed with less and less capital, in accord with the law of diminishing marginal returns; simultaneously, the MPPK rises because more labor is applied to each unit of capital. Symmetric logic applies when moving up along an isoquant -- the MPPK declines while the MPPL rises as more and more capital is used with less

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and less labor. Thus isoquants are convex. Figure 7-10 The Marginal Rate of Technical Substitution at Thrasher Snowboards

The marginal rate of technical substitution of labor for capital (MRTSLK) is the rate at which labor must be substituted for capital to maintain constant output. The MRTSLK equals the absolute value of the slope of the appropriate isoquant and can therefore, be calculated as |∆K/∆L|. Between points a and b, the MRTSLK equals |-2/1| or 2. Between points b and c, the MRTSLK equals |-1/2| or 1/2. Calculating the MRTSLK at a single point on an isoquant requires determining the slope of a line just tangent to the point. The MRTSLK at point b equals the slope of the line of tangency or 1.

Returns to Scale

How output changes when all productive inputs are increased simultaneously helps identify returns to scale.

Returns to scale refer to whether output responds more or less than proportionally as all inputs are increased by a given proportion.

More specifically, returns to scale can be broken down into three categories: constant, increasing, or decreasing.

Constant returns to scale occur when output grows by the same proportion as the increase in inputs.

For example, Stone Age Steel experiences constant returns to scale if output grows by

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50% in conjunction with a 50% increase in labor and capital inputs.

Increasing returns to scale (economies of scale) occur when output grows by a larger proportion than inputs.

Drop Dead Aeroplanes enjoys increasing returns to scale when a 10% increase of both capital and labor results in a 25% increase in output.

Decreasing returns to scale (diseconomies of scale) occur when output grows by a smaller proportion than inputs.

Belch Frozen Dinners realizes decreasing returns to scale when doubling all of their inputs only boosts output by 50%.

The gamut of returns to scale is illustrated in Figure 7-11. When you double both your capital and labor inputs from 10 to 20 units, your production of snowboards more than doubles (from 100 to 300), so increasing returns to scale are present. Doubling your inputs again (from 20 to 40 units each), results in production also doubling (300 to 600), signaling constant returns to scale. Lastly, when you double your inputs from 40 to 80 units, output grows (from 600 to 1,000), but less than double, indicating decreasing returns to scale.

Figure 7-11 Returns to Scale at Thrasher Snowboards

When capital and labor inputs are doubled from 10 to 20 units, output increases from 100 to 300 snowboards weekly. Since output grows at a proportionally greater rate than inputs over this range, increasing returns to scale are realized. Doubling capital and labor inputs from 20 to 40 units doubles output (300 to 600 snowboards weekly), indicating constant returns to scale. Decreasing returns to scale occur when capital and labor are doubled from 40 to 80 units because output grows by less than a multiple of 2 (600 to 1,000 snowboards weekly).

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You may question why returns to scale would ever be anything other than constant. Logically, doubling the number of workers and the capital they work with should double output, so why would you get proportionally more or less output? The division and specialization of labor is one explanation for increasing returns to scale. For example, increasing labor at Thrasher Snowboards might allow all individuals to specialize in one task instead of performing many. This will boost worker productivity because time will not be lost switching from task to task, and specialized individuals become more proficient at their appointed jobs. Increasing returns to scale can also arise when increased size allows the employment of more specialized capital. Conveyor belts and a computer-controlled assembly line will do little for you if you only employ a few employees, but if you are producing on a larger scale and have many employees, this specialized capital will greatly increase your productive capacity.

As firms grow in size, it can become increasingly difficult to organize, manage and coordinate all activities. Once a firm surpasses some threshold, organizational problems retard output and lead to decreasing returns to scale. If your firm becomes large enough, you will have problems communicating and directing the work of your employees. Hiring managers will help initially, but after a while, multiple layers of management will evolve, and coordination will get bogged down as directives slowly work there way down through the layers.

Chapter Review

1. Inputs are productive resources used to produce output, and fall into three broad categories: capital, land, and labor.

2. The short run is a period of time in which at least one input in the production process is fixed. In the long run all inputs can be varied.

3. A production function describes the relationship between inputs and output, and is commonly written as Q = F(K,L) -- Quantity is a function of capital and labor.

4. The quantity of output produced in the short run is also known as total product and is shown graphically by a total product curve.

5. Average product is determined by dividing total product by the quantity of the variable input (usually labor) used: APPL = TP/L. Marginal product is the extra output produced by an additional amount of the variable input (once again usually labor). The marginal physical product of labor (MPPL) is calculated as ∆TP/∆L.

6. Production can be divided into three stages and graphically shown by the average and marginal product curves for labor. Stage III occurs when the MPPL is negative. Stage I occurs from the start of production up to the point where the average physical product of labor (APPL) reaches its maximum. In Stage I the marginal product of capital (MPPK) is negative. Stage II occurs while the APPL is declining and the MPPL is positive. Firms will always want to produce in stage II.

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Ralph T. Byrns Modern Microeconomics ©2001

10/29/01 11:56AM Chapter 7 Production page 18

7. Isoquants are a graphically depiction of the long run choices faced by firms. More specifically, isoquants show the various combinations of two productive inputs (usually capital and labor) that produce the same quantity of output. Isoquants further from the origin show a greater level of output than isoquants closer to the origin.

8. Ridge lines separate the positively and negatively sloped portions of isoquants as well as delineating the three stages of production. Rational firms will never produce on the positively sloped portion of an isoquant because the same quantity of output could be produced with fewer inputs. Stage I of production occurs above the top ridge line, while stage III of production occurs to the right of the bottom ridge line. Stage II, which defines the economically relevant range of production, occurs between the two ridge lines.

9. The marginal rate of technical substitution (MRTS) is the rate at which one input must be increased to keep output constant, when another input is reduced. The MRTS is given by the absolute value of the slope of the isoquant (|∆K/∆L|) or by the ratio of the marginal productivities of the two productive inputs (i.e. MPPL/MPPK).

10. Returns to scale are a measurement of the growth of output that occurs when all productive inputs are increased by some proportion. When output grows in greater proportion than inputs, increasing returns to scale result. Constant returns to scale occur when output grows in the same proportion as inputs. When output grows by a smaller proportion than inputs, decreasing returns to scale occur.

Appendix: The Cobb-Douglas Production Function Some points about Cobb-Douglas production functions include:

1. Functional form: Q = AKaLb

2. Output elasticities: alpha gives output elasticity of labor and beta gives output elasticity of capital.

3. Returns to scale: if

alpha + beta > 1; then increasing returns to scale alpha + beta = 1; then constant returns to scale alpha + beta < 1; then decreasing returns to scale