At the start of the chapter, we made several assumptions about a firm’s production behavior. The third assumption was that the firm’s goal is to minimize the cost of producing whatever quantity it chooses to make. (How a firm determines that quantity is the subject of Chapter 8, Chapter 9, Chapter 10 and Chapter 11.) The challenge of producing a specific amount of a particular good as inexpensively as possible is the firm’s cost-minimization problem.

The firm’s production decision is another constrained optimization problem. Remember from our discussion in Chapter 4 that these types of problems are ones in which an economic actor tries to optimize something while facing a constraint on her choices. Here, the firm’s problem is a constrained minimization problem. The firm wants to minimize the total costs of its production. However, it must hold to a constraint in doing so: It must produce a particular quantity of output. That is, it can’t minimize its costs just by refusing to produce as much as it would like (or refuse to produce anything, for that matter). In this section, we look at how a firm uses two concepts, isoquants (which tell the firm the quantity constraint it faces) and isocost lines (which tell the firm the various costs at which the firm can produce its quantity) to solve its constrained minimization problem.

When learning about the consumer’s utility function in Chapter 4, we looked at three variables: the quantities of the two goods consumed and the consumer’s utility. Each indifference curve showed all the combinations of the two goods consumed that allowed the consumer to achieve a particular utility level.

We can do the same thing with the firm’s production function. We can plot as one curve all the possible combinations of capital and labor that can produce a given amount of output. Figure 6.3 does just that for the production function we have been using throughout the chapter; it displays the combinations of inputs that are necessary to produce 1, 2, and 4 units of output. These curves are known as isoquants. (The Greek prefix iso- means “the same,” and quant is a shortened version of the word “quantity.”)

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Just as with indifference curves, isoquants further from the origin correspond to higher output levels (because more capital and labor lead to higher output), isoquants cannot cross (because if they did, the same quantities of inputs would yield two different quantities of output), and isoquants are convex to the origin (because using a mix of inputs generally lets a firm produce a greater quantity than it could by using an extreme amount of one input and a tiny amount of the other).

The Marginal Rate of Technical Substitution The slope of the isoquant plays a key role in analyzing production decisions because it captures the tradeoff in the productive abilities of capital and labor. Look at the isoquant in Figure 6.4. At point A, the isoquant is steeply sloped, meaning that the firm can reduce the amount of capital it uses by a lot while increasing labor only a small amount, and still maintain the same level of output. In contrast, at point B, if the firm wants to reduce capital just a bit, it will have to increase labor a lot to keep output at the same level. Isoquants’ curvature and convexity to the origin reflect the fact that the capital–labor tradeoff varies as the mix of the inputs changes.

The negative of the slope of the isoquant is called the marginal rate of technical substitution of one input (on the x-axis) for another (on the y-axis), or MRTS_XY. It is the quantity change in input Y necessary to keep output constant if the quantity of input X changes by 1 unit. For the most part in this chapter, we will be interested in the MRTS of labor for capital or MRTS_LK, which is the amount of capital needed to hold output constant if the quantity of labor used by the firm changes.

If we imagine moving just a little bit down and to the right along an isoquant, the change in output—which we know must add up to zero because we’re moving along an isoquant— equals the marginal product of labor times the change in the units of labor due to the move, plus the marginal product of capital times the change in the amount of capital. (This change in capital is negative because we’re taking away capital when we move down along the isoquant.) So we can write the total output change (which, again, is zero along an isoquant) as

ΔQ = MP_L × ΔL + MP_K × ΔK = 0

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If we rearrange this to find the slope of the isoquant, ΔK/ΔL, we have

Therefore, the MRTS_LK at any point on an isoquant tells you the relative marginal products of capital and labor at that point.

The concepts underlying the marginal rate of technical substitution are the same as those of the marginal rate of substitution (MRS) for consumers, which we learned about in Chapter 4. The two are so closely tied together, in fact, that the names are essentially the same—the word “technical” is tacked on to distinguish the producer case. Both the MRTS and MRS are about marginal tradeoffs. The MRS is about a consumer’s willingness to trade one good for another while still obtaining the same utility level. The MRTS is about a firm’s ability to trade one input for another while still producing the same quantity of output. The shape of the curves in both cases tells you about the rate at which one good/input can be substituted for the other.

The nature of the marginal tradeoffs embodied in the MRS and MRTS implies similar things about the shape of the curves from which they are derived. On the consumer side, indifference curves are convex to the origin because the MRS varies with the amount of each good the consumer is consuming. On the production side, isoquants are convex to the origin because the MRTS varies with the amount of each input the firm uses to produce output. When the firm uses a lot of capital and just a little labor (point A in Figure 6.4), it can replace a lot of capital with a little more labor and still produce the same quantity of output. At this point, labor has a high marginal product relative to capital, and the firm’s isoquant is steep. At point B, the firm uses a lot of labor and only a little capital, so capital has a relatively high marginal product and MRTS_LK is small. A smaller MRTS_LK means a flatter isoquant at that input mix.

Substitutability How curved an isoquant is shows how easily firms can substitute one input for another in production. For isoquants that are almost straight, as in panel a of Figure 6.5, a firm can replace a unit of one input (capital, e.g.) with a particular amount of the second input (labor) without changing its output level, regardless of whether it is already using a lot or a little of capital. Stated in terms of the marginal rate of technical substitution, MRTS_LK doesn’t change much as the firm moves along the isoquant. In this case, the two inputs are close substitutes in the firm’s production function, and the relative usefulness of either input for production won’t vary much with how much of each input the firm is using.

Highly curved isoquants, such as those shown in panel b of Figure 6.5, mean that the MRTS_LK changes a lot along the isoquant. In this case, the two inputs are poor substitutes. The relative usefulness of substituting one input for another in production depends a great deal on the amount of the input the firm is already using.

Perfect Substitutes and Perfect Complements in Production In Chapter 4, we discussed the extreme cases of perfect substitutes and perfect complements in consumption. For perfect substitutes, indifference curves are straight lines; for perfect complements, the curves are “L”-shaped right angles. The same holds for inputs: It is possible for them to be perfect substitutes or perfect complements in production. Isoquants for these two cases are shown in Figure 6.6.

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If inputs are perfect substitutes as in panel a, the MRTS doesn’t change at all with the amounts of the inputs used, and the isoquants are perfectly straight lines. This characteristic means the firm can freely substitute between inputs without suffering diminishing marginal returns. An example of a production function where labor and capital are perfect substitutes is Q = f(K, L) = 10K + 5L; 2 units of labor can always be substituted for 1 unit of capital without changing output, no matter how many units of either input the firm is already using. In this case, imagine that capital took the form of a robot that behaved exactly like a human when doing a task, but did the work twice as quickly as a human. Here, the firm can always substitute one robot for two workers or vice versa, regardless of its current number of robots or workers. This is true because the marginal product of labor is 5 (holding K constant, a 1-unit increase in L causes output to grow by 5). At the same time, holding L constant, a 1-unit rise in K will increase output by 10 units, making the MP_K equal to 10. No matter what levels of L and K the firm chooses,

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If inputs are perfect complements, isoquants have an “L” shape. This implies that using inputs in any ratio outside of a particular fixed proportion—that at the isoquants’ corners—yields no additional output. Cabs and drivers on a given shift are fairly close to perfect complements in the production of cab rides. Anything other than a 1 to 1 ratio of cabs to drivers is unlikely to produce any additional cab rides. If a cab company has, say, 30 drivers and 1 cab, it will not be able to offer any more rides than if it had 1 driver and 1 cab. Nor could it offer more rides if it had 1 driver and 30 cabs. Therefore, the production function would be Q = min(L, K), where “min” indicates that output (Q) is determined by the minimum level of either labor (L) or capital (K). Of course, the cab company could offer more rides if it had 30 drivers and 30 cabs, because this would preserve the 1 to 1 driver-to-cab ratio.4

Up to this point in the chapter, we have focused on various aspects of the production function and how quantities of inputs are related to the quantity of output. These aspects play a crucial part in determining a firm’s optimal production behavior, but the production function is only half of the story. As we discussed earlier, the firm’s objective is to minimize its costs of producing a given quantity of output. We’ve said a lot about how the firm’s choices of inputs affect its output but we haven’t talked about the costs of those choices. That’s what we do in this section.

The key concept that brings costs into the firm’s decision is the isocost line. An isocost line connects all the combinations of capital and labor that the firm can purchase for a given total expenditure on inputs. As we saw earlier, iso- is a prefix meaning “the same,” so “isocost” is a line that shows all of the input combinations that yield the same cost. Mathematically, the isocost line corresponding to a total expenditure level of C is given by

C = RK + WL

where R is the price (the rental rate) per unit of capital, W is the price (the wage) per unit of labor, and K and L are the number of units of capital and labor that the firm hires. It’s best to think of the cost of capital as a rental rate in the same type of units as the wage (e.g., per hour, week, or year). Because capital is used over a long period of time, we can consider R to be not just the purchase price of the equipment but also the economic user cost of capital. The user cost takes into account capital’s purchase price, as well as its rate of depreciation and the opportunity cost of the funds tied up in its purchase (foregone interest).

Figure 6.7 shows isocost lines corresponding to total cost levels of $50, $80, and $100, when the price of capital is $20 per unit and labor’s price is $10 per unit. There are a few things to notice about the figure. First, isocost lines for higher total expenditure levels are further from the origin. This reflects the fact that, as a firm uses more inputs, its total expenditure on those inputs increases. Second, the isocost lines are parallel. They all have the same slope, regardless of what total cost level they represent. To see why they all have the same slope, let’s first see what that slope represents.

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We can rewrite the equation for the isocost line in slope-intercept form, so that the value on the vertical axis (capital) is expressed as a function of the value on the horizontal axis (labor):

C = RK + WL

RK = C – WL

This means that the y-intercept of the isocost line is C/R, while the slope is the (negative of the) inputs’ price ratio, –W/R.

As is so often the case in economics, the slope tells us about tradeoffs at the margin. Here, the slope reflects the cost consequences of trading off or substituting one input for another. It indicates how much more of one input a firm could hire, without increasing overall expenditure on inputs, if it used less of the other input. If the isocost line’s slope is steep, labor is relatively expensive compared to capital. If the firm wants to hire more labor without increasing its overall expenditure on inputs, it is going to have to use a lot less capital. (Or if you’d rather, if it chose to use less labor, it could hire a lot more capital without spending more on inputs overall.) If the price of labor is relatively cheap compared to capital, the isocost line will be relatively flat. The firm could hire a lot more labor and not have to give up much capital to do so without changing expenditures.

Because of our assumption (Assumption 8) that the firm can buy as much capital or labor as it wants at a fixed price per unit, the slopes of isocost lines are constant. That’s why they are straight, parallel lines: Regardless of the overall cost level or the amount of each input the firm chooses, the relative tradeoff between the inputs in terms of total costs is always the same.

If these ideas seem familiar to you, it’s because the isocost line is analogous to the consumer’s budget line we saw in Chapter 4. The consumer’s budget line expressed the relationship between the quantities of each good consumed and the consumer’s total expenditure on those goods. Isocost lines capture the same idea, except with regard to firms and their input purchases. We saw that the negative of the slope of the consumer’s budget constraint was equal to the price ratio of the two goods, just as the negative of the slope of the isocost line equals the price ratio of the firm’s two inputs.

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Isocost Lines and Input Price Changes Just like budget lines for consumers, when relative prices change, the isocost line rotates. In our example, say labor’s price (W) rose from $10 to $20. Now if the firm only hired labor, it would only be able to hire half as much. The line becomes steeper as in Figure 6.8. The isocost line rotates because its slope is –W/R. When W increases from $10 to $20, the slope changes from to –1, and the isocost line rotates clockwise and becomes steeper.

Changes in the price of capital also rotate the isocost line. Figure 6.9 shows what happens to the $100 isocost line when the price of capital increases from $20 to $40 per unit, and the wage stays at $10 per unit of labor. If the firm hired only capital, it could afford half as much, so the slope flattens. A drop in the capital price would rotate the isocosts the other way.

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As we have discussed, a firm’s goal is to produce its desired quantity of output at the minimum possible cost. In deciding how to achieve this goal, a firm must solve a cost-minimization problem: It must achieve an objective given a constraint. The objective is the firm’s total cost of inputs, RK + WL. The firm chooses capital and labor inputs K and L to minimize these expenditures. What constrains the firm’s cost-minimizing decision? The quantity of output the firm has chosen to produce. The firm must hire enough capital and labor inputs to produce a certain level of output. The production function relates input choices to the quantity of output, so we can sum up a firm’s cost-minimization problem as follows: Choose K and L to minimize total costs, subject to the constraint that enough K and L must be chosen to produce a given quantity of output. (Remember that at this point in our analysis, quantity has already been chosen. Now it’s the firm’s task to figure out how to optimally produce that quantity.)

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We can combine information about the firm’s costs and the firm’s production function on the same graph to analyze its decision. The isocost lines show the firm’s costs. Then to represent the firm’s production function, we plot the isoquants. These tell us, for a given production function, how much capital and labor it takes to produce a fixed amount of output.

Before we work through a specific example, think about the logic of the firm’s cost-minimization problem. The firm’s objective is to minimize costs subject to the constraint that it has to produce a particular quantity of output, . The cost-minimization part means the firm wants to be on an isocost line that is as close to the origin as possible (because isocost lines closer to the origin correspond to lower levels of expenditure) but still on the isoquant that corresponds to . Figure 6.10 shows the isoquant for the firm’s desired output quantity . The firm wants to produce this quantity at minimum cost. How much capital and labor should it hire to do so? Suppose the firm is considering the level inputs shown at point A, which is on isocost line C_A. That point is on the isoquant, so the firm will produce the desired level of output. However, there are many other input combinations on the isoquant that are below and to the left of C_A. These all would allow the firm to produce but at lower cost than input mix A. Only one input combination exists that is on the isoquant but for which there are no other input combinations that would allow the firm to produce the same quantity at lower cost. That combination is at point B, on isocost C_B. There are input combinations that involve lower costs than B—for example, any combination on isocost line C_C—but these input levels are all too small to allow the firm to produce .

The total costs of producing a given quantity are minimized where the isocost line is tangent to the isoquant. Point B is the point of tangency of the isocost line C_B to the isoquant.

With this tangency property, we once more see a similarity to optimal consumer behavior, which was also identified by a point of tangency. Another important feature of the tangency result is that the isocost line and isoquant have the same slope at a tangency point (that’s the definition of a tangency point). We know what these slopes are from our earlier discussion. The slope of the isocost line is the negative of the relative price of the inputs, –W/R. For the isoquant, the slope is the negative of the marginal rate of technical substitution (MRTS_LK), which is equal to the ratio of MP_L to MP_K. The tangency therefore implies that at the combination of inputs that minimizes the cost of producing a given quantity of output (like point B), the ratio of input prices equals the MRTS:

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This condition has an important economic interpretation that might be easier to see if we rearrange the condition as follows:

The way we’ve written it, each side of this equation is the ratio of an input’s marginal product to its price (capital on the left, labor on the right). One way to interpret these ratios is that they measure the marginal product per dollar spent on each input, or the input’s “bang for the buck.” Alternatively, we can think of each of these ratios as the firm’s marginal benefit-to-cost ratio of hiring an input.

Why does cost minimization imply that each input’s benefit-to-cost ratio is equal? If the firm produced with an input bundle where this wasn’t true, if, say, , then the firm’s benefit-to-cost ratio for capital is higher than for labor. This would mean that the firm could replace some of its labor with capital while keeping its output quantity the same but reducing its total costs. Or if it wanted to, the firm could substitute capital for labor in a way that kept its total costs constant but raised its output. These options are possible because capital’s marginal product per dollar is higher than labor’s. If the sign of the inequality were reversed so , the firm could reduce the costs of producing its current quantity (or raise its production without increasing costs) by substituting labor for capital because the marginal product per dollar spent is higher for labor. Only when the benefit-to-cost ratios of all the firm’s inputs are the same is the firm unable to reduce the cost of producing its current quantity by changing its input levels.

Again, this logic parallels that from the consumer’s optimal consumption choice in Chapter 4. There, at the optimal point the marginal rate of substitution between goods equals the goods’ price ratio. Here, the analog to the MRS is the MRTS (the former being a ratio of marginal utilities, the latter a ratio of marginal products), and the price ratio is now the input price ratio.

There is one place where the parallel between consumers and firms is not exact. The budget constraint plays a big role in the consumer’s utility-maximization problem but doesn’t really exist in the firm’s production problem. As we discuss in later chapters, firms’ desires to maximize their profits lead to a particular quantity of output they want to produce. If for some reason they don’t have enough resources to pay for the inputs necessary to produce this quantity, then someone should always be willing to lend them the difference, because the lender and the firm can split the extra profits that result from producing the profit-maximizing output, making both parties better off. This outcome means that with well-functioning capital markets, firms should never be limited to a fixed amount of total expenditure on inputs in the same way a consumer is constrained to spend no more than her income. That’s the idea embodied in our Assumption 9.

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We’ve established that the cost-minimizing input combination is at the point of tangency between an isocost line and the isoquant that corresponds to the output quantity the firm wants to produce. In other words, a firm is producing that level of output at the lowest cost when the marginal product per dollar spent is equal across all inputs. Given this result, a useful question to ask is how changes in input prices affect the firm’s optimal input mix.

A Graphical Representation of the Effects of an Input Price Change We saw that differences in input costs show up as differences in the slopes of the isocost lines. A higher relative cost of labor (from an increase in W, decrease in R, or both) makes isocost lines steeper. Decreases in labor’s relative cost flatten them. A cost-minimizing firm wants to produce using the input combination where the slope of the isocost line equals the slope of the isoquant. This requirement means that when the inputs’ relative price changes, the point of tangency between the isocost line (now with a new slope) and the isoquant must also change. Input price changes cause the firm to move along the isoquant corresponding to the firm’s desired output level to the input combination where an isocost line is tangent to the isoquant.

Figure 6.11 shows an example of this. The initial input price ratio gives the slope of isocost line C₁. The firm wants to produce the quantity , so initially the cost-minimizing combination of inputs occurs at point A. Now suppose that labor becomes relatively more expensive (or equivalently, capital becomes relatively less expensive). This change causes the isocost lines to become steeper. With the steeper isocost lines, the point of tangency shifts to point B. Therefore, the increase in the relative cost of labor causes the firm to shift to an input mix that has more capital and less labor than before.

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The implication of this outcome makes sense: If a firm wants to minimize its production costs and a particular input becomes relatively more expensive, the firm will substitute away from the now more expensive input toward the relatively less expensive one.

This is why we sometimes observe different production methods used to make the same products. For example, if you spend a growing season observing a typical rice-farming operation in Vietnam, you will see dozens of workers tending small paddies attending plants one-by-one and using only basic farm tools. If you visit a rice farm in Texas, on the other hand, a typical day might involve a single farmer operating various types of large machinery that do the same tasks as the Vietnamese workers. Both farms sell rice to a world market. A key reason for the differences in their production methods is that the relative prices of capital and labor are very different. In Vietnam, labor is relatively cheap compared to capital. Therefore, the tangency of Vietnamese rice farms’ isoquants and isocost lines is at a point such as point A in Figure 6.11. At point A, a lot of labor and only a little capital are used to grow rice. In Texas, on the other hand, labor is relatively expensive. This implies steeper isocost lines for Texas farms, making their cost-minimizing input mix much more capital-intensive, as at point B.

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