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Chapter 6

Costs

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Table of Contents

6.1

6.2

6.3

6.4

6.5

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The Nature of Costs

Short-Run Costs

Long-Run Costs

The Learning Curve

The Costs of Producing Multiple Goods

Introduction

Managerial Problem

In the United States, firms use relatively capital-intensive technology.

Will that same technology be cost minimizing if firms move their production abroad?

Solution Approach

First, a firm must determine which production processes are technically efficient so that production has no waste. Second, a firm should pick from these technically efficient processes the one that is also economically efficient. By minimizing costs, a firm can increase its profit.

Empirical Methods

When considering costs, a good manager includes opportunity costs.

To minimize costs, a manager should distinguish short-run from long-run costs.

Firms may reduce costs overtime based on experience or its learning curve.

If a firm produces several goods, individual cost may depend on the cost of producing multiple goods.

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6.1 The Nature of Costs

Financial accounting statements correctly measure costs for tax purposes and to meet other legal requirements.

To make sound managerial decisions, good managers need more information and a perspective about explicit and implicit costs.

Explicit costs are direct, out-of-pocket payments for labor, capital, energy, and materials.

Implicit costs reflect only a foregone opportunity rather than explicit, current expenditure.

Opportunity Costs

The opportunity cost of a resource is the value of the best alternative use of that resource.

Maoyong owns and manages a firm. He pays himself only $1k per month but could work for another firm and make $11k per month. Working for another firm is the best alternative use of his time, so his opportunity cost of time is $11k.

Assume monthly revenue is $49k and explicit costs are $40k, including Maoyong’s monthly wage. The accounting profit is $9k and Maoyong collects $10k per month (profit + wage). However, his opportunity cost is $11k. So, he incurs an economic loss of $1k.

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6.1 The Nature of Costs

Costs of Durable Inputs

Durable inputs are usable for a long period, perhaps for many years.

Capital such as land, buildings, or equipment are durable inputs.

Two problems with the costs of durable inputs

How to allocate the initial purchase cost over time?

What to do if the value of the capital changes over time?

Solutions to the calculation of durable inputs (truck example)

If there is a rental market: The accountant may expense the truck’s purchase price or may amortize it over the life of the truck, following IRS rules. The firm’s opportunity cost of using the truck is the amount that the firm would earn if it rented the truck to others.

If there is no rental market: The opportunity cost of capital of using the truck a year would be the interest forgone in a year.

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6.1 The Nature of Costs

Sunk Costs

Sunk cost is a past expenditure that cannot be recovered.

If an expenditure is sunk, it is not an opportunity cost. So we should not consider it for managerial decisions.

However, sunk costs appear in financial accounts.

Managers Should Ignore Sunk Costs

A firm paid $300k for a parcel of land but the market value is now $200k. If the firm builds a plant on this land, the value for the firm becomes $240k.

Is it worth carrying out production on this land or should the land be sold for its market value of $200k?

The land’s opportunity cost is $200k and the market value loss of $100k is a sunk cost. The sunk cost cannot be recovered and should not be considered in the decision. The values to compare are $240 versus $200. Certainly, the firm should carry out production on this land.

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6.2 Short-Run Costs

All firms use the same basic cost measures, and these should be based on input’s opportunity costs.

Fixed Cost (F) does not vary with the level of output; includes expenditures on land, office space, production facilities, and other overhead expenses; are often sunk costs, but not always.

Variable Cost(VC) changes as the quantity of output changes; refers to the costs of variable inputs.

Total Cost (C) is the sum of fixed and variable costs.

F and VC should be based on inputs’ opportunity costs.

Common Measures of Cost: Fixed, Variable and Total

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Fixed Cost

Variable Cost

Total Cost

C = F + VC

6.2 Short-Run Costs

Average Fixed Cost (AFC) falls as output rises because the fixed cost is spread over more units.

Average Variable Cost (AVC) or variable cost per unit of output may either increase or decrease as output rises.

Average Cost (AC) or average total cost may either increase or decrease as output rises.

In Table 6.1, AFC falls with output and AVC eventually rises with output. AC falls until output of 8 units and then rises.

Average Cost

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Average Fixed Cost

AFC = F/q

Average Variable Cost

AVC = VC/q

Average Cost

AC = C/q = AFC + AVC

6.2 Short-Run Costs

Table 6.1 How Cost Varies with Output

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6.2 Short-Run Costs

Marginal Cost

MC = ΔC/Δq

Marginal cost (MC) is the amount by which a firm’s cost changes if the firm produces one more unit of output

∆C is the change in cost when the change in output, ∆q, is 1 unit.

MC = ΔVC/Δq

Marginal cost also equals the change in variable cost from a one-unit increase in output.

ΔVC/ is the change in variable cost when the change in output, ∆q, is 1 unit.

In Table 6.1, if the firm increases output from 2 to 3 units, the marginal cost is $20.

Marginal Cost using Calculus: MC = dC/dq = dVC/dq

Marginal cost is the rate of change of cost as we make an infinitesimally small change in output.

MC=dVC/dq because dF/q=0.

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6.2 Short-Run Costs

Cost Curves

Panel a of Figure 6.1 shows the VC, F, and C curves that correspond to Table 6.1

The fixed cost curve, F, is a horizontal line at $48.

The variable cost curve, VC, is zero at zero units of output and rises with output.

The total cost curve, C, is the vertical sum of the VC and F curves, so it is $48 higher than the VC curve at every output level. VC and C curves are parallel.

Panel b of Figure 6.1 shows the AFC, AVC, AC, and MC curves.

The marginal cost curve, MC, cuts the average variable cost, AVC, and average cost, AC, curves at their minimums.

The height of the AC curve at point a equals the slope of the line from the origin to the cost curve at A.

The height of the AVC at b equals the slope of the line from the origin to the VC curve at B.

The height of the MC is the slope of either the C or VC curve at that quantity

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6.2 Short-Run Costs

Figure 6.1 Cost Curves

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6.2 Short-Run Costs

Production Functions and the Shapes of Short-Run Costs Curves

The production function determines the shape of a firm’s cost curves.

In the short run, diminishing marginal returns to labor determine the shape of the production function.

the firm increases output by using more labor. However, each extra worker increases output by a smaller amount.

The production function determines the shape of the variable cost curve and its related curves.

As output increases, variable cost increases more than proportionally because of diminishing marginal returns.

The production function determines the shape of the marginal cost, average variable cost, and average cost curves.

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6.2 Short-Run Costs

The Variable Cost Curve

If input prices are constant, the firm’s production function determines the shape of the variable cost curve.

The VC and the total product curve have the same shape, as it is shown in Figure 6.2.

The firm faces a constant input price for labor, $10 per hour.

The total product curve uses the horizontal axis measuring hours of work.

The variable cost curve uses the horizontal axis measuring labor cost: VC = wL.

The VC of 6 units of output is $240 ($10*24).

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6.2 Short-Run Costs

Figure 6.2 Variable Cost and Total Product

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6.2 Short-Run Costs

The Marginal Cost Curve

MC = ΔVC/Δq

Marginal cost (MC) is the change in variable cost as output increases by 1 unit.

The MC curve is U-shaped because of diminishing marginal returns.

MC = ΔVC/Δq = w(∆L/∆q)

In the short run, capital is fixed. So, the change in variable cost as output increases by 1 unit must be the change in the cost of labor.

Marginal cost equals the wage times the extra labor necessary to produce 1 more unit of output.

In Figure 6.2, to increase Q from 5 to 6 units, MC = $40 (4 x $10).

MC = w/MPL

Remember MPL = Δq/ΔL. So, ΔL/Δq is just the inverse of MPL.

Marginal cost equals wage divided by marginal product of labor.

Marginal product of labor and marginal cost move in opposite directions as output changes.

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6.2 Short-Run Costs

The Average Cost Curves

AVC = VC/q

Average variable cost is just the variable cost divided by output.

Given the shape of the VC is determined by the production function, the diminishing marginal returns to labor also determines the shape of the AVC cost curve.

So, the VC and AVC curves are both U-shaped.

AVC = VC/q = wL/q

In the short run, capital is fixed. So the variable cost is wL.

The average variable cost is cost of labor divided by output.

AVC = w/APL

Remember APL = q/L. So, L/q is just the inverse of APL.

Average cost equals wage divided by average product of labor.

In Figure 6.2, at 6 units of Q, the AVC = $10/0.25 = $40

Average product of labor and average variable cost move in opposite directions as output changes.

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6.2 Short Run Costs

Short-Run Cost Summary

In the short run, the cost associated with fixed inputs is fixed, while the cost from inputs that can be adjusted is variable.

Given that input prices are constant, the shapes of the variable cost and the cost-per-unit curves are determined by the production function.

Where there are diminishing marginal returns, the variable cost and cost curves become relatively steep as output increases, so the average cost, average variable cost, and marginal cost curves rise with output.

The average cost and average variable cost curves fall when marginal cost is below them and rise when marginal cost is above them, so the marginal cost cuts both these average cost curves at their minimum points.

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6.3 Long Run Costs

In the long run, the firm adjusts all its inputs so that its cost of production is as low as possible.

The firm can change its plant size, design, build new machines, and otherwise adjust inputs that were fixed in the short run.

Fixed costs are avoidable in the long run. They are not sunk costs, as they are in the short run. For instance, the rent a restaurant pays is a fixed cost and this rent can be avoided in the long run if the restaurant does not renew the rental agreement.

Input Choice

Technically and Economically Efficient

From among the technically efficient combinations of inputs that can be used to produce a given level of output, a firm wants to choose that bundle of inputs with the lowest cost of production, which is the economically efficient combination of inputs.

To do so, the firm combines information about technology from the isoquant with information about the cost of production.

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6.3 Long Run Costs

The Isocost Line: = w L + r K

An isocost represents all the combinations of inputs that have the same total cost, .

Five of the many combinations of labor and capital that the firm can buy for $200 are in Table 6.2.

In Figure 6.3, the $200 isocost line represents all the combinations of labor and capital that the firm can buy for $200.

Properties of Isocosts

The points at which the isocost lines hit the capital and labor axes depends on the firm’s cost, and on the input prices.

Isocost lines that are farther from the origin have higher costs than those closer to the origin.

The slope of each isocost line is the same: ∆K/∆L = –w/r, the rate at which the firm can trade capital for labor in input markets.

All these properties can be verified by looking the isocots in Figure 6.3

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6.3 Long Run Costs

Table 6.2 Bundles of Labor and Capital That Cost the Firm $200

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6.3 Long Run Costs

Figure 6.3 A Family of Isocost Lines

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6.3 Long Run Costs

Combining Cost and Production Information

The firm minimizes its cost by using the combination of inputs on the isoquant that is on the lowest isocost line that touches the isoquant.

Isocost and Isoquant Combined: Graph Analysis

In Figure 6.4, the lowest possible isoquant that will allow the beer manufacturer to produce 100 units of output is tangent to the $2,000 isocost line.

At x, the bundle of inputs are L = 50 workers and K = 100 units of capital.

At x, the isocost is tangent to the isoquant, so the slope of the isocost, –w/r = –3, equals the slope of the isoquant, which is the negative of the marginal rate of technical substitution.

Notice, y and z also produce 100 units of output but at a cost of $3,000. The x input combination is economically efficient.

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6.3 Long Run Costs

Figure 6.4 Cost Minimization

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6.3 Long Run Costs

There are three equivalent rules to minimize costs in the long run: the lowest isocost rule, the tangency rule and the last-dollar rule.

The Lowest Isocost Rule

The firm minimizes its cost by using the combination of inputs on the isoquant that is on the lowest isocost line that touches the isoquant.

The Tangency Rule: MRTS = - w/r

At the minimum-cost bundle, x, the isoquant is tangent to the isocost line. The slope of the isoquant (MRTS) and the slope of the isocost are equal.

The Last-Dollar Rule: (MPL/w) = (MPK/r)

Cost is minimized if inputs are chosen so that the last dollar spent on labor adds as much extra output as the last dollar spent on capital. Thus, spending one more dollar on labor at x gets the firm as much extra output as spending the same amount on capital.

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6.3 Long Run Costs

Factor Price Changes

How should the firm change its behavior if the cost of one of the factors changes?

If one factor becomes relatively cheaper, the firm should substitute factors considering the slopes of the isoquant and isocost curves.

Graph Analysis:

In Figure 6.5, the initial wage = $24 and the rental rate of capital = $8. The lowest isocost line ($2,000) is tangent to the q = 100 isoquant at x(L = 50, K = 100).

When the wage falls from $24 to $8, the isocost lines become flatter: Labor is relatively less expensive than capital now.

The slope of the isocost lines falls from –w/r = –24/8 = –3 to –8/8 = –1.

The new lowest isocost line ($1,032) is tangent at v (L = 77, K = 52).

Thus, when the wage falls, the firm uses more labor and less capital to produce a given level of output, and the cost of production falls from $2,000 to $1,032.

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6.3 Long Run Costs

Figure 6.5 Effect of a Change in Factor Price

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6.3 Long-Run Costs

The Shapes of Long-Run Cost Curves

The shapes of the long-run AC and MC curves depend on the shape of the long-run TC curve.

In Figure 6.6, panel a, the long-run cost curve rises less rapidly than output at levels below q* and more rapidly at higher q* levels. As a consequence, the MC and AV costs curves are U-shaped. Why?

In the short-run, U-shaped forms are explained by the influence of average fixed costs and diminishing marginal returns. Both arguments are not valid in the long-run.

In the long run, returns to scale determine the shape of the production function, and the production function, in turn, determines the shape of the LRAC curve and other cost curves.

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6.3 Long-Run Costs

Figure 6.6 Long-Run Cost Curves

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6.3 Long-Run Costs

The LRAC curve must be U-shaped if a production function has

increasing returns to scale at low levels of output,

constant returns to scale at intermediate levels of output, and

decreasing returns to scale at high levels of output.

LRAC curves can have many different shapes depending whether the production process has economies of scale or diseconomies of scale.

A cost function shows economies of scale if the AC falls as output expands. Increasing returns to scale in production are sufficient for it.

If an increase in output has no effect on AC, the production process has no economies of scale.

A cost function shows diseconomies of scale if AC rises when output increases.

All these relationships are illustrated in Table 6.3.

Perfectly competitive firms typically have U-shaped LRAC curves.

Noncompetitive markets may be U-shaped, L-shaped, everywhere downward sloping, everywhere upward sloping or have other shapes.

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6.3 Long-Run Costs

Table 6.3 Returns to Scale and Long-Run Costs

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6.3 Long Run Costs

LRAC as the Envelope of SRAC Curves

The long-run average cost is always equal to or below the short-run average cost. Any input combination in the short run is also available in the long run. However, changing capital levels to reduce costs are only available in the long run.

In the long run, the firm chooses the plant size that minimizes its cost of production, so it picks the plant size that has the lowest average cost for each possible output level.

Graph Analysis

At q1, in Figure 6.7, the firm opts for the small plant size, whereas at q2, it uses the medium plant size.

If there are only three possible plant sizes, with short-run average costs SRAC1, SRAC2, and SRAC3, the long-run average cost curve is the solid, scalloped portion of the three short-run curves (envelope curve).

LRAC is the smooth and U-shaped long-run average cost curve.

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6.3 Long Run Costs

Figure 6.7 Long-Run Average Cost as the Envelope of Short-Run Average Cost Curves

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6.4 The Learning Curve

Average cost may fall over time because of increasing returns to scale, technological progress and learning by doing.

Learning by doing refers to the productive skills and knowledge that workers and managers gain from experience.

Workers add speed with practice. Managers learn how to organize production more efficiently, assign tasks based on worker’s skills, and reduce inventory costs. Engineers optimize product designs with experimentation.

For these and other reasons, the average cost of production tends to fall over time, and the effect is particularly strong with new products.

The learning curve is the relationship between average costs and cumulative output. The cumulative output is the total number of units of output produced since the product was introduced. Panel a of Figure 6.8 shows the learning curve for Intel central processing unit.

If a firm is operating in the economies of scale section of its average cost curve, expanding output lowers its average cost for two reasons: economies of scale, and learning by doing.

In panel b of Figure 6.8, economies of scale are seen in each isocost line (AC1, AC2 and AC3), and learning by doing by the distance between those lines.

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Figure 6.8 Learning by Doing

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6.5 The Costs of Producing Multiple Goods

If a firm produces two or more goods that are linked by a single input, the cost of one good may depend on the output level of another.

For example, cattle provide beef and hides.

A cost function exhibits economies of scope if it is less expensive to produce goods jointly than separately.

It is less expensive to produce beef and hides together than separately, so there are economies of scope.

Economies of Scope: SC = [C(q1,0) + C(0,q2) - C(q1, q2)]/C(q1, q2)

The costs of producing q1 of the first good, q2 of the second good and both goods together are C(q1, 0), C(0, q2) and C(q1, q2).

If SC is zero, C(q1,0) + C(0,q2) = C(q1, q2). There are no economies of scope.

If SC is positive, it is cheaper to produce goods jointly. There are economies of scope.

If SC is negative, it is cheaper to produce goods separately. There are diseconomies of scope.

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Managerial Solution

Managerial Problem

In the United States, firms use a relatively capital-intensive technology

Will that same technology be cost minimizing if firms move their production abroad?

Solution

The answer depends on relative factor prices and whether the firm’s isoquant is smooth.

If the isoquant is smooth, even a slight difference in relative factor prices will induce the firm to shift along the isoquant and use a different technology with a different capital-labor ratio.

If the isoquant has kinks, the firm will use a different technology only if the relative factor prices differ substantially.

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Additional Images

Figure 6.9 Technology Choice

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Figure 6.9 Technology Choice

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