Intermediate Microeconomics ECN321 - Topic 7 by Charbel BASSIL
Outline
« The Cost of Production » Microeconomics Seventh Edition by Pindyck and Robinfeld
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Measuring Cost: Which Costs Matter?
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Cost in the Short Run
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Cost in the Long Run
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Long-Run versus Short-Run Cost Curves
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Production with Two Outputs-Economies of Scope
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Dynamic Changes in Costs-The Learning Curve
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Estimating and Predicting Cost
MEASURING COST: WHICH COSTS MATTER? 1- Economic Cost versus Accounting Cost
Economists think of cost differently from financial accountants. • Financial accountants take a retrospective view. Accounting cost includes the actual expenses plus depreciation expenses for capital equipment. • Economists take a forward-looking view. They are concerned with the allocation of scarce resources. Economic cost includes the cost to a firm of utilizing economic resources in production, including opportunity cost.
MEASURING COST: WHICH COSTS MATTER? 2- Opportunity cost
• Opportunity cost is the cost associated with opportunities that are forgone when a firm’s resources are not put to their best alternative use. Example · A firm that owns a building and pays no rent for office space. · Working in a family business without a pay. • Opportunity cost should be taken into account when making economic decisions.
MEASURING COST: WHICH COSTS MATTER? 3- Sunk cost
• Sunk cost is the expenditure that has been made and cannot be recovered. Example · · · ·
A ticket for a bad movie. A bad meal. A prospective sunk cost. The cost of R&D
• Sunk cost should be ignored when making future economic decisions because it cannot be recovered. • If an equipment has no alternative use and induces a sunk cost, its opportunity cost is zero and shouldn’t be included as part of the economic costs. • If an equipment has alternative use and induces a sunk cost it involves an economic cost, namely the opportunity cost of using it.
MEASURING COST: WHICH COSTS MATTER? 4- Fixed costs and variable costs
• Total cost (TC or C) is the total economic cost of production, consisting of fixed and variable costs. • Variable cost (VC) is the cost that varies as output varies. Example: Expenditures for wages, salaries and raw materials, . . . • Fixed cost (FC) is the cost that does not vary with the level of output and that can be eliminated only by shutting down. Example: Expenditures for plant maintenance, insurance, heat and electricity, . . . • Shutting down doesn’t necessarily mean going out of business. Example: A clothing company closing one of its several factories.
MEASURING COST: WHICH COSTS MATTER? 4- Fixed costs and variable costs
How do we know which costs are fixed and which are variable? The answer depends on the time horizon. • Over a very short time horizon-say, a few months-most costs are fixed. Over such a short period, a firm is usually obligated to pay for contracted shipments of materials. • Over a very long time horizon-say, ten years-nearly all costs are variable. Workers and managers can be laid off (or employment can be reduced by attrition), and much of the machinery can be sold off or not replaced as it becomes obsolete and is scrapped. • Fixed costs affect the firm’s decisions, whereas sunk costs do not.
MEASURING COST: WHICH COSTS MATTER? 5- Marginal and average costs
• Marginal cost (MC) (incremental cost) is the increase in cost resulting from the production of one extra unit of output. MC =
∆VC ∆q
=
∆TC ∆q
• Average total cost (ATC) is the firm’s total cost divided by its level of output. ATC =
TC q
= AFC + AVC
• Average fixed cost (AFC) is the fixed cost divided by the level of output. AFC =
FC q
• Average variable cost (AVC) is the variable cost divided by the level of output. AVC =
VC q
COST IN THE SHORT RUN 1- The determinants of short-run cost
• Suppose that labor is the only input. • How do VC and TC increase with output in the short run? The rate at which these costs increasse depends on the extent to which production involves diminishing marginal returns to variable inputs. I
We know that diminishing marginal returns to labor occur when the marginal product of labor is decreasing.
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If the marginal product of labor is decreasing rapidly, to produce more output, the firm must hire more labor. In this case, TC and VC increase at a high rate when output increases.
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If the marginal product of labor is decreasing slowly, in this case, TC and VC will not rise so quickly when the rate of output increases.
COST IN THE SHORT RUN 1- The determinants of short-run cost
What is the relation between production and cost? MC
= =
∆TC ∆q ∆VC ∆FC ∆VC ∆(VC + FC ) = + = +0 ∆q ∆q ∆q ∆q
(1)
Suppose that the labor wage is w . Thus, ∆VC = w ∆L
(2)
We substitute equation (14) in equation (13) to obtain MC =
w ∆L ∆q
(3)
COST IN THE SHORT RUN 1- The determinants of short-run cost
Marginal product of labor is ∆q ∆L
(4)
1 ∆L = MPL ∆q
(5)
MPL = Thus
We substitute equation (5) in equation (16) to obtain MC =
w MPL
When there is only one variable input, MC is equal to the price of the input divided by its MP.
(6)
COST IN THE SHORT RUN 1- The determinants of short-run cost
• Diminishing marginal returns means that the marginal product of labor declines as the quantity of labor employed increases. • As a result, when there are diminishing marginal returns, we can see from equation (6) that marginal cost increases as output increases (output levels from 4 through 11 in the table).
COST IN THE SHORT RUN 2- The shapes of the cost curves
Figure: A firm’s costs
COST IN THE SHORT RUN 2- The shapes of the cost curves
• TC is the sum of FC and VC . • ATC is the sum of AVC and AFC . • When MC < AC the AC curves fall. • When MC > AC the AC curves rises. ⇒ MC crosses the AVC and ATC curves at their minimum points.
COST IN THE SHORT RUN 2- The shapes of the cost curves
Formal proof that MC crosses the AVC at its minimum: AVC =
TVC TC − FC = Q Q
(7)
To minimize AVC we derive it with respect to Q: δAVC Q
= =
MC × Q − TC + FC Q2 MC × Q − TVC Q2
(8)
COST IN THE SHORT RUN 2- The shapes of the cost curves
Then we equate the derivative to zero: δAVC Q MC TVC − Q Q2 MC AVC × Q − Q Q2 AVC
= 0 = 0 = 0 = MC
(9)
COST IN THE SHORT RUN 2- The shapes of the cost curves
Formal proof that MC crosses the ATC at its minimum: ATC =
TC TVC + FC = Q Q
(10)
To minimize ATC we derive it with respect to Q: δATC Q
= = = = =
MC × Q − TVC − FC Q2 AVC FC MC − − 2 Q Q Q MC AVC AFC − − Q Q Q MC AVC + AFC − Q Q MC ATC − Q Q
(11)
COST IN THE SHORT RUN 2- The shapes of the cost curves
Then we equate the derivative to zero: δATC Q ATC
= 0 = MC
(12)
COST IN THE SHORT RUN 2- The shapes of the cost curves
Figure: Cost curves for a firm
COST IN THE SHORT RUN 2- The shapes of the cost curves
Minimum point of the ATC curve lies above and to the right of the minimum point of the AVC curve. Why? I
Consider the line drawn from origin to point A. The slope of the line measures the AVC .
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Because the slope of the VC curve is the MC , the tangent to the VC curve at A is the marginal cost of production when output is 7.
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At A, this marginal cost is equal to the average variable cost because average variable cost is minimized at this output.
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We can do the same to proove that the minimum of the ATC is equal to MC and that it is above and to the right of the minimum point of the AVC curve.
COST IN THE LONG RUN 1- The firms’ problem in the long run
• In the long run, a firm can expand its capacity by using more inputs (K and L). The analysis changes from the short run. • In the long run, firms choose the combination of inputs that minimize its cost of producing a given output. • Suppose two inputs capital (K ) and labor (L). • What are the prices of (K ) and (L)?
COST IN THE LONG RUN 2- The price of capital
• Firms can rent or purchase capital (capital goods or equipment). Suppose here that capital is purchased. Example: Delta Airlines bought a Boeing 777 airplane ($150 million). The purchase price was allocated or amortized across the life of the airplane (30 years, $5 million per year.) • If the firm didn’t purchase the airplane it could earn an interest on its $150 million. Thus, buying the airplane generates an opportunity cost.
COST IN THE LONG RUN 2- The price of capital
• The user cost of capital is the annual cost of owning and using a capital asset (i.e airplane) instead of selling it or never buying it. It is equal to economic depreciation plus forgone interest. Example: · In the first year: $5 million + 0.10 × $150 million = $20 million . · In ten years: $5 million + 0.10 × $100 million = $15 million. • We can express the user cost of capital as rate: r = Depreciation rate + Interest rate Example: If depreciation rate is 1/30 = 3.33 then, r = 3.33 + 10 = 13.33 • The price of capital is its user cost, given by: r = Depreciation rate + Interest rate
COST IN THE LONG RUN 2- The price of capital
• What if capital is renteded? In this case, the price of capital is its rental rate -the cost per year for renting a unit of capital. • If the capital market is competitive, the rental rate should be equal to the user cost, r . Why? Firms that own capital expect to earn a competitive return when they rent it. This competitive return is the user cost of capital. • Capital that is purchased can be treated as though it were rented at a rental rate equal to the user cost of capital.
COST IN THE LONG RUN 3- The price of labor
The price of labor is the wage rate, w , in a competitive labor market.
COST IN THE LONG RUN 4- The isocost line
• Isocost line is a graph showing all possible combinations of labor and capital that can be purchased for a given total cost. • Total cost C of producing any particular output is given by the sum of the firm’s labor cost wL and its capital cost rK : C = wL + rK
(13)
• From equation (13) we deduce the isocost line: K=
w C − ( )L r r
(14)
w • It follows that the isocost line has a slope of ∆K ∆L = −( r ), which is the ratio of the wage rate to the rental cost of capital.
COST IN THE LONG RUN 4- The cost minimizing input choice
Figure: Producing a given output at minimum cost
COST IN THE LONG RUN 5- Choosing inputs
• We showed that all the time: MRTS = −
MPL ∆K = ∆L MPK
(15)
• Only at the optimum: MPL w = MPK r
(16)
• We can rewrite equation (16) as follows: MPL MPK = w r
(17)
L • MP w is the additional output that results from spending an additional dollar for labor. K • MP is the additional output that results from spending an additional K dollar for capital.
COST IN THE LONG RUN 6- Change in total cost
• When the expenditure on all inputs increases (C ), the slope of the isocost line does not change because the price of the inputs have not changed but the intercept changes. • The isocost line shifts upward.
COST IN THE LONG RUN 7- Change in the price of labor
• If the price of labor increases while total cost (C ) is fixed, the slope of the isocost line (− wr ) increases in magnitude and the isocost line rotates inward. • If the price of labor decreases while total cost (C ) is fixed, the slope of the isocost line (− wr ) decreases in magnitude and the isocost line rotates outward. • Firms substitute capital for labor.
COST IN THE LONG RUN 7- Change in the price of labor and C
If the price of labor and C increase, the slope of the isocost line (− wr ) increases in magnitude so does the intercept. In this case, the isocost line shifts upward and rotate. It becomes steeper.
Figure: Change in the labor price
Firms substitute capital for labor.
COST IN THE LONG RUN 8- Cost minimization with varying output levels
Suppose w = $10/hour and r = $20/hour . The isocost line is: C = $10L + $20K Lets draw three isocost lines representing a cost of $1000, $2000 and $3000.
(18)
COST IN THE LONG RUN 8- Cost minimization with varying output levels
COST IN THE LONG RUN 8- Cost minimization with varying output levels
• Points A, B and C are the optimal points for the firm (maximum production with the lowest cost.) • The curve that relates A, B and C is the expansion path. • The expansion path describes the combinations of labor and capital that the firm will choose to minimize costs at each output level.
COST IN THE LONG RUN 9- The expansion path and long-run costs
• Long run total cost is the relation between output and total cost. • To move from the expansion path to the cost curve, we follow three steps: I
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Choose an output level represented by an isoquant. Then find the point of tangency of that isoquant with an isocost line. From the chosen isocost line determine the minimum cost of producing the output level that has been selected. Graph the output-cost combination.
• Because we have constant returns to scale, the long-run total cost is a straight line.
COST IN THE LONG RUN 9- The expansion path and long-run costs
LONG-RUN vs SHORT-RUN COST CURVES 1- The inflexibility of short-run production
• In the long run all inputs are flexible. In the short run some inputs are fixed (inflexible). • This flexibility allows the firm to produce at a lower average cost than in the short run. Why?
LONG-RUN vs SHORT-RUN COST CURVES 1- The inflexibility of short-run production
A
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LONG-RUN vs SHORT-RUN COST CURVES 1- The inflexibility of short-run production
• If capital is fixed at K1 in the short run: I
to produce q1 the firm minimizes cost by choosing labor equal to L1 . Its cost C0 is given by the isocost line AB.
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to produce q2 the firm chooses labor equal to L3 . Its cost C3 is given by the isocost line EF .
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and the short-run expansion path is the line OAP.
• If capital wasn’t fixed at K1 as in the long run: I
to produce q2 the firm chooses K2 and L2 . Its cost C2 is given by the isocost line CD.
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the firm’s long-run expansion path is the straight line from the origin that corresponds to the expansion path.
• Conclusion: C3 > C2
LONG-RUN vs SHORT-RUN COST CURVES 2- Long-run average and marginal cost
Definitions: • Long-run average cost curve (LAC) is a curve relating average cost of production to output when all inputs, including capital, are variable. • Short-run average cost curve (SAC) is a curve relating average cost of production to output when level of at least one input is fixed (i.e capital). • Long-run marginal cost curve (LMC) is a curve showing the change in long-run total cost as output is increased incrementally by 1 unit.
LONG-RUN vs SHORT-RUN COST CURVES 2- Long-run average and marginal cost
• The shape of the LAC and LMC is determined by the scale of the firm’s operation. I
With constant returns to scale LAC and LMC are constant, thus the curves are horizontal.
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With increasing returns to scale LAC and LMC fall with output.
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With decreasing returns to scale LAC and LMC increase with output.
• Like the SAC, the LAC is U-shaped, but the source of the U-shape is increasing and decreasing returns to scale, rather than diminishing returns to a factor of production.
LONG-RUN vs SHORT-RUN COST CURVES 2- Long-run average and marginal cost
Same comment as before
LONG-RUN vs SHORT-RUN COST CURVES 3- Economies and diseconomies of scale
As output increases, the firm’s average cost of producing that output is likely to decline, at least to a point. This can happen for the following reasons: I
If the firm operates on a larger scale, workers can specialize in the activities at which they are most productive.
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Scale can provide flexibility. By varying the combination of inputs utilized to produce the firm’s output, managers can organize the production process more effectively.
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The firm may be able to acquire some production inputs at lower cost because it is buying them in large quantities and can therefore negotiate better prices. The mix of inputs might change with the scale of the firm’s operation if managers take advantage of lower-cost inputs.
LONG-RUN vs SHORT-RUN COST CURVES 3- Economies and diseconomies of scale
At some point, however, it is likely that the average cost of production will begin to increase with output. There are three reasons for this shift: I
At least in the short run, factory space and machinery may make it more difficult for workers to do their jobs effectively.
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Managing a larger firm may become more complex and inefficient as the number of tasks increases.
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The advantages of buying in bulk may have disappeared once certain quantities are reached. At some point, available supplies of key inputs may be limited, pushing their costs up.
LONG-RUN vs SHORT-RUN COST CURVES 3- Economies and diseconomies of scale
• As we have seen, the firm’s costs depend on its production. Thus, on its scale operation. • We define: I
Economies of scale: the situation in which output can be doubled for less than a doubling of cost.
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Diseconomies of scale: the situation in which a doubling of output requires more than a doubling of cost.
• Do not confound returns to scale (inputs are used in constant proportions) and economies of scale (input proportions are variable).
LONG-RUN vs SHORT-RUN COST CURVES 3- Economies and diseconomies of scale
A firm can exhibit constant or increasing returns to scale and have economies of scale. Example: A dairy farm I
Increasing Returns to Scale: Output more than doubles when the quantities of all inputs are doubled.
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Economies of Scale: A doubling of output requires less than a doubling of cost.
LONG-RUN vs SHORT-RUN COST CURVES 3- Economies and diseconomies of scale
Economies of scale are measured in terms of a cost-output elasticity: Ec =
∆C C ∆q q
=
∆C ∆q C q
=
MC AC
(19)
It is the percentage change in the cost of production resulting from a 1% increase in output. I
If EC > 1 then MC > AC : diseconomies of scale.
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If EC < 1 then MC < AC : economies of scale.
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If EC = 1 then MC = AC : nothing, we may have constant returns if inputs have increased proportionately,
LONG-RUN vs SHORT-RUN COST CURVES 4- Relationship between SR and LR cost
Figure: Long-run cost with economies and diseconomies of scale
LONG-RUN vs SHORT-RUN COST CURVES 4- Relationship between SR and LR cost
• In the long run, the firm can change the size of its plant. It will always choose the plant that minimizes the average cost of production. • The LAC is given by the crosshatched portions of the SAC curves. • If there are more than three plant sizes, the LAC is the envelope of the SAC curves. The LAC exhibits ecnomies of scale initially and diseconomies of scale at higher output levels. • LMC is not the envelope of the SMC curves. It applies to all plant sizes. Each point on the LMC is the SMC associated with the most-efficient plant.
PRODUCTION WITH TWO OUTPUTS-ECONOMIES OF SCOPE 1- Product transformation curves
• The product transformation curve is a curve showing the various combinations of two different outputs (products) that can be produced with a given set of inputs. • It has a negative slope because to produce more of one output, the firm must give up some of the other output.
PRODUCTION WITH TWO OUTPUTS-ECONOMIES OF SCOPE 1- Product transformation curves
Figure: Product transformation curve
PRODUCTION WITH TWO OUTPUTS-ECONOMIES OF SCOPE 2- Economies and diseconomies of scope
• Economies of scope is the situation in which joint output of a single firm is greater than output that could be achieved by two different firms when each produces a single product. • Diseconomies of scope is the situation in which joint output of a single firm is less than output that could be achieved by separate firms when each produces a single product. • There is no direct relation between economies of scale and economies of scope.
PRODUCTION WITH TWO OUTPUTS-ECONOMIES OF SCOPE 3- The degree of diseconomies of scope
• Degree of economies of scope (SC) is the percentage of cost savings resulting when two or more products are produced jointly rather than individually. It is measured by: SC =
C (q1 ) + C (q2 ) − C (q1 , q2 ) C (q1 , q2 )
(20)
• With economies of scope the joint cost is less than the sum of the individual costs. Thus SC > 0. • The larger the value of SC , the greater the economies of scope. • With diseconomies of scope the joint cost is greater than the sum of the individual costs. Thus SC < 0.
THE LEARNING CURVE
• As a firm gains experience in the production of a good, its average cost of production usually declines. The learning curve shows the decline in the average input cost of production with rising total outputs of the firm over time. • Average cost decline at a decreasing rate so that the learning curve is convex to the origin. That is, firms usually achieve the largest decline in average input costs when the production process is relatively new and smaller declines as the firm matures.
THE LEARNING CURVE
Figure: The learning curve
THE LEARNING CURVE The learning curve equation is: L = a + bN −β
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L = labor input per unit of output.
(21)
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N = cumulative units of output produced.
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a, b and β are positive parameters and 0 < β < 1.
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When N = 1, L = a + b measures the labor input required to produce the first unit of output. When β = 0, labor input per unit of output remains the same as the cumulative level of output increases; there is no learning.
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When β > 0 and N gets larger and larger, L becomes close to a.
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The larger β is, the more important the learning effect.
THE LEARNING CURVE
• The reduction in LAC due to increasing returns to scale is shown by a movement along the LAC curve as output increases. • The reduction in LAC due to learning is shown by the downward shift in the LAC curve for a given level of output.
THE LEARNING CURVE
Figure: Economies of scale versus learning
COST MINIMIZATION: A MATHEMATICAL APPROACH • Firms minimize the cost of producing an output. • Inputs have positive but decreasing marginal products. Therefore: MPK (K , L)
=
MPL (K , L)
=
δF (K , L) > 0, δK δF (K , L) > 0, δL
δ 2 F (K , L) <0 δK 2 δ 2 F (K , L) <0 δL2
(22)
• A competitive firm takes the prices of both labor w and capital r as given. The cost minimizing problem is: min C = wL + rK
(23)
• Subject to a constraint that a fixed output q0 be produced: F (K , L) = q0
(24)
COST MINIMIZATION: A MATHEMATICAL APPROACH The frim’s demand for K and L are those that minimize (23) subject to (24). To do it we use the method of Lagrange multiplier. The latter can be written as: φ = wL + rK − λ[(F (K , L) − q0 )] (25) The resulting equations are: δφ = r − λMPK (K , L) = 0 δK δφ = w − λMPL (K , L) = 0 δL δφ = q0 − F (K , L) = 0 δλ where MP is short for marginal product.
(26) (27) (28)
COST MINIMIZATION: A MATHEMATICAL APPROACH Thus MPK (K , L) r MPL (K , L) w q0
1 λ 1 = λ = F (K , L) =
(29) (30) (31)
From (29) and (30) we derive: 1 MPK (K , L) MPL (K , L) = = λ r w
(32)
To minimize its costs the firm will choose its factor inputs in order to equate the ratio of the marginal product of each factor divided by its price. We can rewrite (32) as: MPK (K , L) r = MPL (K , L) w
(33)
COST MINIMIZATION: A MATHEMATICAL APPROACH We already said that the optimum is the point of tangency between the isocost line and the isoquant. Lets use some calculus. At the same isoquant curve we have: MPK (K , L)dK + MPL (K , L)dL = dq = 0
(34)
Rearranging −
MPL (K , L) dK = = MRTSLK dL MPK (K , L)
(35)
From equations (33) and (35) we find that the optimum is given by: MPL (K , L) w = = MRTSLK MPK (K , L) r
(36)
PRODUCTION MAXIMIZATION: A MATHEMATICAL APPROACH
Suppose a production function: F (K , L) = q0
(37)
C0 = wL + rK
(38)
and an isocost function: The firm’s demand for K and L are those that maximize (37) subject to (38). To do it we use the method of Lagrange multiplier. The latter can be written as: φ = F (K , L) − µ(wL + rK − C0 ) (39)
PRODUCTION MAXIMIZATION: A MATHEMATICAL APPROACH The resulting equations are: δφ = MPK (K , L) − µr δK δφ = MPL (K , L) − µw δL δφ = −wL − rK + C0 δµ
= 0
(40)
= 0
(41)
= 0
(42)
From equations (40) to (42) we can write: MPK (K , L) r MPL (K , L) µ= w
µ=
Thus
MPK (K , L) MPL (K , L) = r w
(43) (44)
(45)
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Read Chapter 8 of Pindyck and Robinfeld
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Next time: Chapter 8 of Pindyck and Robinfeld « Profit Maximization and Competitive Supply »
