Microeconomics. Lecture Outline. Claudia Vogel. Winter Term 2009/2010. Part II Producers, Consumers, and Competitive Markets

Transcription

1 Microeconomics Claudia Vogel EUV Winter Term 2009/2010 Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Lecture Outline Part II Producers, Consumers, and Competitive Markets 7 Measuring Cost: Which Costs Matter? Cost in the Short Run Cost in the Long Run Long-Run versus Short-Run Cost Curves Production with Two Outputs - Economies of Scope Mathematical Appendix Summary Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

2 Measuring Cost: Which Costs Matter? Measuring Cost: Which Costs Matter? accounting cost: Actual expenses plus depreciation charges for capital equipment. economic cost: Cost to a rm utilizing economic resources in production, including opportunity cost. opportunity cost: Cost associated with opportunities that are forgone when a rm's resources are not put to their best alternative use. sunk cost: Expenditure that has been made and cannot be recovered. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Measuring Cost: Which Costs Matter? Fixed Costs and Variable Costs total costs (TC or C): Total economic cost of production, consisting of xed and variable costs. xed cost (FC): Cost that does not vary with the level of output and that can be eliminated only by shutting down. variable cost (VC): Cost that varies as output varies. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

3 Measuring Cost: Which Costs Matter? Marginal and Average Cost marginal cost (MC): Increase in cost resulting from the production of one extra unit of output. Because xed cost does not change as the rm's level of output changes, marginal cost is equal to the increase in variable cost or the increase in total cost that results from an extra unit of output. We can therefore write marginal cost as MC = VC q = TC q average total costs (ATC): Firm's total cost divided by its level of output. average xed cost (AFC): Fixed cost divided by the level of output. average variable cost (AVC): Variable cost divided by the level of output. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Measuring Cost: Which Costs Matter? Example: A Firm's Cost Rate of Fixed Variable Total Marginal Average Average Average Output Cost Cost Cost Cost Fixed Cost Variable Cost Total Cost (U. per Y.) ($ per Y.) ($ per Y.) ($ per Y.) ($ per Y.) ($ per U.) ($ per U.) ($ per U.) (FC) (VC) (TC) (MC) (AFC) (AVC) (ATC) (1) (2) (3) (4) (5) (6) (7) Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

4 Cost in the Short Run The Determinants of Short-Run Cost The change in variable cost is the per-unit cost of the extra labor w times the amount of extra labor needed to produce the extra output L. Because VC = w L, it follows that MC = VC q = w L q The extra labor needed to obtain an extra unit of output is L q = 1 MP L. As a result, MC = w MP L Diminishing Marginal Returns and Marginal 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, marginal cost will increase as output increases. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Cost in the Short Run The Shapes of the Cost Curves Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

5 Cost in the Short Run Example: The Short-Run Cost of Aluminium Smelting 1/2 Production Costs for Aluminium Smelting ($/ton) (based on an output of 600 tons/day Per-ton costs that are constant Output 600 Output > 600 for all output levels tons/day tons/day Electricity $316 $316 Alumina Other Raw Materials Plant Power and fuel Subtotal $820 $820 Per-ton costs that increase when output exceeds 600 tons/day Labor $150 $225 Maintenance Freight Subtotal $320 $480 Total per-ton production costs $1140 $1300 Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Cost in the Short Run Example: The Short-Run Cost of Aluminium Smelting 2/2 Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

6 Cost in the Long Run User Cost of Capital user cost of capital: Annual cost of owning and using capital asset, equal to economic depreciation plus forgone interest. User Cost of Capital = Economic Depreciation + (Interest Rate)(Value of Capital) We can also express the user cost of capital (i.e. the price of capital) a a rate per dollar of capital: r = Depreciation rate + Interest rate rental rate: Cost per year of renting one 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. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Cost in the Long Run The Isocost Line isocost line: 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: C = wl + rk Rewrite the total cost equation: K = C r w r L The isocost line has a slope of K L = w r, which is the ratio of the wage rate to the rental capital. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

7 Cost in the Long Run Choosing Inputs MRTS = K L = MP L MP K When a rm minimizes the cost of producing a particular output, the following condition holds: MP L MP K = w r We can rewrite this condition as follows: MP L w = MP K r Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Cost in the Long Run Example: The Eect of Euent Fees on Input Choices When a rm is not charged for dumping its wastewater in a river, it chooses to produce a given output using gallons of wastewater and 2000 machine-hours of capital at A. However, an euent fee raises the cost of wastewater, shifts the isocost curve from FC to DE, and causes the rm to produce at B - a process that results in much less euent. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

8 Cost in the Long Run Cost Minimization with Varying Output Levels expansion path: Curve passing through points of tangency between a rm's isocost lines and its isoquants. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Long-Run versus Short-Run Cost Curves The Inexibility of Short-Run Production Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

9 Long-Run versus Short-Run Cost Curves Long-Run Average Cost long-run average cost curve (LAC): Curve relating average cost of production to output when all inputs, including capital, are variable. short-run average cost curve (SAC): Curve relating average cost of production to output when level of capital is xed. long-run marginal cost curve (LMC): Curve showing the change in long-run total cost as output is increased incrementally by 1 unit. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Long-Run versus Short-Run Cost Curves Economies and Diseconomies of Scale 1/3 As output increases, the rm's average cost of producing that output is likely to decline, at least to a point. This can happen for the following reasons: 1 If the rm operates on a larger scale, workers can specialize in the activities at which they are most productive. 2 Scale can provide exibility. By varying the combination of inputs utilized to produce the rm's output, managers can organize the production process more eectively. 3 The rm 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 rm's operation if managers take advantage of lower-cost inputs. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

10 Long-Run versus Short-Run Cost Curves Economies and Diseconomies of Scale 2/3 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: 1 At least in the short run, factory space and machinery may make it more dicult for workers to do their jobs eectively. 2 Managing larger rms may become more complex and inecient as the number of tasks increases. 3 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. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Long-Run versus Short-Run Cost Curves Economies and Diseconomies of Scale 3/3 economies of scale: Situation in which output can be doubled for less than a doubling of cost. diseconomies of scale: Situation in which a doubling of output requires more than a doubling of cost. Increasing Returns to Scale: Output more than doubles when the quantities of all inputs are doubled. Economies of Scale: A doubling of output requires less than a doubling of cost. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

11 Long-Run versus Short-Run Cost Curves Cost-Output Elasticity Economies of scale are often measured in terms of a cost-output elasticity, E C. E C is the percentage change in cost of production resulting from a 1-percent increase in output: E C = C/C q/q To see how E C relates to our traditional measures of cost, rewrite the equation as follows: E C = C/ q C/q = MC AC Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Long-Run versus Short-Run Cost Curves The Relationship between Short-Run and Long-Run Cost Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

12 Production with Two Outputs - Economies of Scope Product Transformation Curve product transformation curve: Curve showing the various combinations of two dierent outputs (products) that can be produced with a given set of inputs. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Production with Two Outputs - Economies of Scope Economies and Diseconomies of Scope economies of scope: Situation in which joint output of a single rm is greater than output that could be achieved by two dierent rms when each produces a single product. diseconomies of scope: Situation in which joint output of a single rm is less than could be achieved by separate rms when each produces a single product. To measure the degree to which there are economies of scope, we should ask what percentage of the cost of production is saved when two (or more) products are produced jointly rather than individually. SC = C (q 1) + C (q 2 ) C (q 1, q 2 ) C (q 1, q 2 ) degree of economies of scope (SC): Percentage of cost savings resulting when two or more products are produced jointly rather than individually. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

13 Mathematical Appendix Cost Minimization Firm's Minimization problem: C = wl + rk min Firm's constraint: F (K, L) = q 0 Solving by Lagrange Multiplier Method: The Lagrangian: φ (L, K, λ) = wl + rk λ (F (K, L) q 0 ) Derivatives of the Lagrangian: φ(l,k,λ) K φ(l,k,λ) L F (K,L) = r λ = w λ φ(l,k,λ) = F (K, L) q λ 0 = 0 Solving the Resulting Equations: MP K (K,L) r = MP L(K,L) w K = r λmp K (K, L) = 0 F (K,L) L = w λmp L (K, L) = 0 r = λmp K (K, L) = 0 λ = r = λmp L (K, L) = 0 λ = r MP K (K,L) w MP L (K,L) Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Mathematical Appendix Output Maximization Firm's Maximization problem: F (K, L) max Firm's constraint: wl + rk = C 0 Solving by Lagrange Multiplier Method: The Lagrangian: φ (K, L, λ) = F (K, L) µ (wl + rk C 0 ) Derivatives of the Lagrangian: φ(l,k,λ) = MP K K (K, L) µr = 0 µ = MP K (K,L) r φ(l,k,λ) = MP L L (K, L) µw = 0 µ = MP L(K,L) w φ(l,k,λ) = wl + rk C λ 0 = 0 Solving the Resulting Equations: MP K (K,L) r = MP L(K,L) w Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

14 Mathematical Appendix The Cobb-Douglas-Function 1/2 Cobb-Douglas-Function: F (K, L) = AK α L β A, α, β > 0; α, β < 1 Firm's Minimization problem: C = wl + rk min Firm's constraint: F (K, L) = AK α L β = q 0 Solving by Lagrange Multiplier Method: The Lagrangian: φ (L, K, λ) = wl + rk λ ( AK α L β q 0 ) Derivatives of the Lagrangian: φ(l,k,λ) = r λ ( βak α L β 1) = 0 λ = K φ(l,k,λ) = r λ ( αak α 1 L β) = 0 L φ(l,k,λ) = AK α L β q λ 0 = 0 Solving the Resulting Equations: I in II: L = rβ wα in III: K = ( q0 A ) 1 α+β ( rβ K expansion path wα in L: L = ( q0 A ) 1 α+β ( rβ wα ) β α+β ) α α+β w βak α L β 1 Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Mathematical Appendix The Cobb-Douglas-Function 2/2 Cost-Function: C = wl + rk C = w C = w ( q0 β α+β A ) 1 ( ) α α+β rβ α+β wα + r α α+β ( α β constant returns to scale: α + β = 1 ) β α+β ( q0 ) 1 ( ) β α+β rβ α+β + r A wα + C = w β + r α [ (α β ) β + ( ) α α+β α β ( α β ) α] ( q0 ( q 0 A A ) ) 1 α+β Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

15 Summary Summary 1/3 Managers, investors, and economists must take into account the opportunity cost associated with the use of a rm's resources: the cost associated with the opportunities forgone when the rm uses its resources in its next best alternative. A sunk cost is an expenditure that has been made and cannot be recovered. After it has been incurred, it should be ignored when making future economic decisions. In the short run, one or more of the rm's inputs are xed. Total cost can be divided into xed cost and variable cost. A rm's marginal cost is the additional variable cost associated with each additional unit of output. The average variable cost is the total variable cost divided by the number of units of output. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Summary Summary 2/3 In the short run, when not all inputs are variable, the presence of diminishing returns determines the shape of the cost curves. In particular, there is an inverse relationship between the marginal product of a single variable input and the marginal cost of production. The average variable cost and average total cost curves are U-shaped. The short-run marginal cost curve increases beyond a certain point, and cuts both average cost curves from below at their minimum points. In the long-run, all inputs to the production process are variable. As a result, the choice of inputs depends both on the relative costs of the factors of production and on the extent to which the rm can substitute among inputs in its production process. The cost-minimizing input choice is made by nding the point of tangency between the isoquant representing the level of desired output and an isocost line. The rm's expansion path shows how its cost-minimizing input choices vary as the scale or output of its operation increases. As a result, the expnasion path provides useful information relevant for long-run planning decisions. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

16 Summary Summary 3/3 A rm enjoys economies of scale when it can double its output at less than twice the cost. Correspondingly, there are diseconomies of scale when a doubling of output requires more than twice the cost. Scale economies and diseconomies apply even when input proportions are variable; returns to scale applies only when input proportions are xed. When a rm produces two (or more) outputs, it is important to note whether there are economies of scope in production. Economies of scope arise when the rm can produce any combination of the two outputs more cheaply than could two independent rms that each produced a single product. The degree of economies of scope is measured by the percentage reduction in cost when one rm produces two products relative to the cost of producing them individually. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Problem 1 Exerxises 6 Please explain whether the following statements are true or false. 1 If the owner of a business pays himself no salary, then the accounting cost is zero, but the economic cost is positive. 2 A rm that has positive accounting prot does not necessarily have positive economic prot. 3 If a rm hires a currently unemployed worker, the opportunity cost of utilitizing the worker's services is zero. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

17 Exerxises 6 Problem 2 1 Assume that the marginal cost of production is increasing. Can you determine whether the average variable cost is increasing or decreasing? Explain. 2 Assume that the marginal cost of production is greater than the average variable cost. Can you determine whether the average variable cost is increasing or decreasing? Explain. 3 Distinguish between economies of scale and economies of scope. Why can one be present without the other? 4 What is the dierence between economies of scale and returns to scale? Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Problem 3 Exerxises 6 1 Is the rm's expansion path always a straight line? 2 Suppose the economy takes a downturn, and that labor costs fall by 50 percent and are expected to stay at that level for a long time. Show graphically how this change in the relative price of labor and capital aects the rm's expansion path. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36

18 Exerxises 6 Problem 4 Suppose that a rm's production function is q = 10L 1 2 K 1 2. The cost of a unit of labor is $20 and the cost of a unit of capital is $80. 1 The rm is currently producing 100 units of output and has determined that the cost-minimizing optimal quantities of labor and capital are 20 and 5, respectively. Graphically illustrate this using isoquants and isocost lines. 2 The rm now wants to increase output to 140 units. If capital is xed in the short run, how much labor will the rm need? Illustrate this point graphically and nd the rm's new total cost. 3 Graphically identify the optimal cost-minimizing level of capital and labor in the long run if the rm wants to produce 140 units. 4 If the marginal rate of technical substitution is K, nd the optimal level of L capital and labor required to produce the 140 units of output. Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36 Problem 5 Exerxises 6 Suppose the process of producing lightweight parkas by Polly's Parkas is described by the function q = 10K 0.8 (L 40) 0.2 where q is the number of parkas produced, K the number of computerized stitching-machine hours, and L the number of person-hours of labor. In addition to capital and labor, $10 worth of raw material is used in the production of each parka. 1 By minimizing cost subject to the production function, derive the cost-minimizing demands for K and L as a function of output(q), wage rates (w), and rental rates on machines (r). Use these results to derive the total cost function: that is, costs as a function of q, r, w, and the constant $10 per unit material cost. 2 This process requires skilled workers, who earn $32 per hour. The rental rate on the machines used in the process is $64 per hour. At these factor prices, what are total costs as a function of q? Does this technology exhibit decreasing, constant, or increasing returns to scale? 3 Polly's Parkas plans to produce 2000 parkas per week. At the factor prices given above, how many workers should the rm hire (at 40 hours per week) and how many machines should it rent (at 40 machine-hours per week)? What are the marginal and average costs at this level of production? Claudia Vogel (EUV) Microeconomics Winter Term 2009/ / 36