Week 11 Micro
The Conditions for Cost Minimization
*Condition 1:* tangency of the isoquant and isocost curves - slope of the isoquant =-MRTS - slope of the isocost =-w/r It follows that the tangency rule can be written as MRTS=w/r For example, for the production function f(L,K)=LK, MP_L=K and MP_K=L, which implies MRTS=〖MP〗_L∕〖MP〗_K =K/L. If w=$10 per hour and r=$20 per hour, the tangency condition is K/L=10/20. Two equivalent ways to get MRTS: 1. By definition: set q ̅=LK. Then K=q ̅/L. Differentiate K w.r.t. L yields dK/dL=-q ̅/L^2 =-K/L, where the last equality follows from K=q ̅/L. 2. Use the ratio of MP's: (MP_L)/(MP_K )=K/L *Condition 2:* the combination is on the given isoquant. If Q=f(L,K) is the known production function, and q ̅ is the output the firm wants, then the cost minimizing combination (L,K) must produce q ̅, f(L,K)=q ̅ For example, for f(L,K)=LK and a given output level 100, the condition is simply LK=100. Combine conditions 1 and 2 allows us to solve for the cost minimizing bundle.
Total Cost (TC)
All costs of production: the sum of variable cost and fixed cost.
Example 7.4 The Effect of Effluent Fees on Input Choices
An effluent fee is a per-unit fee that a steel firm must pay for the effluent that goes into the river. What is the effect of an effluent fee on the amount of wastewater dumped by the steel firm? [6.2 Slide 22] When the firm is not charged for dumping its wastewater in a river, it chooses to produce a given output using 10,000 gallons of wastewater and 2000 machine-hours of capital at A. However, an effluent fee raises the cost of wastewater, rotates the isocost curve from FC to DE, and causes the firm to produce at B—a process that results in much less effluent. The manager estimates that a machine-hour costs $40 and that dumping each gallon of wastewater in the river costs $10. The total cost of production is therefore $180,000: $80,000 for capital and $100,000 for wastewater.
The Isocost Curve
An isocost curve shows all possible combinations of labor and capital that can be purchased for a given total cost. The 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* For each different level of total cost, the equation above describes a different isocost line. If we rewrite the total cost equation as an equation for a straight line, we get K= (C / r) - (w / r)L. It follows that the isocost line has a slope of ΔK/ΔL = -w∕r, the ratio of the wage rate to the rental rate of capital.
Long-Run Marginal Cost (LMC)
Analogously to the short-run case, long-run marginal cost (LMC) is the slope of the long-run total cost curve: LMC=ΔLTC/ΔQ. In words, LMC is the cost to the firm, in the long run, of expanding its output by 1 unit.
Economies of Scale
As output increases, the firm's average cost of producing that output is likely to decline, at least to a point. *Economies of scale:* Situation in which the LAC declines as output increases. Economies of scale can happen for the following reasons: 1. If the firm operates on a larger scale, workers can specialize in the activities at which they are most productive. 2. 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.
Diseconomies of Scale
At some point, however, it is likely that the average cost of production will begin to increase with output. *Diseconomies of scale:* Situation in which the LAC increase as output increases. Diseconomies of scale can happen for the following reasons: 1. Managing a larger firm may become more complex and inefficient as the number of tasks increases 2. 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.
Effect of Changes in Input Prices
Changes in relative price alters the slope of isocost lines, causing the lowest workable isocost line to rotate around the given isoquant. Consequence: the firm uses more of the cheaper input.
Curvature of the VC Curve
Concave production function for L<4 → convex VC for Q<43 Increasing returns to labor for L<4: increments in L produce successively larger increments in Q in that region. Put another way, a given increase in output Q requires successively smaller increments in the variable input, L, i.e. VC=wL grows at a diminishing rate for output levels less than 43. Convex production function for L<4 → convave VC for Q<43 Diminishing returns to labor for L>4: ... Successively larger increments in L are required to produce a given increment in Q, i.e., VC grows at an increasing rate for output levels greater than 43. We can also describe the curvature of VC more rigorously using the following result: MC=ΔVC/ΔQ=wΔL/ΔQ=w/(ΔQ/ΔL)=w/(MP_L ) where the last equality holds because ΔQ/ΔL=MP_L, the marginal product of labor. The first "=": the marginal cost (MC) is the change in variable cost (VC) for a 1-unit change in output (q), i.e., MC=ΔVC/Δq. The second "=": the change in variable cost is the per unit cost of the extra labor, the wage rate w, times the amount of extra labor needed to produce the extra output, ΔL. That is, ΔVC=wΔL. The last "=": the marginal product of labor MP_L is the change in output resulting form a 1-unit change in labor input, or MP_L=Δq/ΔL. Therefore, the extra labor needed to obtain an extra unit of output is ΔL/Δq=1/MP_L.
Fixed Cost (FC)
Cost that does not vary with the level of output in the short run (the cost of all fixed factors of production). Equipment rental payment, property taxes, insurance payments, interest on loans
Variable Cost (VC)
Cost that varies with the level of output in the short run (the cost of all variable factors of production). Labor cost
Fixed Cost Curve
FC is flat. The vertical distance between TC and VC is everywhere FC: TC=FC+VC Note that the TC of producing zero output is equal to FC. (FC) Because fixed costs do not vary with the level of output, their graph is simply a horizontal line. (TC) Because TC=FC+VC, the TC curve can be obtained by parallelly shifting the VC curve upward by the amount of the FC. Put another way, the vertical distance between the VC and TC curves is everywhere equal to FC. The total cost curve is parallel to the variable cost curve and lies FC units above it. Note in particular that the total cost of producing zero output is equal to fixed costs, FC. This means that the FC is there no matter you produce or not. It also means that if you are given a TC function, which is a function of Q, you can find the FC by setting Q=0.
The Long-Run Expansion Path
Given sufficient time to adjust, the firm can always buy the cost-minimizing input bundle that corresponds to any particular output level and relative input prices. To see how the firm's costs vary with output in the long run (i.e. the long-run total cost curve), we need only compare the costs of the respective optimal input bundles. *Output expansion path* the locus of tangencies (minimum-cost input combinations) traced out by an isocost curve of given slope as it shifts outward into the isoquant map for a production process. This definition for output expansion path can be extended to allow changes in input prices. Such an extension reflects the possible changes in input prices due to increased bargaining power in the input markets as scale increases. [6.2 Slide 25] With fixed input prices r and w, bundles S,T, U, and others along the locus EE represent the least costly ways of producing the corresponding levels of output
The Importance of MC
In terms of its role in the firm's decision of how much output to produce, by far the most important of the seven cost curves is the marginal cost curve. The firm's typical operating decision involves the question of whether to expand or contract its current level of output. To make this decision intelligently, the firm must compare the relevant costs and benefits. The cost of expanding output (or the savings from contracting) is by definition equal to marginal cost. In terms of its role in the firm's decision of how much output to produce, by far the most important of the seven cost curves is the marginal cost curve. The reason, as we will see in the coming chapters, is that the firm's typical operating decision involves the question of whether to expand or contract its current level of output. To make this decision intelligently, the firm must compare the relevant costs and benefits. The cost of expanding output (or the savings from contracting) is by definition equal to marginal cost.
Flexibility in Long-Run Production
In the long run, a firm has much more flexibility. It can expand its capacity by expanding existing factories or building new ones; it can expand or contract its labor force, and in some cases, it can change the design of its products or introduce new products. Such flexibility gives rise to a cost minimization problem: how a firm can choose its combination of inputs to minimize its cost of producing a given output. For simplicity, we will work with two variable inputs: labor (L) and capital (K). It is useful to have an object that can describe all the input combinations that produces the same output level. That object is the isocost curve.
Choosing Inputs to Minimize Total Cost
Suppose we wish to produce q_1. Given the isoquant curve q_1, there are many different bundles of (K,L) to produce q_1. How can we do so at minimum cost? Look at the firm's production isoquant, labeled q_1, in the graph on the next slide. The problem is to choose the point on this isoquant that minimizes total cost. The point of tangency of the isoquant q1 and the isocost line C1 at point A gives us the cost-minimizing choice of inputs, L1 and K1
Example: Tesla's Battery Costs
The average battery production cost was about $400 per kWh in 2016. The battery for Tesla's Model 3 has a 50 kWh capacity, which at $400 per kWh implies a cost of $20,000 per battery. However, that cost can be reduced substantially by producing batteries in large volumes. A high volume of production is the objective of Tesla's Gigafactory. Tesla's electric cars, with prices around $85,000 have been unaffordable for most people. However, in 2017, Tesla will be producing a new "mass market" car, with a starting price of about $35,000. To achieve such a dramatic reduction in price, the company will rely on scale economies in battery production in its new $5 billion "Gigafactory" in Nevada. Battery costs are expected to decrease by one-third (to about $250 per kWh of energy storage), and fall further as production rises. A kilowatt hour (kWh)
Long-Run Average Cost
The ratio of long-run total cost to output: LAC=LTC/Q
LAC and Industry Structure
The shape of LAC is important because of its effect on the structure of industry. Downward sloping throughout (U-shaped LAC, with Q_0 greater than the whole industry output): *a single firm* *natural monopoly:* an industry whose market output is produced at the lowest cost when production is concentrated in the hands of a single firm. U-shaped LAC, with Q_0 constituting a substantial share of industry output: *a small handful of big firms* U-shaped LAC, with Q_0 constituting a small fraction of industry output: *numerous small firms*
Recast the Tangency Condition: Equal Marginal Product per Dollar
The tangency condition can be written as (MP_L)/w=(MP_K)/r. - MP_L/w, *the marginal product of labor per dollar,* is the additional output that results from spending an additional dollar for labor. - MP_K/r, *the marginal product of capital per dollar,* is the additional output that results from spending an additional dollar for capital. MRTS = w/r mPl/mPk = w/r Therefore, a cost-minimizing firm should choose its quantities of inputs so that the last dollar's worth of any input yields the same amount of extra output. If the wage rate is $10 and adding a worker will increase output by 20 units, then the additional output per dollar spent on an additional worker will be 20/10=2 units of output per dollar
Long-Run Total Cost (LTC)
To go from the long-run expansion path to the *long-run total cost (LTC)* curve, we simply plot the relevant quantity-cost pairs from the last figure. LTC=LTC(Q)
The Inflexibility of Short-Run Production
When a firm operates in the short run, its cost of production may not be minimized because of inflexibility in the use of fixed inputs. - The firm is unable to substitute the relatively inefficient inputs (i.e. the inputs with low marginal products per dollar) for more efficient ones. In the long run, the firm has the flexibility to adjust all inputs to produce any given input in a cost-efficient way. - Equal marginal products per dollar for all inputs. [6.3 Slide 6] Suppose output is initially at level q1, (using L1, K1). In the short run, output q2 can be produced only by increasing labor from L1 to L3 because capital is fixed at K1. In the long run, q_2 can be produced cost efficiently by increasing labor from L1 to L2 and capital from K1 to K2. Why is the cost of production higher when capital is fixed? Because the firm is unable to substitute relatively inexpensive capital for more costly labor when it expands production. This inflexibility is reflected in the short-run expansion path, which begins as a line from the origin and then becomes a horizontal line when the capital input reaches K1. The laundry shop example: In the short run, Kelly can only use the exiting washer and dryers. In the long run, she can consider buying more washers and dryers.
Equal Marginal Products per Dollar
Why must the rule of equal marginal products per dollar hold for cost minimization? If the rule does not hold, the firm can produce more without increasing cost. - E.g., suppose w=$10, r=$2, and MP_L=MP_K=20 bags of laundry. (MP_L)/w=2<5=(MP_K)/r - Reallocate the last dollar to produce more: reduce one dollar's worth of labor input (output drops by 2 bags) and use that dollar to add one dollar's worth of capital input (output increases by 5 bags) Keep doing the reallocation until (MP_L)/w=(MP_K)/r, which will occur due to diminishing marginal returns. Intuitively, labor and capital are equally productive, but capital is cheaper, so the firm wants to use more capital and less labor. If the firm reduces labor and increases capital, its marginal product of labor will rise and its marginal product of capital will fall. Eventually, the point will be reached at which the production of an additional unit of output costs the same regardless of which additional input is used. At that point, the firm is minimizing its cost.
The Long-Run Total, Average, and Marginal Cost Curves
[6.2 Slide 27] The long-run total cost curve will always pass through the origin. The long-run average and long-run marginal cost curves are derived from the long-run total cost curve in a manner completely analogous to the short-run case. The relationship between average and marginal quantities still apply. In the long run, the firm always has the option of ceasing operations and ridding itself of all its inputs. This means that the long-run total cost curve (top panel) will always pass through the origin.
Producing A Given Output at Minimum Cost
[6.2 slide 11] Isocost curve C_1 is tangent to isoquant q1 at A: (L_1,K_1) Hence, C_1 is the minimum cost to produce q1: - Some input combinations other than A - e.g., (L_2,K_2) and (L_3,K_3) - can yield the same output but at higher cost. - Input combinations that cost less C_1 -- e.g., any combination on isocost C_0 -- will not allow the firm to achieve output q1.
Input Substitution When an Input Price Changes
[6.2 slide 20] When the price of labor increases, the isocost curves become steeper. Output q1 is now produced at point B on isocost curve C2 by using L2 units of labor and K2 units of capital.
The Isocost Line
[6.2 slide 7] The isocost line C_0 describes all possible combinations of labor and capital that cost a total of C0 to hire. A higher level of total cost C_1 corresponds to a higher isocost line. Given w and r, all the isocost lines are parallel. The slope ΔK/ΔL = -w∕r means that if the firm reduces capital input by w/r units (which saves (w/r)r=w dollars) and hires an additional unit of labor input (which costs w), its total cost of production would remain the same (move down along a given isocost curve).
The Relationship between Short-Run and Long-Run Cost Curves
[6.3 Slide 16] The long-run average cost (LAC) curve is the envelope of the short-run average cost curves (SAC). For the output level at which a given SAC is tangent to the LAC, the long-run marginal cost (LMC) of producing that level of output is the same as the short-run marginal cost (SMC) Each point along a given SAC curve, except for the tangency point, lies above the corresponding point on the LAC curve. - Too much capital to the left of the tangency, and too little capital to the right of the tangency. At the minimum point on the LAC curve (q_2), the long-run and short-run marginal and average costs all take exactly the same value
The Shape of LAC
[6.3 slide 9] As output level increases, a firm's long-run average cost can increase, remain constant, or increase. minimum efficient scale: the level of production required for LAC to reach its minimum level
Average Fixed Cost (AFC)
is fixed cost divided by the quantity of output: AFC=FC/Q AFC Curve: [6.1 slide 23] AFC falls with output: "spreading overhead costs." Geometrically, AFC at any level of output Q is the slope of the ray to the FC curve at Q. As output shrinks toward zero, AFC grows without bounds, and it falls ever closer to zero as output increases. Like all other AFC curves, it takes the form of a rectangular hyperbola.
Marginal Cost
is the change in total cost that results from producing an additional unit of output. MC=ΔTC/ΔQ=ΔVC/ΔQ. The second equality holds since ΔFC/ΔQ is zero. MC Curve: [6.1 Slide 26] MC at any level of output Q is the slope of the TC (or VC) curve at that Q. The MC curve is downward sloping up to the inflection point of the TC curve at Q_1 and upward sloping thereafter. The MC curve intersects the ATC and AVC curves at their respective minimum points. When MC < average cost (either ATC or AVC), MC drags down the average cost curve; when MC > average cost, MC pulls up the average cost curve. (slope) Geometrically, marginal cost at any level of output may be interpreted as the slope of the total cost curve at that level of output. And since the total cost and variable cost curves are parallel, marginal cost is also equal to the slope of the variable cost curve. (Recall that the variable cost component is all that varies when total cost varies, which means that the change in total cost per unit of output must be the same as the change in variable cost per unit of output.) (minimum) Notice in the top panel in Figure 9.5 that the slope of the total cost curve decreases with output up to Q1, and rises with output thereafter. This tells us that the marginal cost curve, labeled MC in the bottom panel, will be downward sloping up to Q1 and upward sloping thereafter. Q1 is the point at which diminishing returns set in for this production function, and diminishing returns are what ac-count for the upward slope of the short-run marginal cost curve. (intersection) At the output level Q3, the slope of the total cost curve is exactly the same as the slope of the ray to the total cost curve (the ray labeled R1 in the top panel in Figure 9.5). This tells us that marginal cost and average total cost will take precisely the same value at Q3. To the left of Q3, the slope of the total cost curve is smaller than the slope of the corresponding ray, which means that marginal cost will be smaller than average total cost in that region. For output levels in excess of Q3, the slope of the total cost curve is larger than the slope of the corresponding ray, so marginal cost will be larger than average total cost for output levels larger than Q3. These relationships are reflected in the average total cost and marginal cost curves shown in the bottom panel in Figure 9.5. Notice that the relationship between the MC and AVC curves is qualitatively similar to the relationship be-tween the MC and ATC curves. One common feature is that MC intersects each curve at its minimum point. Both average cost curves have the additional property that when MC is less than average cost (either ATC or AVC), the average cost curve must be decreasing with output; and when MC is greater than average cost, average cost must be increasing with output.
Average Total Cost (ATC)
is total cost divided by the quantity of output: ATC=TC/Q=AFC+AVC ATC Curve: [6.1 slide 25] Still similarly, ATC at any level of output Q is the slope of the ray to the TC curve at Q. Top panel: The slope of a ray to the TC curve declines with output up to the output level Q_3; thereafter it begins to increase. Bottom panel: ATC reaches its minimum value at Q_3, the output level at which the ray R_3 is tangent to the TC curve. ATC=AFC+AVC Q_2, the output level at which the ray R_2 is tangent to the variable cost curve (ATC=AFC+AVC) so the vertical distance between the ATC and AVC curves at any level of output will always be the corresponding level of AFC.
Average Variable Cost (AVC)
is variable cost divided by the quantity of output: AVC=VC/Q AVC Curve [6.1 slide 24] Similarly, AVC at any level of output Q is the slope of the ray to the VC curve at Q. Top panel: The slope of a ray to the VC curve declines with output up to the output level Q_2; thereafter it begins to increase. Bottom panel: AVC reaches its minimum value at Q_2, the output level at which the ray R_2 is tangent to the VC curve. Q_2, the output level at which the ray R_2 is tangent to the variable cost curve
