Chapter 7: Types of Cost and Short-Run Cost Curves

Types of Cost and Short-Run Cost Curves

  • The optimal cost-minimizing combination of inputs is chosen by a firm.

Measuring Cost - Which Costs Matter?

Economics Cost Versus Accounting Cost

  • Accounting Cost – Actual expenses plus depreciation charges for capital equipment.
  • Tend to take a retrospective view of finances/operations.
  • Economic Cost – Cost to a firm of utilizing economic resources in production.
  • Tend to take a forward-looking view, as economists are concerned with the allocation of scarce resources.
  • They care about costs in the future, and ability to rearrange resources to lower cost and improve profitability.

Opportunity Cost

  • Opportunity Cost – Cost associated with opportunities forgone when a firm’s resources are not put to their best alternative use.
  • E.g. Forgone rent by not leasing an office space is the opportunity cost.
  • Economic Cost = Opportunity Cost
  • Concept of opportunity cost is useful if the alternatives forgone don’t reflect actual monetary outlays.
  • Opportunity cost may refer to forgone future value \rightarrow The actual economic cost may not be the difference between values. (i.e. sell toys on market for 15 or to another retailer for 15 → The opportunity cost is 15.)
  • Accountants and economic costs may differ in treatment of depreciation → economics are concerned with capital cost of plant and machinery (including wear and tear), but accountants use tax rules to apply allowable depreciation.

Sunk Costs

  • Sunk Cost – Expenditure that has been made and cannot be recovered.
  • Opposite of Opportunity → Usually visible, but after it has been incurred, it should be ignored when making future decisions.
  • Cannot be recovered – Shouldn’t influence the firm’s decisions.
  • If it can be put to other use or sold, there is economic cost of using it rather than selling/renting.
  • Prospective Sunk Cost – An investment where firms must consider if the decision is economical (it will lead to a large flow of revenues to justify cost).
  • Economic analysis removes the sunk cost of the option from the analysis.

Fixed Costs and Variable Costs

  • Total Cost (TC) – Total economic cost of production consisting of fixed and variable costs.
  • Fixed Cost (FC) – Cost that doesn’t vary with level of output and that can only be eliminated by shutting down.
  • Variable Cost (VC) – Cost that varies proportionately with output.

Shutting Down

  • To reduce fixed costs, a company must reduce the output to zero and shut down the factory – the company can still stay in business with other factories.

Fixed Or Variable

  • Time horizon may decide the nature of costs
  • Over a short time period, most costs are fixed due to obligations to pay for contracted transactions and cannot easily lay off workers.
  • Over a long time period many costs become variable due to higher flexibility in workforce, and capital.
  • When a firm plans to increase/decrease production, it will want to know how change affects its costs.

Fixed Versus Sunk Cost

  • Fixed costs are costs paid by operating firms and can be avoided if a firm shuts down production or goes out of business.
  • Affects firm’s decisions going forward.
  • Sunk costs are costs that have been incurred and cannot be recovered.
  • Doesn’t affect firm’s decisions going forward.
  • Prospective sunk costs do affect firm’s decisions.

Amortizing Sunk Costs

  • Amortization – Policy of treating a one-time expenditure as an annual cost spread out over some number of years.
  • Amortizing capital expenditures and spreading them out over many years and treating them as fixed costs is useful for evaluating long-term profitability.
  • It can also simplify the economic analysis of a firm’s operation – make it easier to understand the trade-off of labour and capital, etc.

Marginal and Average Cost

  • Marginal Cost (MC) – Known as incremental cost, is the increase in cost resulting from producing one extra unit of output.
  • Calculation = Increase in variable cost or total cost / increase of quantity
  • MC=ΔVCΔq=ΔTCΔqMC = \frac{ΔVC}{Δq} = \frac{ΔTC}{Δq}
  • Tells us how much it will cost to expand output by one unit.
  • Average Total Cost (ATC) – Known as AC or Average economic cost, it is the firm’s total cost divided by level of output, TCq\frac{TC}{q}
  • Tells us the per-unit cost of production.
  • ATC has two components
  • Average Fixed Cost (AFC) – Fixed cost divided by the level of output.
  • FCq\frac{FC}{q}
  • Average Variable Cost (AVC) – Variable cost divided by the level of output.

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Cost in the Short Run

The Determinants of Short-Run Cost

  • Variable and total costs increase with output in the short run.
  • The rate of increasing costs depends on the nature of the production process and the extent to which production involves diminishing marginal returns on variable factors.
  • Greater expenditures are required to produce output at a higher rate – variable and total costs increase as more output is produced.
  • If Marginal product decreases only slightly, then costs don’t rise as quickly.
  • Relationship between Production and Cost:
  • MC=ΔVCΔq=wΔLΔqMC = \frac{ΔVC}{Δq} = \frac{w ΔL }{Δq}
  • Where w = wage and ΔL is the extra output.
  • Extra labour to obtain one more unit of labor is 1MPL\frac{1}{MP_L} , so:
  • MC=wMPLMC = \frac{w}{MP_L}
  • If there is only one variable input, marginal cost is equal to the price of the input divided by its marginal product.

Diminishing Marginal Returns and Marginal Cost

  • Diminishing marginal returns = Declining Marginal Product of Labor as Quantity of labor employed increases.
  • If there are diminishing marginal returns, marginal cost increases as output increases.

The Shapes of the Cost Curves

  • Graphically: Total Cost TC is the vertical sum of fixed cost FC and variable cost VC
  • Average total cost ATC is the sum of average variable cost AVC and average fixed cost AFC.
  • Marginal Cost MC crosses average variable cost and average total cost curves at their minimum points.

  • Graph A Costs
  • FC doesn’t vary with output, it’s horizontal.
  • VC is zero at zero output and increases continuously as output increases.
  • TC is determined by adding the fixed curve to the variable cost curve.
  • Graph B Average Costs
  • AFC falls continuously from ATC.
  • Whenever MC < AC, the AC curve rises.

Image result for love heart The Average-Marginal Relationship

  • Average total cost ATC is the sum of AVC and AFC → and it follows the direction of MC.
  • If the AFC curve declines everywhere, the vertical distance between ATC and AVC decreases as output increases.
  • AVC reaches a minimum point at a lower output than the ATC curve because MC = ATC at its minimum point.
  • To visualize the average variable cost, we can draw a line from the origin to a certain point on the cost curve.
  • Because the slope of the VC curve is the marginal cost, the tangent to the VC curve is the marginal cost.

Total Cost as a Flow

  • Firm’s output is measured as a flow – Output amount is measured as units per year.
  • Average and Marginal are measured in dollars per unit.
  • We can refer to total cost as Cost (C) and average total cost as Average Cost (AC).

Long-Run Production Theory - Capital, isocost line, cost min theory,

and Long-run Costs Curves

Cost in the Long Run

  • In the LR, a firm is more flexible – It can choose a combination of inputs to minimize cost of producing a certain output.

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The User Cost of Capital

  • Treat capital as if it is rented so that the company can amortize the purchase price over the life of capital.
  • Where the “cost per year” is the annual economic depreciation.
  • There is an opportunity cost of interest that could have been earned.
  • User Cost of Capital – annual cost of owning and using a capital asset, equal to economic depreciation + forgone interest.
  • User Cost of Capital = Economic Depreciation + (Interest Rate)(Value of Capital)
  • Also expressed as a rate per dollar of capital: r = Depreciation rate + Interest Rate
  • As the capital depreciates, its value declines with opportunity cost of financial capital.

The Cost-Minimizing Input Choice

  • Firm’s problem: How to select inputs to produce a given output at minimum cost.
  • Variable Inputs: Labor in hours of work per year and Capital measured in hours of use of machinery per year.
  • The Price of labor is wage rate, w.

The Price of Capital

  • In the LR, firm can adjust the amount of capital used → But large initial expenditures on capital are necessary.
  • Express capital expenditure as a flow by amortizing expenditure over lifetime of the capital (User Cost of Capital) at r.

The Rental Rate of Capital

  • Rental rate – Cost per year of renting one unit of capital.
  • In a competitive market, rental rate = user cost (r).
  • Firms that own capital expect to earn a competitive return when they rent it – rate of return investing elsewhere + compensation for depreciation.
  • Capital that is purchased can be treated as though it were rented at a rental rate equal to the user cost of capital.

The Isocost Line (Budget Lines for Firms)

  • Graph showing all possible combinations of labor and capital that can be purchased for a given total cost.
  • The Total Cost = Labor Cost + Capital Cost.
  • Total Cost C=wL+rK C = wL + rK
  • We can rewrite total cost equation to a straight line.
  • Straight Line Total Cost K=Cr(wr)LK = \frac{C}{r} – (\frac{w}{r})L 
  • Slope is ΔKΔL=(wr)\frac{ΔK}{ΔL} = -(\frac{w}{r}) or the ratio of the wage rate to the rental cost of capital.
  • Similar to the slope of a budget line.
  • If firm gives up a unit of labor (recovering w dollars) to buy w/r units of capital at a cost of r dollars per unit, its total cost of production is the same.
  • E.g. Wage is $10 and rental cost is $5, so we can give up one labor unit for two capital units.

Choosing Inputs

  • Choose a point on the isoquant to minimize total cost.
  • The Isocost curve is tangent to the isoquant, showing an output that can be produced at minimum cost with labor input and capital input.
  • Slopes of isoquant = Isocost line.

  • If expenditure on all inputs increases, slope of Isocost line doesn’t change because prices haven’t changed – but intercept does change.

Input Price Change

  • If price of an input increases, the slope of Isocost –(w/r) increases in magnitude so the Isocost line becomes steeper.
  • Output is produced on the Isocost line with a higher level of the lower price input and less of the higher price input. (substitution effect)

Production Technology Ratio

  • MRTS=ΔKΔL=MPLMPKMRTS = -\frac{ΔK}{ΔL} = \frac{MP_L}{MP_K}
  • Because the Isocost line has a slope equal to change in capital over change in labor, MPLMPK=wr\frac{MP_L}{MP_K} = \frac{w}{r}
  • Condition: MPLw=MPKr\frac{MP_L}{w} = \frac{MP_K}{r}
  • MPLw\frac{MP_L}{w} represents additional output that results from spending an additional dollar for labor
  • MPKr\frac{MP_K}{r} represents additional output that results from spending an additional dollar for capital.
  • A firm that minimizes cost should choose quantities of inputs so that the last dollar’s worth of any input added to the production process yields the same amount of extra output.

Cost Minimization with Varying Output Levels

  • Lowest cost way to produce a certain output is at tangency between isocost curve and isoquant.
  • Expansion Path – Curve passing through points of tangency between a firm’s isocost lines and its isoquants.
  • It describes the combinations of labor and capital the firm will choose to minimize costs at each output level.
  • it is usually upward sloping because both inputs increase with output.
  • Slope of Expansion Path = ΔKΔL\frac{ΔK}{ΔL}

The Expansion Path and Long-Run Costs

  • Expansion path contains the same info as a LR total cost curve C(q).
  • Moving Curves
  • Choose an output level represented by an isoquant.
  • From the chosen isocost line, determine the minimum cost of producing the output level selected. (tangency point)
  • Graph the output-cost combination.

Long-Run versus Short-Run Cost Curves

  • SR average cost curves are U-shaped, but LR average cost curves depend on the economic factors.

The Inflexibility of Short-Run Production

  • Cost of production may not be minimized in SR because of inflexibility in the use of capital inputs.
  • A new output can be produced in SR by increasing labor.
  • Horizontal expansion path.
  • A new output can be produced in LR by increasing labor and increasing capital at a cheaper cost.
  • Upward-sloping Expansion Path.

Long-Run Average Cost

  • Used to analyze how costs vary as the firm moves along its expansion path in the LR.
  • Determinant of the shape of the LR average and marginal cost curves → relationship between scale of operation and inputs.
  • If there are constant returns to scale → Input prices remain unchanged, so average cost is the same at all output levels.
  • If there are increasing returns to scale, average cost of production falls with output.
  • If there are decreasing returns to scale, average cost of production increases with output.
  • Long-Run Average Cost Curve (LAC) – Curve relating average cost of production to output when all inputs, including capital, are variable.
  • U-shaped due to increasing/decreasing returns to scale.
  • Short-Run Average Cost Curve (SAC) – Curve relating average cost of production to output when level of capital is fixed.
  • U-shaped due to diminishing returns to a factor of production.
  • Long-Run Marginal Cost Curve (LMC) – Curve showing the change in long-run total cost as output is increased incrementally by 1 unit.
  • LMC lies below the LAC when LAC is falling and above it when LAC is rising.
  • These two curves intersect when LAC is at the minimum.
  • Special Case: if LAC is minimum, LAC and LMC are equal.

Economies and Diseconomies of Scale

  • As output increases, firm’s average cost will probably decline for several reasons:
  • At a larger scale, workers specialize in productive activities.
  • Scale provides flexibilities → Varying combinations of inputs may organize the production process.
  • Firm can acquire inputs at a lower cost because it buys at large quantities. Mix of inputs may change if managers take advantage of lower-cost inputs.
  • At some point, the average cost of production begins to increase with output:
  • In the SR, factory space and machinery may make it difficult for efficient work.
  • Managing a larger firm may become more complex and inefficient with more and more tasks.
  • Advantages of buying in bulk may disappear after certain quantity levels. Also, available supplies for inputs may be limited.
  • Economies of Scale – Situation in which output can be doubled for less than a doubling of cost.
  • Dis-economies of Scale – Doubling of output requires more than twice the cost.

Image result for scaleGenerally → The U-shaped average cost curve characterizes the firm facing economies of scale for lower output levels and eventually diseconomies for higher levels of output.

  • Difference between returns to scale (inputs are used in constant proportions is increased) and economies of scales (input proportions are variable).
  • E.g. A dairy farm doubles cows to double milk production (returns to scale) and large dairy farms may use milking machines and less cows.

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.

  • Economies of scale are measured in terms of cost-output elasticity EC.
  • EC=(ΔCC)(Δqq)=MCACEC = (\frac{ΔC}{C}) (\frac{Δq}{q}) = \frac{MC}{AC}
  • Relating to traditional cost measures: (ΔCΔq)(Cq)=MCAC(\frac{ΔC}{ Δq}) (\frac{C}{q}) = \frac{MC}{AC}
  • Elasticity is equal to 1 if marginal costs equal average ones – costs increase proportionally with output and there are no economies/diseconomies of scale.
  • With Economies of scale (costs increase less than proportionately), MC<ACMC < AC andEC<1 EC < 1 .
  • With Dis-economies of scale (costs increase more than proportionately), MC>ACMC > AC and EC>1EC > 1 .

The Relationship between Short-Run and Long-Run Cost

  • Decision to expand capital is important as it may be difficult to change for some time.
  • Graphically - LAC is the envelope of the SAC curves – with economies and dis-economies of scale, the minimum points of the Short Run average cost curves do not lie on the long-run average cost curve.

  • LAC → Firm can change capital expenditure, so it chooses the amount that minimizes average cost of production.
  • It shows economies of scale initially but exhibits dis-economies at higher output levels.
  • The LAC curve never lies above any of the short-run average cost curves, and economies/dis-economies of scale prevent the points of minimum average cost to lie on the long-run average cost curve.
  • A long-run marginal cost curve is not the envelope of the short-run marginal cost curves. Because short-run marginal costs only apply to specific capital expenditures whereas long-run marginal costs apply to a variable amount.

Economies of Scope, Lagrange Multipliers (cost min)

Production with Two outputs → Economies of Scope

  • Firms typically produce more than one product → and they may be linked or have no relation.
  • Firms enjoy production or cost advantages when it produces tow or more products → as production may yield an automatic by-product.

Product Transformation Curves

  • Curve showing the various combinations of two different outputs (products) that can be produced with a given set of inputs.
  • Transformation Curves are bowed out (concave) because there are economies of scope → curves are more far out at higher inputs.
  • Negative slope → to get more of one output, the firms gives up some of the other output.
  • Joint Production has advantages that enable a single company to produce more goods with the same resources than two separate companies.

Economies and Dis-economies of Scope

  • Economies of Scope – 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.
  • Dis-economies of Scope – Situation in which joint output of a single firm is less than could be achieved by separate firms when each produces a single product.
  • No direct relation between economies of scale and economies of scope.
  • E.g. Two-output firm may still enjoy economies of scope even if production process involves dis-economies of scale.

The Degree of Economies of Scope

  • Percentage of cost savings resulting when two or more products are produced jointly rather than individually.
  • Degree of Economies of Scope (SC) =

  • C(q1)C(q1) represents cost of producing only output q1,C(q2)q1, C(q2) represents the cost of producing only output q2,q2, and C(q1,q2)C(q1,q2) represents the joint cost of producing both outputs.
  • With physical units of output added, this becomes C(q1+c2)C(q1 + c2)
  • If SC>0SC > 0 , there are economies of scope. (Joint Cost < Sum of Individual Costs)
  • If SC<0SC < 0 , there are dis-economies of scope (Joint Cost > Sum of Individual Costs)

Production and Cost Theory → a Mathematical Treatment

Cost Minimization

  • Given two inputs capital K and Labor L, the production function F(K,L) describes the maximum output that can be produced for every combination of inputs.
  • MPK(K,L)MP_K(K, L)Marginal Product of Capital
  • MPL(K,L)MP_L(K, L)marginal Product of Labor
  • Cost minimization problem:
  • Minimize C=wL+rKC = wL + rK subject to F(K,L)=q0F(K,L) = q_0
  • Where C is cost of producing the fixed level of output q0q_0 .
  • “Choose values of K and L that minimize Cost subject to Production Function.”

Lagrangian Method

Set up the Lagrangian

  • Φ=wL+rKλ[F(K,L)q0]Φ = wL + rK – λ[F(K, L) – q_0]

F.O.C → Differentiate the Lagrangian with respect to K, L, and λ and equate the resulting derivatives to 0 to obtain necessary conditions for a minimum.

Solve equations to obtain optimizing values of L, K, and λ.

  • Combine the first two conditions:
  • MPK(K,L)r=MPL(K,L)w\frac{MP_K(K,L)}{ r} = \frac{MP_L(K, L) }{ w}
  • Rewrite the first two conditions to evaluate the Lagrange multiplier.

  • If output increases by one unit, the first equation above measures the additional input cost of producing an additional unit of output by increasing capital.
  • The second one measures additional input of producing an additional unit of output by increasing labour.
  • Lagrange Multiplier = Marginal Cost of Production.

Marginal Rate of Technical Substitution

  • Isoquant → Curve that represents the set of all input combinations that give the firm the same level of output (i.e. q0q_0 )
  • Production Isoquant → F(K,L)=q0 F(K,L) = q_0 .
  • MPK(K,L)dK+MPL(K,L)MPK(K,L)MP_K(K, L)dK + \frac{MP_L(K, L)}{ MP_K(K, L)}
  • Where MRTSL,KMRTS_{L,K} is the firm’s Marginal Rate of technical Substitution between labour and capital.
  • Rewrite: MPL(K,L)MPK(K,L)=wr\frac{MP_L(K, L)}{MP_K(K, L)} = \frac{w}{r}
  • Left side = negative of the slope of the isoquant, so the MRTS which trades off inputs while keeping output constant is equal to the ratio of prices of inputs (slope of isocost).
  • Alternatively MPLw=MPKr\frac{MP_L}{ w} = \frac{MP_K}{ r}
  • Marginal products of all production inputs must be equal when marginal products are adjusted by unit cost of each input.

Duality in Production and Cost Theory

  • Firm’s input decision has a dual nature → the optimal choice can be analyzed as:
  • Minimize C=wL+rKC = wL + rK subject to F(K,L)=q0 F(K,L) = q_0
  • Maximize F(K, L) subject to wL+rL=C0wL + rL = C_0
  • Alternative Solution is to maximize output given the budget.

Lagrangian Method

Set up the Lagrangian

  • Φ=F(K,L)μ(wL+rKC0)Φ = F(K,L) – μ(wL + rK – C_0)

F.O.C. Differentiate Lagrangian with respect to K, L, and μ and set the resulting equation = 0 to find necessary conditions for a maximum.

Use equations to solve for K and L. IN particular, we can combine first two equations to equate the Marginal Products and unit costs of inputs. (like before)

The Cobb-Douglas Cost and Production Functions

  • Cobb-Douglas Production Function – Production Function of the form q=AKαLβ q = AK^αL^β
  • q is rate of output, K is quantity of capital, and L is the quantity of labor.
  • A, α, and β are positive constants which implies decreasing marginal products.
  • Returns to Scale:
  • If α + β = 1, there are constant returns to scale because doubling K and L doubles F.
  • If α + β > 1, there are increasing returns to scale.
  • If α + β < 1, there are decreasing returns to scale.
  • Cobb-Douglas function is used to accommodate differences in returns to scale and changes in technology or productivity through changes in A (like the macroeconomics z value).

In Practice

  • Write the LagrangianΦ=wL+rKλ[AKαLβq0] Φ = wL + rK – λ[AK^αL^β – q_0]
  • FOC:

  • Expansion Path - Find the value of Lambda and equate the first two equations:


  • Where L is the Expansion Path.
  • Factor Demand for Capital Sub in the Expansion Path into the Constraint to get a new equation.

  • Cost minimizing quantity of capital.
  • Factor Demand for Labor - Sub in the factor demand for capital into the Expansion Path.
  • If wage rate w rises relative to price of capital r, the firms uses more capital and less labour and vice versa.

  • Cost Function - Total cost of producing any output q can be obtained by subbing in the two factor demand equations into the original cost function.

  • Cost function tells us how total cost of production increases with level of output, and how cost changes as input prices change.
  • Constant Returns to Scale Cost Function (α + β = 1)

  • If wage doubles, cost of producing q0 less than doubles because α <1 and β < 1.

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