# 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
- $MC = \frac{ΔVC}{Δq} = \frac{ΔTC}{Δq}$
- 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, $\frac{TC}{q}$- Tells us the per-unit cost of production.
- ATC has two components
**Average Fixed Cost (AFC) –**Fixed cost divided by the level of output.- $\frac{FC}{q}$
**Average Variable Cost (AVC) –**Variable cost divided by the level of output.

### 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:
- $MC = \frac{ΔVC}{Δq} = \frac{w ΔL }{Δq}$
- Where w = wage and ΔL is the extra output.
- Extra labour to obtain one more unit of labor is $\frac{1}{MP_L}$ , so:
- $MC = \frac{w}{MP_L}$
- 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.

### 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.

#### 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**→**We can rewrite total cost equation to a***straight line*.**Straight Line Total Cost**→**Slope is $\frac{ΔK}{ΔL} = -(\frac{w}{r})$ 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

- $MRTS = -\frac{ΔK}{ΔL} = \frac{MP_L}{MP_K}$
- Because the Isocost line has a slope equal to change in capital over change in labor, $\frac{MP_L}{MP_K} = \frac{w}{r}$
- Condition: $\frac{MP_L}{w} = \frac{MP_K}{r}$
*$\frac{MP_L}{w}$ represents additional output that results from spending an additional dollar for labor**$\frac{MP_K}{r}$ 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 =**$\frac{ΔK}{ΔL}$

#### 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.

Generally → 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 E
_{C}. - $EC = (\frac{ΔC}{C}) (\frac{Δq}{q}) = \frac{MC}{AC}$
- Relating to traditional cost measures: $(\frac{ΔC}{ Δq}) (\frac{C}{q}) = \frac{MC}{AC}$
- 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), $MC < AC$ and$EC < 1$ .
- With Dis-economies of scale (costs increase more than proportionately), $MC > AC$ and $EC > 1$ .

**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) =**

- $C(q1)$ represents cost of producing only output $q1, C(q2)$ represents the cost of producing only output $q2,$ and $C(q1,q2)$ represents the joint cost of producing both outputs.
- With physical units of output added, this becomes
*$C(q1 + c2)$* - If $SC > 0$ , there are economies of scope. (Joint Cost < Sum of Individual Costs)
- If $SC < 0$ , 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. **$MP_K(K, L)$ –**Marginal Product of Capital**$MP_L(K, L)$ –**marginal Product of Labor- Cost minimization problem:
$F(K,L) = q_0$*Minimize*$C = wL + rK$ subject to- Where C is cost of producing the fixed level of output $q_0$ .
- “Choose values of K and L that minimize Cost subject to Production Function.”

### Lagrangian Method

#### Set up the Lagrangian

- $Φ = wL + rK – λ[F(K, L) – q_0]$

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.

S**olve equations to obtain optimizing values of L, K, and λ.**

- Combine the first two conditions:
- $\frac{MP_K(K,L)}{ r} = \frac{MP_L(K, L) }{ w}$
- 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. $q_0$ )
- Production Isoquant → $F(K,L) = q_0$ .
- $MP_K(K, L)dK + \frac{MP_L(K, L)}{ MP_K(K, L)}$
- Where $MRTS_{L,K}$ is the firm’s Marginal Rate of technical Substitution between labour and capital.
- Rewrite: $\frac{MP_L(K, L)}{MP_K(K, L)} = \frac{w}{r}$
- 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*→*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 $C = wL + rK$ subject to $F(K,L) = q_0$
- Maximize F(K, L) subject to $wL + rL = C_0$
- Alternative Solution is to maximize output given the budget.

### Lagrangian Method

#### Set up the Lagrangian

- $Φ = F(K,L) – μ(wL + rK – C_0)$

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 $q = AK^αL^β$- 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 Lagrangian$Φ = wL + rK – λ[AK^αL^β – q_0]$
- FOC:

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

or

- Where L is the Expansion Path.
**Factor Demand for Capital**→

- 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 q
_{0}less than doubles because α <1 and β < 1.

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