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- University of Wisconsin - Milwaukee
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- Business Administration 320
- Trick
- Chapter 9: Applications of Cost Theory

Alicia B.

Short Run Cost Function

- Can be represented by a cubic relationship
- Both capital (K) and labor (L) are fixed

Long Run Cost Function

- A quadratic curve of a cross section of ACs for various firms is used
- Capital (K) is not fixed, labor (L) is fixed

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Log Linear LR Cost Curves

Another functional form to the LR cost curve.

[LnTC = a + b(LnQ) + c(LnW)

- Coefficients are the elasticities
- "b" is the output elasticity of TC
- "b" can never be negative

Significance of "b"

- If
**b = 1**, then constant returns to scale LR cost function - If
**b < 1**, then increasing returns to scale LR cost function - If
**b > 1**, then decreasing returns to scale LR cost function

Testing Returns to Scale: T-test

H_{0}: b = 1

H_{a}: b not equal to 1

t= coefficient/standard error

If critical t < estimated t: reject the null hypothesis

If critical t > estimated t: do not reject the null hypothesis

4 Functional Shapes

Constant

- horizontal line

- 45 degree angle

- U-shaped
- AC is often quadratic
- MC is often quadratic

- S-shaped
- TC is often cubic

To Calculate Average Cost

AC = TC/Q

- quadratic curve

To Calculate Marginal Cost

MC = dTC/dQ

- quadratic curve

Economies of Scope

Economies of scope occur when producing two or more products jointly by one firm is less than the cost of producing them separately.

- Bank offers economies of scope by having people who can help customer with checking, savings, credit cards, etc.

Engineering Cost Approach

- An alternative to regression analysis
- Evaluates cost using knowledge about the efficiency of machinery
- Size and volume are mathematically related (engineering relationships)

Survivor Technique

Examines what size of firms are tending to succeed over time, and what sizes of firms are declining.

- Darwinian survival test for firm size

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Break-even Analysis

If:

- TR=P(Q)
- TC=F+V(Q)

Then at Break-even: TR=TC

Solving for Break-even Quantity

Q_{BE} = F/(P-V)

- How much Q to produce so that TR = TC

Contribution Margin

CM = P-V

- How much each unit sold contributes to the fixed cost

Solving for Break-even Sales Volume

R_{BE} = Q_{BE}(P)

Or R_{BE} = F/(1-V/P)*P

- Amount of sales revenue that breaks even

Variable Cost Ratio

V/P

- Exists in the calculation of Break-even Sales Volume

Solving for Target Profit Quantity

Q_{target∏}_{ }= (F+∏)/(P-V)

- Quantity needed to attain target profit (∏)

Contribution Analysis

Find if added sales through an ad campaign or new product is justified.

- Look at incremental contributions and incremental additions to cost

Do the project if:

- Added contribution > added cost
- CM(ΔQ) > Δdirect FC

Limitations of Break-even and Contribution Analysis

Sometimes the assumptions do not hold:

- Costs may be semi-variable (increases in cost at different increments)
- Many times firms sell multiple products, or small, medium, and large varieties (allocation of costs)
- There is uncertainty as to the price, VC, and FC in the problem
- The planning horizon may be longer than a year

DOL = elasticity of operating profit

- operating profit ∏ = TR - TC
- a measure of the importance of fixed cost or business risk to fluctuations in output

Interpretation: a 1% increase in Q will cause an X% increase in operating profit

Break-even Analysis and Risk Assessment

One approach to risk, is the probability of losing money.

- z is the number of standard deviations away from the mean.

z = (Q_{BE} – Q)/standard deviation

- 68% of time within 1SD
- 95% of time within 2SD
- 99% of time within 3SD

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