Econ 102 > Cost Minimization > Flashcards

Cost Minimization Flashcards

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1
Q

What is the primary goal of cost minimization?

A

To find the cheapest way to produce a given level of output.

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2
Q

What are conditional factor demand functions?

A

Functions showing the optimal input quantities as a function of input prices and output: x1∗(w1,w2,y)x1∗​(w1​,w2​,y) and x2∗(w1,w2,y)x2∗​(w1​,w2​,y).

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3
Q

What is the cost function?

A

The minimum cost of producing output y: c(w1,w2,y)=w1x1∗(w1,w2,y)+w2x2∗(w1,w2,y)c(w1​,w2​,y)=w1​x1∗​(w1​,w2​,y)+w2​x2∗​(w1​,w2​,y).

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4
Q

What is an isocost line?

A

All combinations of inputs that yield the same total cost.

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5
Q

What is the equation for an isocost line?

A

C=w1x1+w2x2C=w1​x1​+w2​x2​ or rearranged: x2=Cw2−w1w2x1x2​=w2​C​−w2​w1​​x1​.

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6
Q

What is the slope of an isocost line?

A

−w1w2−w2​w1​​ (the negative of the input price ratio).

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7
Q

How does cost change as we move to higher isocost lines?

A

Cost increases as we move to higher isocost lines.

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8
Q

What is the cost-minimization condition?

A

The slope of the isocost line equals the slope of the isoquant: −w1w2=−MP1MP2−w2​w1​​=−MP2​MP1​​.

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9
Q

What does the tangency condition mean economically?

A

The rate at which the firm can trade inputs in the market (price ratio) equals the rate at which technology allows trading inputs while maintaining output (TRS).

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10
Q

What is the cost function for Perfect Complements?

A

c(w1,w2,y)=w1ya+w2ybc(w1​,w2​,y)=w1​ay​+w2​by​.

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11
Q

What is the cost function for Perfect Substitutes?

A

c(w1,w2,y)=min⁡(w1ya,w2yb)c(w1​,w2​,y)=min(w1​ay​,w2​by​) - use whichever input is cheaper per unit of output.

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12
Q

What is the general form of the cost function for Cobb-Douglas?

A

c(w1,w2,y)=Kw1aa+bw2ba+by1a+bc(w1​,w2​,y)=Kw1a+ba​​w2a+bb​​ya+b1​.

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13
Q

How do Constant Returns to Scale affect costs?

A

Costs increase linearly with output. Average Cost is constant.

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14
Q

How do Increasing Returns to Scale affect costs?

A

Costs increase less than proportionally with output. Average Cost decreases as output increases.

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15
Q

How do Decreasing Returns to Scale affect costs?

A

Costs increase more than proportionally with output. Average Cost increases as output increases.

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16
Q

For Cobb-Douglas, how do we determine returns to scale from the exponents?

A

CRS if a+b=1, IRS if a+b>1, DRS if a+b<1.

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17
Q

What is the key difference between short run and long run cost minimization?

A

Short run: Some factors are fixed. Long run: All factors are variable.

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18
Q

How do short-run costs compare to long-run costs?

A

Short-run cost is always greater than or equal to long-run cost.

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19
Q

When are short-run and long-run costs equal?

A

Only when the fixed input in the short run happens to be at its long-run cost-minimizing level.

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20
Q

What are fixed factors?

A

Factors that must be paid even if output is zero.

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21
Q

What are quasi-fixed factors?

A

Factors that must be paid in a fixed amount whenever output is positive, but can be avoided if output is zero.

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22
Q

What is the difference between sunk costs and recoverable costs?

A

Sunk costs cannot be retrieved, while recoverable costs can be recovered (e.g., through resale of equipment).

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23
Q

What are the three steps to find the minimal total cost function?

A

1) Find conditional input demands
2) Substitute into cost equation
3) Simplify

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24
Q

What is the difference between conditional factor demands and profit-maximizing factor demands?

A

Conditional demands depend on output level, while profit-maximizing demands depend on output price.

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25
If input prices double, what happens to the slope of the isocost line?
Nothing - the slope depends on the ratio of input prices, which doesn't change when both prices double.
26
If a firm has IRS technology, what should it do to reduce average costs?
Increase production to spread fixed costs over more units of output.
27
Why might a firm operate at higher cost in the short run than in the long run?
Because some inputs are fixed in the short run, preventing optimal adjustment.
28
What is the fundamental objective of cost minimization?
To identify the cheapest input combination that can produce a specified output level, given input prices.
29
What are conditional factor demand functions mathematically?
x1∗(w1,w2,y)x1∗​(w1​,w2​,y) and x2∗(w1,w2,y)x2∗​(w1​,w2​,y) - showing optimal input quantities as functions of input prices and target output.
30
How is the cost function derived from conditional demands?
c(w1,w2,y)=w1x1∗(w1,w2,y)+w2x2∗(w1,w2,y)c(w1​,w2​,y)=w1​x1∗​(w1​,w2​,y)+w2​x2∗​(w1​,w2​,y) - the minimum expenditure to produce output y.
31
What does an isocost line represent?
All combinations of inputs (x1,x2)(x1​,x2​) that require the same total expenditure.
32
What is the general form of the isocost equation?
C=w1x1+w2x2C=w1​x1​+w2​x2​ or solved for x₂: x2=Cw2−w1w2x1x2​=w2​C​−w2​w1​​x1​.
33
What is the economic interpretation of the isocost slope?
−w1w2−w2​w1​​ represents the market trade-off between inputs - how much x₂ must be sacrificed for one more unit of x₁ while maintaining the same cost.
34
How does changing the cost level C affect the isocost line?
Increasing C creates a parallel outward shift; decreasing C creates a parallel inward shift.
35
What is the graphical solution to cost minimization?
Find the lowest isocost line that just touches (is tangent to) the target isoquant.
36
State the tangency condition mathematically.
MP1MP2=w1w2MP2​MP1​​=w2​w1​​ or equivalently MP1w1=MP2w2w1​MP1​​=w2​MP2​​ - the marginal product per dollar should be equal across all inputs.
37
What is the economic intuition behind the tangency condition?
The rate at which technology allows input substitution (TRS) equals the rate at which the market allows input substitution (price ratio).
38
When might the tangency condition not hold?
With corner solutions (perfect substitutes) or kinked solutions (perfect complements).
39
How do conditional factor demands differ from profit-maximizing factor demands?
Conditional demands are functions of output level y, while profit-maximizing demands are functions of output price p.
40
What is the key property of conditional factor demands?
They are homogeneous of degree 0 in input prices - doubling all input prices doesn't change optimal input quantities.
41
For Perfect Complements f(x1,x2)=min⁡{ax1,bx2}f(x1​,x2​)=min{ax1​,bx2​}, what are the conditional demands?
x1c=yax1c​=ay​, x2c=ybx2c​=by​ - inputs are used in fixed proportions regardless of prices.
42
What is the cost function for Perfect Complements?
c(w1,w2,y)=w1ya+w2ybc(w1​,w2​,y)=w1​ay​+w2​by​ - cost increases linearly with output.
43
For Perfect Substitutes f(x1,x2)=ax1+bx2f(x1​,x2​)=ax1​+bx2​, what determines input choice?
Compare aw1w1​a​ vs bw2w2​b​ - use only the input with higher output per dollar.
44
What is the cost function for Perfect Substitutes?
c(w1,w2,y)=min⁡(w1ya,w2yb)c(w1​,w2​,y)=min(w1​ay​,w2​by​) - use the cheaper input exclusively.
45
For Cobb-Douglas f(x1,x2)=Ax1ax2bf(x1​,x2​)=Ax1a​x2b​, what is the general cost function form?
c(w1,w2,y)=Kw1aa+bw2ba+by1a+bc(w1​,w2​,y)=Kw1a+ba​​w2a+bb​​ya+b1​ where K is a constant depending on A, a, b.
46
Under Constant Returns to Scale, how do costs behave?
c(w1,w2,y)=k⋅yc(w1​,w2​,y)=k⋅y - costs are linear in output, average cost is constant.
47
Under Increasing Returns to Scale, how do costs behave?
c(w1,w2,y)=k⋅yac(w1​,w2​,y)=k⋅ya with a < 1 - costs increase at a decreasing rate, average cost falls.
48
Under Decreasing Returns to Scale, how do costs behave?
c(w1,w2,y)=k⋅ybc(w1​,w2​,y)=k⋅yb with b > 1 - costs increase at an increasing rate, average cost rises.
49
For Cobb-Douglas, how do we determine returns to scale from exponents?
Sum a+b: =1 (CRS), >1 (IRS), <1 (DRS).
50
What defines the short run in production theory?
At least one input is fixed - the firm cannot adjust all factors of production.
51
What defines the long run in production theory?
All inputs are variable - the firm can adjust all factors of production.
52
How do short-run costs compare to long-run costs?
cs(y)≥c(y)cs​(y)≥c(y) for all y - short-run cost is always at least as large as long-run cost.
53
When are short-run and long-run costs equal?
Only when the fixed input in the short run happens to be at its long-run optimal level for that output.
54
Why is the long-run cost curve called the "envelope" of short-run cost curves?
Because it envelopes (lies below or touches) all possible short-run cost curves.
55
What are fixed factors?
Factors that must be paid regardless of output level (e.g., lease payments, salaried staff).
56
What are quasi-fixed factors?
Factors that must be paid in fixed amounts when producing positive output, but can be avoided if output is zero.
57
What is the key difference between sunk costs and recoverable costs?
Sunk costs are irrecoverable (e.g., advertising), while recoverable costs can be recouped (e.g., equipment resale).
58
If all input prices double, what happens to cost-minimizing input quantities?
No change - conditional demands are homogeneous of degree 0 in input prices.
59
If a firm experiences IRS, what is the optimal production strategy?
Increase scale to benefit from falling average costs - potentially leading to natural monopoly.
60
Why might a rational firm operate at a loss in the short run?
If revenue covers variable costs and contributes to fixed costs, it's better than shutting down completely.
61
How does the concept of opportunity cost affect the economic cost function?
All inputs are valued at their current market prices, including implicit costs of owned resources.
62
What is Shephard's lemma?
The derivative of the cost function with respect to an input price gives the conditional demand for that input: ∂c∂wi=xic∂wi​∂c​=xic​.
63
How does the cost function respond to input price changes?
It is non-decreasing and homogeneous of degree 1 in input prices.
64
What is the relationship between average cost and returns to scale?
Falling AC indicates IRS, constant AC indicates CRS, rising AC indicates DRS.
Econ 102
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