Cost Curves

• How the firm chooses the inputs it uses to produce a given output at the lowest possible cost.
  
• How the firm’s production costs change as it changes the number and mix of its inputs.

Basic Cost Concepts

• Opportunity cost is the cost of a good measured by the alternative uses foregone by producing the good.
  
• Accounting cost is the actual cost paid for inputs.
  
• Economic cost is the amount required to keep an input in its present use or the amount that input would be worth in its next best alternative use.

Basic Cost Concepts (cont.)

• Labor costs
  • Wage payments are explicit costs.
    
  • Assume that the wage rate equals the amount a worker would earn in their next best alternative employment.
• Capital costs
  • An accountant uses the historical cost of capital and some depreciation rule.
    
  • An economist treats the cost of capital as a sunk cost — an expenditure that once made can’t be recovered.
    
  • The implicit cost of capital equals its rental rate.
• Entrepreneurial costs
  • Owners of the firm are entitled to the difference between revenue and costs: accounting profit.
    
  • Entrepreneurial cost is the salary an owner could earn in their next best employment.
    
  • A firm’s economic profit = accounting profit − entrepreneurial cost.

Economic Profits and Cost Minimization

• Two assumptions:
  1. Two inputs: labor \(L\) and capital \(K\).
     
  2. Inputs are hired in perfectly competitive markets.

\[ TC = wL + vK \] \[ \pi = Pq - wL - vK = P f(K,L) - wL - vK \]

• Economic profit depends only on the amounts of labor and capital hired.

Cost-Minimizing Input Choice

How does a firm produce a given output at the lowest possible cost?

Three equivalent conditions:

1. \(RTS = w/v\)

2. Isoquant tangent to the total cost line.

3. \(\frac{MPL}{w} = \frac{MPK}{v}\)

Example: \(RTS = w/v\)

• Suppose producing \(q\) requires \(10K\) and \(10L\).

• Let \(RTS = 2\), \(w = 1\), \(v = 1\).

Then: \[ TC = 10(1) + 10(1) = 20 \] But \(RTS = 2 \neq w/v = 1\).
So the firm can replace 2 units of \(K\) with 1 of \(L\) to reduce cost.

\[ TC' = 11(1) + 8(1) = 19 \]

Thus, whenever \(RTS \neq w/v\), the firm can change the allocation of inputs in a way that achieves lower cost.

Thu \(RTS \neq w/v\), the firm can change the allocation of inputs in a way that achieves lower cost.

Cost Minimization (contd.)

Cost minimization occurs where the isoquant is tangent to an isocost line.

More on: \(\frac{MPL}{w} = \frac{MPK}{v}\)

\[ RTS_{L,K} = \frac{MPL}{MPK} = \frac{w}{v} \quad \Rightarrow \quad \frac{MPL}{w} = \frac{MPK}{v} \]

• Firm gets the same “bang for the buck” from each input at the cost-minimizing point.

• For example: If \(\frac{MPL}{w} > \frac{MPK}{v}\):

  → Labor yields more output per dollar.

  → Firm should hire more labor, use less capital.

Short Run vs. Long Run

• Short run: the period of time in which a firm must consider some inputs (e.g. capital) to be fixed.

• Long run: the period of time in which a firm may consider all of its inputs to be variable.

Total cost: Total Fixed Cost + Total Variable Cost

• Fixed Costs (FC): costs associated with inputs that are fixed in the short run.
  • Fixed cost F, independent of the level of output
  • A coffee shop must pay its rent even if it does not open.
  • AFC is the fixed cost per unit of output
  • As the numebr of unit of output increase, AFC falls steeply at first and then gradually
• Variable Costs (VC): costs associated with inputs that can be varied in the short run.
  • Changes with output.
  • Example: Amount spent on raw materials for cake depends on how many cakes are being baked.

SR Total Cost Curves

• TFC curve is a straight line because TFC is constant at all levels of output

• TC and TVC curves are inversely S-shaped because they rise initially at a decreasing rate, then at a constant rate and finally, at an increasing rate. This shape is determined by the law of variable proportions:

  • At Q=0, TC=TFC because there’s no vairable cost at zero level of output.
  • So, TC and TVC curves start from the same point, which is above the origin.

SR Average and Marginal Cost Functions

• Average Cost: total cost per unit.
  
• Marginal Cost: cost of one additional unit.

\[ AFC(Q) = \frac{FC}{Q}, \quad AVC(Q) = \frac{VC(Q)}{Q}, \quad ATC(Q) = AFC(Q) + AVC(Q) \]

\[ AC = \frac{TC}{Q}, \qquad MC = \frac{dTC}{dQ} \]

\[ MC(Q) = \frac{dTC(Q)}{dQ} = \frac{dVC(Q)}{dQ} \]

SR Average and Marginal Costs (contd.)

• MC<AVC \(\rightarrow\) average falling, MC>AVC \(\rightarrow\) average rising
• \(MC\) intersects \(AVC\) at its minimum.
  
• \(MC\) intersects \(ATC\) at its minimum.
  
• If \(MC < AC\): average falls; if \(MC > AC\): average rises.

SR Average and Marginal Costs (contd.)

• As output increases,the gap btw AC and AVC falls.

• As output increases, the gap between AC and AFC increases.

• AVC never intersects AC due to the gap of AFC.

• AFC needs to be 0 so as to make aVC intersect AC.

• MC intersects AC and AVC at their lowest points.

Numerical Example

Q	FC	VC	TC	MC	AFC	AVC	ATC
0	50	0	50	–	–	–	–
1	50	40	90	40	50	40	90
2	50	70	120	30	25	35	60
3	50	90	140	20	16.7	30	46.7
4	50	100	150	10	12.5	25	37.5
5	50	120	170	20	10	24	34

• \(MC\) decreases then increases (diminishing \(MPL\)).
  
• \(AVC\) U-shaped.
  
• \(ATC\) minimized where \(MC = ATC\).
  
• \(AFC\) falls monotonically.

Link to Production

\[ VC(Q) = w \cdot L(Q) \] \[ MC(Q) = \frac{dVC}{dQ} = w \frac{dL}{dQ} = \frac{w}{MPL} \]

• Diminishing \(MPL\) → rising \(MC\) beyond some \(Q\).

Short Run vs. Long Run: Input Inflexibility

• In the short run, with capital fixed, firm may not achieve the cost-minimizing input combination.

• In the short run, capital is fixed so firm may use non-optimal labor.

Shifts in Cost Curves

Factors that shift cost curves:

1. Changes in input prices (e.g. wages).
   

• An increase or decrease in input prices will cause the cost curves to shift up/down
• Will depend on the ability fo the firm to substitute inputs

2. Technological innovation → shifts curves downward.
   

• Cost curves will shift down with tech innovation
• Tech change may be “biased”

3. Economies of scope → multiproduct interactions.

• Relaltes to multiproduct firms
• Expansion of one product may improve the ability to produce another product

A Numerical Example: Hamburger Heaven

\[ TC = 5K + 5L, \quad q = 40 \]

Output (q)	Workers (L)	Grills (K)	Total Cost (TC)
40	1	16.0	£85.00
40	2	8.0	50.00
40	3	5.3	41.50
40	4	4.0	40.00
40	5	3.2	41.00
40	6	2.7	43.50
40	7	2.3	46.50
40	8	2.0	50.00
40	9	1.8	54.00
40	10	1.6	58.00

Short-Run Numerical Example

Suppose we now fix the number of grills to 4.

Output (q)	L	K	STC	SAC	SMC
10	0.25	4	21.25	2.125	—
20	1.00	4	25.00	1.25	0.50
30	2.25	4	31.25	1.04	0.75
40	4.00	4	40.00	1.00	1.00
50	6.25	4	51.25	1.025	1.25
60	9.00	4	65.00	1.085	1.50
70	12.25	4	81.25	1.160	1.75
80	16.00	4	100.00	1.250	2.00
90	20.25	4	121.25	1.345	2.25
100	25.00	4	145.00	1.450	2.50

SRAC and SRMC curves (for 4 grills)

Short-Run Cost Components

\[ TC(Q) = FC + VC(Q) \]

• Fixed Cost \(FC > 0\) even if \(Q=0\).
  
• Variable Cost \(VC(Q)\) increases with \(Q\).

Behavior of SR Curves:

• \(AFC\) strictly decreasing in \(Q\).
  
• \(AVC\) typically U-shaped (increasing then diminishing returns).
  
• \(ATC\) U-shaped as sum of \(AFC + AVC\).

Long-Run Costs

\[ LRTC(Q): \text{ minimum total cost when all inputs variable} \] \[ LRAC(Q) = \frac{LRTC(Q)}{Q}, \quad LRMC(Q) = \frac{dLRTC(Q)}{dQ} \]

The Firm’s Expansion Path

The expansion path shows cost-minimizing \((K,L)\) combinations for different output levels.

Cost Curves

Relationship between output and total cost depends on returns to scale:

1. Constant returns: costs rise proportionally with output.
   
2. Decreasing returns: costs rise faster.
   
3. Increasing returns: costs rise slower.

Cost Minimization and Expansion Path

Minimize \(C = wL + rK\) subject to \(f(K,L)=Q\).
At optimum:

\[ MRTS_{LK} = \frac{MPL}{MPK} = \frac{w}{r} \]

Expansion path traces \((K^*, L^*)\) as \(Q\) increases → defines \(LRTC(Q)\).

Economies and Diseconomies of Scale

• Economies of scale: \(LRAC'(Q) < 0\).
  
• Diseconomies of scale: \(LRAC'(Q) > 0\).

LRAC as Lower Envelope of SRATC

Conclusion

• Cost min: \(RTS= w/v\) and equal \(MP_i/p_i\), across inputs

• Expansion path \(\rightarrow\) min-cost input combinations

• Avg and marginal cost curves derived from total cost

• In SR, some inputs fixed so that costs not minimal

• Diminishing productivity \(\rightarrow\) rising short run costs

• Cost curves shift with input prices, tech, and economies of scope