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## The Firm’s Production and Selling Decisions

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### 4

The Firm’s Production and Selling Decisions

Outline

- Production and Input Choice, with One Variable Input
- Multiple Input Decisions: The Choice of Optimal Input Combinations
- Cost and Its Dependence on Output
- Economies of Scale

Outline

- Price and Quantity: One Decision, Not Two
- Total Profit: Keep your Eye on the Goal
- Marginal Analysis and Maximization of Total Profit
- Generalization: The Logic of Marginal Analysis and Maximization

Production and Input Choice, with 1 Variable Input

- Arkansas chicken farmer named Florence, who owns a small poultry business.
- She knows Q corn she feeds her chickens will impact Q meat.
- She could also buy more T, growth hormones, and L to ↑Q meat. But for now, let’s focus on the relationship between poultry meat and corn.

Production and Input Choice, with 1 Variable Input

- Total Physical Product (TPP) = amount of output that can be produced as 1 input changes, with all other inputs held constant.
- Table 1 shows TPP or how much chicken Flo can produce with different Q corn, holding all other inputs fixed.
- If Q corn = 0 → Q meat = 0. Each add. bag of corn yields more poultry. 4 bags → 100 lbs. After 9 bags, ↑corn → ↓output –chickens are so overfed they become ill.

Production and Input Choice, with 1 Variable Input

- Average Physical Product (APP) = TPP/(Q of input) = measures output per unit of input.
- E.g., 4 bags corn → 100 lbs meat, so APP = 25.
- Marginal Physical Product (MPP) = additional output resulting from a 1 unit increase in the input, holding all other inputs constant.
- E.g., ↑corn from 4 to 5 bags, the 5th bag yields an add. 30 lbs of meat.

Graph of MPP

- Marginal returns to an input typically rise and then fall.
- Area of ↑MPP (1 to 4 bags) → each add. bag of corn adds more to TPP than previous bag. ↑TPP rapidly.
- Area of ↓MPP (between 4 and 9 bags) → each add. bag of corn adds less to TPP than previous bag. ↑TPP at a dim. rate.
- Area of (-)MPP (beyond 9 bags) → each add. bag of corn reduces TPP by more than previous bag. ↓TPP.

The “Law” of Diminishing Marginal Returns

- ↑ Q of any one input, holding Q of all other inputs constant, leads to lower marginal returns to the expanding input.
- E.g., Flo feeds chickens more and more, without giving them extra water, cleaning up after them more, or buying add. chickens. Eventually overfed and become sick.
- Law of dim. marginal returns should hold for most activities.

Can you think of one?

Optimal Purchase Rule for a Single Input

- How does a firm decide on the quantity of an input?
- Assume P corn = $10/40-lb bag and P chicken = $0.75/lb.

Consider purchasing just 1 bag of corn. Does this max profits?

- 1 bag produces 14 lbs of chicken.

TR: $0.75 x 14 = $10.50

TC: $10 x 1 = $10.00

Profit: = $0.50

- Shouldn't stop at 1 bag because 2 bags yield more profit.

TR: $0.75 x 36 = $27.00

TC: $10 x 2 = $20.00

Profit: = $7.00

Optimal Purchase Rule for a Single Input

- Easier way to proceed. Until 9 bags, each add. bag of corn ↑Q chicken. So each bag (1-9) raises TR, but also costs $10. To max profit, Flo should compare revenue that each bag generates against the cost of each bag.
- Marginal Revenue Product (MRP) = MPP x Price of output.
- MRP = add. revenue generated from ↑input by 1 unit.

Optimal Purchase Rule for a Single Input

- Rule: If MRP > P of an input → use more of the input.

If MRP < P of an input → use less of the input.

- Purchase an input where MRP = P of the input.
- E.g., Flo should purchase 7 bags of corn.

Can you explain why she should not buy the 8th bag?

- Note: ↓MPP (bag 4 to 9) → ↓MRP. At 7 bags, Flo is producing where dim. MPP sets in. Flo should stop ↑corn purchases when MRP falls to = P of corn.

Multiple Input Decisions

- Firms seek the method of production that is least costly.
- Consider the choice between L and K in prod. Compared with Mexico, in U.S., L is expensive and K is cheap. So (K/L) U.S. > (K/L) Mexico
- One input can often be substituted for another in production.
- E.g., shoes produced in Mexico are manufactured using more L and less K than shoes in U.S.

Multiple Input Decisions

- A firm can produce same amount of a good with less of one input (say L) as long as it’s willing to use more of another input (like K).
- Actual combos of inputs (such as K and L) depend on relative P of inputs. Firms strive to produce a good using the least expensive method.

Marginal Rule for Optimal Input Proportions

- E.g., Flo can feed chickens soymeal or cornmeal –they are substitutes in production.
- Not perfect substitutes. Soymeal has more protein but fewer carbohydrates than corn.
- Best to feed some combo of 2 meals. ↓Q poultry if Flo relies too much on 1 input. There are dim. returns to substitution among the inputs.

Marginal Rule for Optimal Input Proportions

How much of each input should Flo purchase?

- Feed ↑corn and ↓soy. Soy costs twice as much, but yields only 67% more meat.
- If Flo ↓soy by 1 bag → saves $20. But ↓outputby 50 lbs. So buy 1.67 (or 50/30) bags of corn to make up for ↓output, cost = $16.70. She saves $3.30 while holding Q output fixed.

Marginal Rule for Optimal Input Proportions

- Above: MPPsoy/Psoy < MPPcorn/Pcorn

i.e., 50/$20 < 30/$10

- Soy yields 2.5 lbs. meat per $1 while corn yields 3 lbs. per $1. More output from corn rather than soy at the margin.
- MPP of an input/P of an input = add. output from spending $1 on the input.
- By substituting input with lower output per $1 for input with higher output per $1; firm can reduce costs while holding Q output fixed.

Marginal Rule for Optimal Input Proportions

- Rule: if MPPb/Pb > MPPa/Pa→ spend less on input a and more on input b.
- Optimally, MPPa/Pa = MPPb/Pb
- Above: MPPcorn/Pcorn > MPPsoy/Psoy
- These ratios will equalize at an optimum because of dim. MPP. As Flo uses ↑corn and ↓soy →↓MPP corn and ↑MPP soy, until two ratios are equal.

Marginal Rule for Optimal Input Proportions

- Changes in Input Prices and Input Proportions:
- Optimally, MPPcorn/Pcorn = MPPsoy/Psoy
- What if ↑P corn?
- Then ↑MPP corn to match ↑P corn. How? Flo will use ↓corn and ↑soy until ratios are equal.
- As ↑P input → firms switch to cheaper inputs.

Cost Curves and Input Quantities

- 3 different cost curves –Total Cost (TC), Average Cost (AC), and Marginal Cost (MC).
- Flo’s costs depend on Q of inputs and on P of those inputs.
- To calculate costs, assume:
- P corn is beyond Flo's control.
- Q of all other inputs (except corn) are fixed.
- P corn = $10 per 40 lb. bag

Cost Curves and Input Quantities

- TPP → Q output firm can produce given Q inputs. Q inputs and P inputs → firm can determine TC of producing any Q output.
- TC = P inputs x Q inputs
- AC = TC/Q output
- E.g., TC 100 lbs = $40 → AC = $40/100 = $0.40
- MC = TC when output increases by 1 unit
- E.g., if TC 100 lbs. = $40.00

TC 99 lbs. = $39.70

MC 100th lb. = $0.30

- Note: table above doesn’t show this because ↑output > 1.

FIGURE 4. Flo’s Average Cost and Marginal Cost Curves

AC and MC typically ↓ and then ↑ as the ↑output level.

Fixed and Variable Costs

- TC, AC, and MC can be divided into 2 parts –fixed costs and variable costs.
- Fixed cost is the cost of an input whose Q does not ↑ when ↑output. Input that the firm requires to produce any output. Any other cost is a variable cost.
- E.g., takes at least 1 taxi to run a cab co. and its cost is the same whether 1 or 60 people ride in it. But gas use rises as more people ride. Taxi is a fixed cost and gas is a variable cost.

What are the fixed and variable costs where you work?

Fixed and Variable Costs

- TC = TVC + TFC
- AC = AVC + AFC
- AC = TC/Q output
- AVC = TVC/Q output
- AFC = TFC/Q output

Table 4. Flo’s Total and Average Fixed Costs

Flo pays rent of $5 per week for her chicken coop.

FIGURE 6. Graph of Flo’s Average Fixed Cost

If Flo produces 1 package, TFC is carried by 1. But if she produces 4, TFC gets divided between 4 packages. So ↓AFC as ↑output.

FIGURE 7. Flo’s Total Variable Cost Curve

TVC has same shape as TC because ↑variable costs as ↑output.

Fixed and Variable Costs

- Marginal Cost = Marginal Variable Cost (MVC)

Why doesn't MC have a fixed component (i.e., MC = MVC + MFC)?

Shape of the Average Cost Curve

- AC is generally U shaped –it initially declines and eventually rises with the level of output.
- AC declines for 2 reasons:
- Changing input proportions: at first, Flo feeds chickens more corn while holding all other inputs constant. Output rises rapidly when ↑MPP corn, which tends to ↓AC.
- ↓Average fixed costs as ↑output.

Shape of the Average Cost Curve

- AC eventually rises for 2 reasons:
- Dim MPP: ↑output more slowly as ↓MPP corn, which tends to ↑AC.
- Bureaucratic mess: as firms grow in size they lose personal touch of management and become increasingly bureaucratic, which drives up costs.
- Point where ↑AC varies by industry. AC in auto industry begins ↑ after more units of output than farming. Huge K investment → AFC↓ dramatically.

Short-run versus Long-run Costs

- Cost of changing a firm's output level depends on period of time under consideration. Many input choices are precommitted by past decisions.
- Sunk cost = a cost to which a firm is precommitted for some limited period of time.
- E.g., a 2-year-old machine with a 9-year economic life is a variable cost after 7 years because the machine would have to be replaced anyway.

Short-run versus Long-run Costs

- SR = period of time when some of the firm's cost commitments end.
- LR = period of time when all of the firm's cost commitments end.
- There are no fixed costs in LR –all costs are variable.
- E.g., if # of workers can be altered daily, and # of machines altered yearly, and size of plant every 10 years. Then 10 years is the LR.

Short-run versus Long-run Costs

- Size of a firm may be fixed in SR because it has purchased or leased a particular plant, but firm can alter size of its plant in LR.
- E.g., Flo has already built a chicken coop, which restricts her ability to ∆ output level in SR. In LR, Flo can build a new larger coop to produce more.

Average Cost Curve in the Short and Long Run

- LR AC curve differs from SR AC curve because all inputs are variable in LR.
- E.g., In SR, Flo can only chose how many chickens to squeeze into coop. In LR, she can chose among different coop sizes.

Average Cost Curve in the Short and Long Run

- If Flo expects to sell 40 → she buys a small coop with AC of SL. If Q = 40 → AC = $12 (pt U).
- She is surprised by strong D and can sell 100 with AC = $12 (pt V).
- Now she needs a bigger coop with AC of BG with its lower AC of $9 for Q = 100.
- In SR, Flo is stuck with AC of SL. In LR, she can replace coop and the relevant AC is STG.
- LR AC curve shows the lowest possible SR AC for each output level.

Economies of Scale

- Returns to scale indicates how the output level changes when all the firm's inputs are doubled.
- Increasing Returns to Scale (IRTS): Q output more than doubles.
- IRTS gives a cost advantage to larger firms. Found in industries like telecommunications, electricity, automobiles, and aircraft.
- Constant Returns to Scale (CRTS): Q output doubles.
- Decreasing Returns to Scale (DRTS): Q output less than doubles.
- Gives a cost advantage to smaller firms. Most U.S. industries have DRTS.

Economies of Scale

- Returns to scale impacts the shape of the AC curve.
- AC = TC/Q output = (P input x Q input)/Q output
- E.g., if Q inputs doubles and Q output doubles, then AC is constant.

Increasing returns

Constant returns

Decreasing returns

to scale

to scale

to scale

AC

AC

FIGURE 9. 3 Possible Shapes for the AC CurveLong-Run Average Cost

Long-Run Average Cost

Long-Run Average Cost

Quantity of Output

Quantity of Output

Quantity of Output

(a)

(b)

(c)

Economies of Scale

- Law of dim. marginal returns and IRTS may seem contradictory, but they are unrelated.
- Dim. marginal returns refers to increasing a single input. Returns to scale refers to a doubling of all inputs.
- A firm with dim. returns to a single input could have IRTS, CRTS, or DRTS.

Price and Quantity: One Decision, Not Two

- Critical decision -when Apple decides how many ipods to produce and P it will charge.
- P affects how consumers respond and Q affects K and L costs.
- When firms chose P and Q to max profits → they can pick only one –P or Q.
- Chose P → customers decide Q
- Chose Q → market determines P at which this Q can be sold

FIGURE 10. Demand Curve for Flo’s Poultry Meat

Flo faces a local D curve.

If she picks P = $19 → Qd = 1.

If she picks Q = 9 → P = $11 to find required # of customers.

A

$19

Price per package

B

$11

D

1

9

Quantity of Chicken (20 lb-packages per week)

Price and Quantity: One Decision, Not Two

- Each pt on D curve corresponds to a (P,Q) pair. A firm can pick 1 pair, but it can never pick P from 1 pt on D and a different Q from another pt on D.
- Economists assume that firms pick (P,Q) pair that maximizes profits.

Total Profit: Keep Your Eye on the Goal

- Total profit (or economic profit) = TR – TC (including opportunity cost)
- Opportunity costs include any K or L supplied by the firm’s owners.
- Economic profit = Accounting profit – opportunity cost.
- E.g., if a talented attorney, gives up her salary of $120,000 to start her own law firm and earns $150,000 after paying for all operating costs → accounting profit = $150,000 but economic profit = $30,000
- E.g., if you start a business and earn 6% on money you invested but could have earned 4% in T-Bills → economic profit = 2%.

Total Profit: Keep Your Eye on the Goal

- Total, Average, and Marginal Revenue:
- Total Revenue (TR) = P Q
- Calculated from D curve
- Average Revenue (AR) = TR/Q = (P Q)/Q = P
- AR curve = D curve
- Marginal Revenue (MR) = TR when ↑output by 1 unit.
- Slope of TR curve

Total Profit: Keep Your Eye on the Goal

- Total, Average, and Marginal Cost:
- TC = P inputs x Q inputs
- AC = TC/Q output
- Per unit costs
- MC = ∆TC when ↑output by 1 unit.
- Slope of TC curve

Total Profit: Keep Your Eye on the Goal

- Maximization of Total Profits:
- Profits typically increase with output, then fall.
- Some intermediate level of output generates max profit.

Total Profit: Keep Your Eye on the Goal

- In our example:
- Total profit (Π) is max at 5 or 6 packages, where farm earns its highest profits of $27 per week.
- Any other Q level →↓Π
- E.g., if Q = 3 → Π = $18 or if Q = 8 → Π = $16

Marginal Analysis and Maximization of Total Profit

- Use marginal analysis to find Q that max profits.
- Marginal profit = ∆ total profit when ↑Q by 1 unit.
- Slope of total profit curve
- Rule: if marginal Π > 0 → ↑Q

if marginal Π < 0 → ↓Q

- Profit-max Q is reached when marginal Π = 0.
- Graphically, only reach top of total profit “hill” when marginal profit (its slope) = 0.

FIGURE 13(a). Profit Maximization

Total Profit “hill”

27

20

0

2

3

4

5

6

7

8

9

10

Total Profit per week ($)

1

–20

–30

Output, Packages per week

Total profit has a “hill” shape. At Q = 0, Π = 0. At larger Q levels, firm floods the market, and ↓Π. Only at intermediate Q levels is Π > 0.

Marginal Analysis and Maximization of Total Profit

- Like marginal Π, MR and MC can guide us to Q output where total profit is maximized.
- MR = slope of TR and MC = slope of TC
- Total profit is max when TR and TC are farthest apart.
- Occurs when their slopes are equal, so they are not growing closer together (Π↓) or growing further apart (Π↑).
- Rule: if MR > MC → Q

if MR < MC → Q

- Profit maximizing Q is where MR = MC.

TR

Profit

FIGURE 13(b). Profit MaximizationTotal Π= vertical distance between TR and TC curves

125

99

84

Total Revenue, Total Cost per week ($)

57

$27

20

0

1

2

3

4

5

6

7

8

9

10

Output, packages per week

Marginal Analysis and Maximization of Total Profit

- Finding the Optimal P from Optimal Q:
- Optimal Q is where MR = MC (and marg. Π = 0).
- E.g., at Q = 6; MR = MC = $9.
- Producer picks Q then demand curve P buyers will pay to purchase that level of output.
- E.g., at Q = 6 → P = $14 –only P at which this Q is purchased by consumers.

Logic of Marginal Analysis & Maximization

- Decision makers constantly faced with problem of choosing the magnitude of some variable.
- E.g., how many cars to produce; how many workers to hire, or how many pints of ice cream to buy.
- Generally, larger the number selected → higher the total benefit. However, costs ↑ as number chosen ↑.
- Optimally, decision makers chose Q of some variable where difference between total benefit and total cost is greatest.

Logic of Marginal Analysis & Maximization

- Decide about Q of some variable, then max net benefit = total benefit – total cost by choosing the Q where marginal benefit = marginal cost.
- Rule is true regardless of who the decision maker is.

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