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2.4: Cost and Revenue Functions

ECON 306 · Microeconomic Analysis · Fall 2019

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microf19
microF19.classes.ryansafner.com

Recall: The Firm's Two Problems

  • First Stage: the firm's profit maximization problem:

  1. Choose: < output >

  2. In order to maximize: < profits >

  • Second Stage: the firm's cost minimization problem:

  1. Choose: < inputs >

  2. In order to minimize: < cost >

  3. Subject to: < producing the optimal output >

  • Minimizing costs \(\iff\) maximizing profits

A Competitive Market

  • We assume (for now) the firm is in a competitive industry:

  1. Firms' products are perfect substitutes

  2. Firms are "price-takers", no one firm can affect the market price

  3. Market entry and exit are free1

1 Remember this feature. It turns out to be the most important feature that distinguishes different types of industries!

Profit

  • Recall that profit is is: $$\pi=\underbrace{pq}_{revenues}-\underbrace{(wl+rk)}_{costs}$$

  • We'll first take a closer look at costs, then at revenues

  • Next class we'll put them together to find \(q^*\) that maximizes \(\pi\) (the first stage problem)

Some Costs Concepts

Costs in Economics are Opportunity Costs

  • Costs in economics are different from common conception of "cost"
    • Accounting cost: monetary cost
    • Economic cost: value of next best use of resources given up (opportunity cost)

Costs in Economics are Opportunity Costs

  • Costs in economics are different from common conception of "cost"

    • Accounting cost: monetary cost
    • Economic cost: value of next best use of resources given up

  • This leads to the difference between

    • Accounting profit: revenues minus accounting costs
    • Economic profit: revenues minues accounting & opportunity costs

The Accounting vs. Economic Point of View I

  • Helpful to consider two points of view:
    • "Accounting point of view": are you taking in more cash than you are spending?
    • "Economic point of view: is your product you making the best social use of your resources (i.e. are there higher-valued uses of your resources you are keeping them away from)?

The Accounting vs. Economic Point of View II

  • Social implications: are consumers best off with you using scarce resources (with alternative uses!) to produce your current product?

  • Remember: this is an economics course, not a business course!

    • What might be good/bad for one business might have bad/good consequences for society!

Opportunity Costs

  • Each choice incurs an opportunity cost

Examples:

  • If you choose to start a business, you may give up your salary at your current job
  • If you invest in a factory, you give up other investment opportunities
  • If you use an office building you own, you cannot rent it to other people

Opportunity Costs and Economic Profit

Example:

  • Craig's Consulting has the following revenues and costs:
Revenues $600,000
Supplies ($20,000)
Electricity and Water ($10,000)
Employee Salaries ($300,000)
Craig' Salary ($200,000)

Opportunity Costs and Economic Profit

Example:

  • Craig's Consulting has the following revenues and costs:
Revenues $600,000
Supplies ($20,000)
Electricity and Water ($10,000)
Employee Salaries ($300,000)
Craig' Salary ($200,000)
  • Craig could close his firm and rent out the building he owns for $50,000 per year.
  • Instead of running his own business, Craig could work at a larger consulting firm and expect to earn $300,000 per year.

Opportunity Costs and Economic Profit

Example:

  • Craig's Consulting has the following revenues and costs:
Revenues $600,000
Supplies ($20,000)
Electricity and Water ($10,000)
Employee Salaries ($300,000)
Craig' Salary ($200,000)
  • Craig could close his firm and rent out the building he owns for $50,000 per year.
  • Instead of running his own business, Craig could work at a larger consulting firm and expect to earn $300,000 per year.

  1. What is Craig's Consulting's accounting cost? economic cost?

  2. What is Craig's Consulting's accounting profit? economic profit?

Opportunity Cost is Hard for People

Opportunity Costs vs. Sunk Costs

  • Opportunity cost is a forward-looking concept

  • Choices made in the past with non-recoverable costs are called sunk costs

  • Sunk costs should not enter into future decisions

  • Many people have difficulty letting go of unchangeable past decisions: sunk cost fallacy

Sunk Costs: Examples

Sunk Costs: Examples

Sunks Costs: Examples

The Sunk Cost Fallacy

Common Sunk Costs in Business

  • Licensing fees, long-term lease contracts

  • Specific capital (with no alternative use): uniforms, menus, signs

  • Research & Development spending

  • Advertising spending

Costs in the Short Run

Costs in the Short Run

  • Total cost function, \(C(q)\) relates output \(q\)1 to the total cost of production \(C\)

$$C(q)=f+VC(q)$$

Costs in the Short Run

  • Total cost function, \(C(q)\) relates output \(q\)1 to the total cost of production \(C\)

$$C(q)=f+VC(q)$$

  • Two kinds of costs:

1. Fixed costs, \(f\) are costs that do not vary with output

  • Only true in the short run! (Consider this the cost of maintaining your capital)

Costs in the Short Run

  • Total cost function, \(C(q)\) relates output \(q\)1 to the total cost of production \(C\)

$$C(q)=f+VC(q)$$

  • Two kinds of costs:

1. Fixed costs, \(f\) are costs that do not vary with output

  • Only true in the short run! (Consider this the cost of maintaining your capital)

2. Variable costs, \(VC(q)\) are costs that vary with output (notice the variable in them!)

  • Typically, the more production of \(q\), the higher the cost
  • e.g. firm is hiring additional labor

1 Using optimal combinations of \(l\) and \(k\)!

Fixed vs. Sunk costs

  • What is the difference between fixed and sunk costs?

  • Sunk costs are a type of fixed cost that are not avoidable or recoverable

  • Many fixed costs can be avoided or changed in the long run

  • Common fixed, but not sunk, costs:

    • rent for office space
    • durable equipment
    • operating permits (that are renewed)

  • When deciding to stay in business, fixed costs matter, sunk costs do not!

Cost Functions: Example

Example: Suppose your firm has the following total cost function:

$$C(q)=q^2+q+10$$

  1. Write a function for the fixed costs, \(f\).

  2. Write a function for the variable costs, \(VC(q)\).

Cost Functions: Example, Visualized

\(q\) \(f\) \(VC(q)\) \(C(q)\)
\(0\) \(10\) \(0\) \(10\)
\(1\) \(10\) \(2\) \(12\)
\(2\) \(10\) \(6\) \(16\)
\(3\) \(10\) \(12\) \(22\)
\(4\) \(10\) \(20\) \(30\)
\(5\) \(10\) \(30\) \(40\)
\(6\) \(10\) \(42\) \(52\)
\(7\) \(10\) \(56\) \(66\)
\(8\) \(10\) \(72\) \(82\)
\(9\) \(10\) \(90\) \(100\)
\(10\) \(10\) \(110\) \(120\)

Average Costs

  • Average Fixed Cost: fixed cost per unit of output:

$$AFC(q)=\frac{f}{q}$$

Average Costs

  • Average Fixed Cost: fixed cost per unit of output:

$$AFC(q)=\frac{f}{q}$$

  • Average Variable Cost: variable cost per unit of output:

$$AVC(q)=\frac{VC(q)}{q}$$

Average Costs

  • Average Fixed Cost: fixed cost per unit of output:

$$AFC(q)=\frac{f}{q}$$

  • Average Variable Cost: variable cost per unit of output:

$$AVC(q)=\frac{VC(q)}{q}$$

  • Average (Total) Cost: (total) cost per unit of output:

$$AC(q)=\frac{C(q)}{q}$$

Average Costs

  • Average Fixed Cost: fixed cost per unit of output:

$$AFC(q)=\frac{f}{q}$$

  • Average Variable Cost: variable cost per unit of output:

$$AVC(q)=\frac{VC(q)}{q}$$

  • Average (Total) Cost: (total) cost per unit of output:

$$AC(q)=\frac{C(q)}{q}$$

$$\begin{align*} C(q) &= VC(q)+f\\ \frac{C(q)}{q} &= \frac{VC(q)}{q} + \frac{f}{q}\\ AC(q) &=AVC(q) + AFC(q)\\ \end{align*}$$

Marginal Cost

  • Marginal Cost is the change in cost for each additional unit of output produced:

$$MC(q) = \frac{\Delta C(q)}{\Delta q} \approx \frac{C_2-C_1}{q_2-q_1}$$

  • Calculus: first derivative of the cost function

  • Marginal cost is the primary cost that matters in making decisions

    • All other costs are driven by marginal costs
    • This is the main cost that firms can "see"

The Importance of Marginal Cost

Dazexiang Rebellion against the Qin Dynasty (209 B.C.)

Average and Marginal Costs: Example

Example: A small farm grows strawberries on 5 acres of land that it rents for $200 a week. The farm can hire workers at a wage of $250/week for each worker. The table below shows how the output of strawberries (in truckloads) varies with the number of workers hired:

Output Labor
0 0
1 1
2 3
3 7
4 12
5 18
  1. If labor is the only variable cost, calculate the \(MC(q)\) and \(AC(q)\) for each of the first 5 truckloads.

Average and Marginal Costs: Visualized

\(q\) \(C(q)\) \(MC(q)\) \(AFC(q)\) \(AVC(q)\) \(AC(q)\)
\(0\) \(10\) \(-\) \(-\) \(-\) \(-\)
\(1\) \(12\) \(2\) \(10.00\) \(2\) \(12.00\)
\(2\) \(16\) \(4\) \(5.00\) \(3\) \(8.00\)
\(3\) \(22\) \(6\) \(3.33\) \(4\) \(7.30\)
\(4\) \(30\) \(8\) \(2.50\) \(5\) \(7.50\)
\(5\) \(40\) \(10\) \(2.00\) \(6\) \(8.00\)
\(6\) \(52\) \(12\) \(1.67\) \(7\) \(8.70\)
\(7\) \(66\) \(14\) \(1.43\) \(8\) \(9.40\)
\(8\) \(82\) \(16\) \(1.25\) \(9\) \(10.25\)
\(9\) \(100\) \(18\) \(1.11\) \(10\) \(11.10\)
\(10\) \(120\) \(20\) \(1.00\) \(11\) \(12.00\)

Relationship Between Marginal and Average

  • There is a general mathematical relationship between a marginal and an average value:

  • Whenever marginal \(>\) average, average is increasing

Relationship Between Marginal and Average

  • There is a general mathematical relationship between a marginal and an average value:

  • Whenever marginal \(>\) average, average is increasing

  • Whenever marginal \(<\) average, average is decreasing

Relationship Between Marginal and Average

  • There is a general mathematical relationship between a marginal and an average value:

  • Whenever marginal \(>\) average, average is increasing

  • Whenever marginal \(<\) average, average is decreasing

  • When marginal \(=\) average, average is maximized/minimized

  • When \(MC=AC\), \(AC\) is at a minimum

  • When \(MC=AVC\), \(AVC\) is at a minimum

  • Economic importance (later):

    • Break-even price and shut-down price

Costs: Example

Example: Suppose a firm's cost structure is described by: $$\begin{align*} C(q)&=15q^2+8q+45\\ MC(q)&=30q+8\\ \end{align*}$$

  1. Write expressions for the firm's fixed costs, variable costs, average fixed costs, average variable costs, and average (total) costs.

  2. Find the minimum average (total) cost.

  3. Find the minimum average variable cost.

Costs: Example: Visualized

Costs in the Long Run

Costs in the Long Run

  • In the long run, firm can change all factors of production, and vary the scale of production

  • Long run average cost, LRAC(q): cost per unit of output when the firm can change both \(l\) and \(k\) to make more \(q\)

  • Long run marginal cost, LRMC(q): change in long run total cost as the firm produce an additional unit of \(q\) (by changing both \(l\) and/or \(k\))

  • Don't worry much about these, they are nearly identical to short run cost curves

  • One important idea...

Average Cost in the Long Run

  • In the long run, firm can choose \(k\) (factories, locations, etc)

  • Separate short run average cost (SRAC) curves for each amount of \(k\) potentially chosen

  • Long run average cost (LRAC) curve "envelopes" the lowest (optimal) parts of all the SRAC curves!

"Subject to producing the optimal amount of output, choose l and k to minimize cost"

Average Cost in the Long Run

  • In the long run, firm can choose \(k\) (factories, locations, etc)

  • Separate short run average cost (SRAC) curves for each amount of \(k\) potentially chosen

Long Run Costs & Scale Economies I

  • Further properties about costs based on scale economies of production:

  • Economies of scale: costs fall with output

  • Diseconomies of scale: costs rise with output

  • Constant economies of scale: costs don't change with output

  • Note economies of scale \(\neq\) returns to scale!

    • RTS (last class): a technological relationship between inputs & output
    • EOS (this class): an economic relationship between output and average costs

Long Run Costs & Scale Economies II

  • Minimum Efficient Scale: \(q\) with the lowest \(AC(q)\)

  • Economies of Scale: \(\uparrow q\), \(\downarrow AC(q)\)

  • Diseconomies of Scale: \(\uparrow q\), \(\uparrow AC(q)\)

Long Run Costs and Scale Economies: Example

Example: A firm's long run cost structure is as follows:

$$\begin{align*} LRC(q)&= 32000q-250q^2+q^3\\ LRMC(q)&=32000-500q+3q^2\\ \end{align*}$$

  1. At what levels of output will the firm face economies of scale and diseconomies of scale? (Hint: This firm has a \(U\)-shaped LRAC.)

Long Run Costs and Scale Economies: Example, Visualized

Revenues

Revenues for Firms in Competitive Industries I

Revenues for Firms in Competitive Industries I

Revenues for Firms in Competitive Industries I

  • Demand for a firm's product is perfectly elastic at the market price

Revenues for Firms in Competitive Industries I

  • Demand for a firm's product is perfectly elastic at the market price

  • Where did the supply curve come from? You'll see

Revenues for Firms in Competitive Industries II

  • Total Revenue \(R(q)=pq\)

Average and Marginal Revenues

  • Average Revenue: revenue per unit of output $$AR(q)=\frac{R}{q}$$
    • Is always equal to the price! Why?

Average and Marginal Revenues

  • Average Revenue: revenue per unit of output $$AR(q)=\frac{R}{q}$$

    • Is always equal to the price! Why?

  • Marginal Revenue: change in revenues for each additional unit of output sold: $$MR(q) = \frac{\Delta R(q)}{\Delta q} \approx \frac{R_2-R_1}{q_2-q_1}$$

    • Calculus: first derivative of the revenues function
    • For a competitive firm, always equal to the price!

Average and Marginal Revenues: Example

Example: A firm sells bushels of wheat in a very competitive market. The current market price is $10/bushel.

Average and Marginal Revenues: Example

Example: A firm sells bushels of wheat in a very competitive market. The current market price is $10/bushel.

For the 1st bushel sold:

  • What is the total revenue?

  • What is the average revenue?

Average and Marginal Revenues: Example

Example: A firm sells bushels of wheat in a very competitive market. The current market price is $10/bushel.

For the 1st bushel sold:

  • What is the total revenue?

  • What is the average revenue?

For the 2nd bushel sold:

  • What is the total revenue?

  • What is the average revenue?

  • What is the marginal revenue?

Total Revenue, Example: Visualized

\(q\) \(R(q)\)
\(0\) \(0\)
\(1\) \(10\)
\(2\) \(20\)
\(3\) \(30\)
\(4\) \(40\)
\(5\) \(50\)
\(6\) \(60\)
\(7\) \(70\)
\(8\) \(80\)
\(9\) \(90\)
\(10\) \(100\)

Average and Marginal Revenue, Example: Visualized

\(q\) \(R(q)\) \(AR(q)\) \(MR(q)\)
\(0\) \(0\) \(-\) \(-\)
\(1\) \(10\) \(10\) \(10\)
\(2\) \(20\) \(10\) \(10\)
\(3\) \(30\) \(10\) \(10\)
\(4\) \(40\) \(10\) \(10\)
\(5\) \(50\) \(10\) \(10\)
\(6\) \(60\) \(10\) \(10\)
\(7\) \(70\) \(10\) \(10\)
\(8\) \(80\) \(10\) \(10\)
\(9\) \(90\) \(10\) \(10\)
\(10\) \(100\) \(10\) \(10\)

Recall: The Firm's Two Problems

  • First Stage: the firm's profit maximization problem:

  1. Choose: < output >

  2. In order to maximize: < profits >

  • Second Stage: the firm's cost minimization problem:

  1. Choose: < inputs >

  2. In order to minimize: < cost >

  3. Subject to: < producing the optimal output >

  • Minimizing costs \(\iff\) maximizing profits

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