Lecture 4. Producer Theory (producer theory)
Recommended reading : 【Microeconomics】 Microeconomics Table of Contents
1. Overview
2. Production Function
3. Cost Function
4. Profit
5. Supply Curve
1. Overview
⑴ k: quantity of capital (capital)
⑵ ℓ: quantity of labor (labour)
⑶ q: firm’s total output (seems to be an abbreviation of “quantity”)
⑷ r: return from capital (an abbreviation of “rate,” i.e., interest)
⑸ w: return from labor (wage)
2. Production Function (production function)
⑴ Assume the firm’s output q is a function of k and ℓ, i.e., f(k, ℓ).
① Marginal product (MP): distinguished as MPk and MPℓ
② Average product (AP): distinguished as APk and APℓ
③ At the maximum of the average product, the average product and marginal product coincide.
Figure 1. Total product, average product, marginal product
⑵ Returns to scale (returns to scale)
① Increasing returns to scale (IRS)
○ Definition: when all inputs are multiplied by k, output increases by more than k (similar to an “amplification” effect)
○ For t > 1, f(tk, tℓ) > t f(k, ℓ)
② Decreasing returns to scale (DRS)
○ Definition: when all inputs are multiplied by k, output increases by k or less
○ For t > 1, f(tk, tℓ) < t f(k, ℓ)
○ In this case, the firm should reduce its scale of operation.
③ Constant returns to scale (CRS)
○ Definition: when all inputs are multiplied by k, output increases by exactly k
○ For t > 1, f(tk, tℓ) = t f(k, ℓ)
⑶ Isoquant curve (isoquant curve): differs from the indifference curve in that it always represents a cardinal production level.
① Definition: a curve representing the same level of output. Related to k(ℓ) (∵ it can be separated in an implicit-function representation)
② Feature 1. Downward sloping: since the two inputs are substitutable, if one increases, the other can decrease to keep output constant
③ Feature 2. Law of diminishing marginal product: convexity of the isoquant to the origin; also called the law of diminishing returns○ the degree to which one input can substitute for the other decreases as the usage of that input increases
○ Reason: if there is a fixed input, increasing its use tends to increase inefficiency (e.g., mutual interference)
○ (Note) marginal product can increase at early stages
④ Feature 3. farther from the origin implies higher output
⑤ Feature 4. two isoquants do not intersect
⑥ Marginal rate of technical substitution (MRTS): related to -k’(ℓ)
○ Definition: indicates the degree of substitution between inputs
○ the negative sign is to take the absolute value
○ Law of diminishing marginal rate of technical substitution
⑦ Perfect substitutes
○ Definition: one input can completely substitute for the other
○ isoquants appear as straight lines
⑧ Perfect complements (non-substitutable inputs)
○ Definition: when output does not increase even if one input increases while the other is fixed
○ isoquants appear as right angles; substitution between the two inputs is impossible
⑷ Production possibility curve (PPC) (production possibility curve; PPC)
① Overview
○ Simple definition: a curve showing combinations of goods that an economy can produce at maximum capacity
○ Strict definition: a graph showing combinations of two goods that can be produced in full-employment equilibrium
○ plotted in the first quadrant
② Points farther from the origin than the PPC: not feasible
③ Points closer to the origin than the PPC: inefficient
④ Law of increasing opportunity cost (law of increasing opportunity cost)
○ Meaning 1. in reality, the PPC is concave to the origin
○ Meaning 2. as the output of one good increases, its opportunity cost increases
○ Meaning 3. the slope of the PPC increases
○ Tip: the slope of the PPC represents opportunity cost
○ (Note) if opportunity cost increases → concave; constant → straight line; decreasing → convex
○ Fundamental reason: as output increases, less efficient factors are employed, raising opportunity cost
⑤ Welfare economics: the PPC concept is used extensively
3. Cost Function (cost function)
⑴ Cost (cost)
① Everyday meaning: the amount of budget actually spent/reduced
② Economic meaning: opportunity cost; called economic cost
③ Unless stated otherwise, “cost” refers to the economic meaning
⑵ Assumption: v and w represent the costs of capital k and labor ℓ, respectively.
⑶ Producer’s cost minimization problem: solved using the Lagrange multiplier method
⑷ Contingent (conditional) input demand: refers to the solutions kc(v, w, q) and ℓc(v, w, q) from the cost-minimization problem
⑸ Total cost function (total cost function)
① MC(v, w, q): marginal cost function
○ has a U-shaped graph
○ Initially: marginal cost decreases as marginal product rises
○ Later: marginal cost increases due to diminishing marginal product
② AC(v, w, q): average cost function
○ has a U-shaped graph
○ Initially: fixed costs are spread over more units, so average cost falls
○ Later: once output exceeds a certain level, rising marginal cost raises average cost
○ The average cost curve always passes through the minimum point of the marginal cost curve and the variable marginal cost curve
○ Intuition: when average cost = marginal cost, an additional unit does not change average cost
⑹ Short-run cost minimization problem
① Fixed input (fixed input; also called short-run input)
○ Definition: an input whose quantity cannot be changed in the short run (or within a given period)
○ Capital is the 대표적인 fixed input: in the short run, vk1 is treated as a fixed cost
② Variable input (variable input)
○ Definition: an input whose quantity can be freely adjusted during the period of analysis
○ Labor is the 대표적인 variable input: wℓ is treated as a variable cost
③ Short run (short-run)
○ Definition: a period during which one or more inputs are fixed
○ usually refers to the period when capital is fixed (among capital and labor)
④ Long run (long-run)
○ Definition: a sufficiently long period during which all inputs can be adjusted
○ neither capital nor labor is fixed
⑤ Short-run cost minimization problem (short-run cost minimization problem)
⑥ (Note) which of k and ℓ is fixed is not important
⑦ SC: short-run total cost function
⑧ SAC: short-run average cost function
⑨ SMC: short-run marginal cost function
⑩ SAFC: short-run average fixed cost function (the “fixed” here means k is fixed)
⑪ SAVC: short-run average variable cost function (the “variable” here means ℓ varies)
⑺ Long-run cost function
Figure 2. Deriving the long-run cost function from short-run cost functions
① In the short run, fixed inputs exist; in the long run, all inputs are variable
② The long-run cost function is the pointwise minimum of the short-run cost functions across different facilities
⑻ Economies of scale
① Economies of scale: average production cost decreases as output increases
② Diseconomies of scale: average production cost increases as output increases○ Example: musical instruments, pottery
③ (Related concept) Economies of scope
○ cost advantages from producing multiple products jointly within one firm
○ Example 1. shoes and handbags
○ Example 2. reusing production facilities and distribution networks
○ Example 3. a ramen company and snack products
4. Profit (profit)
⑴ Profit maximization problem (profit maximization problem)
Figure 3. Example of the total revenue curve and total cost curve
① Profit: since it includes cost C, you must first solve the cost-minimization problem
② Revenue (revenue)
③ Marginal revenue (MR): additional sales revenue from producing one more unit○ (Note) in perfect competition, output changes do not affect price, so it equals price
④ Marginal cost (MC): additional cost to produce one more unit
⑤ Solved using the Lagrange multiplier method
○ if marginal cost is a decreasing function, there is no solution to the profit-maximization problem
○ reason: the Lagrange method finds stationary points; in that case it yields a minimum instead
⑵ Optimal output level: denoted q*
5. Supply Curve (supply curve)
⑴ Supply curve
① Definition: a curve with quantity supplied on the horizontal axis and price on the vertical axis
② A perfectly competitive market has a supply curve that is a horizontal line at the common product price
⑵ Producer surplus (producer surplus)
① Definition: with price p = p0, the integral of the price difference between the horizontal price line and the marginal cost curve over quantity q
② i.e., the satisfaction a producer gains by selling at a higher price
③ (Note) “could have sold cheaply but sold expensively” (∵ because the cost curve is increasing, smaller q would be sold at a lower price)
④ Formalization (omitted here)
⑤ (Note) in consumer theory, there is a similar concept called consumer surplus.
: Together, they form total surplus.
Input: 2020.03.30 09:24