Microeconomics Lecture 3: Production and Costs and Producer Choice

Introduction to the Theory of the Firm

• The Model of Producer Choice: Similar to the model of consumer choice, the study of the supply side follows a structured logic. The firm first aims to minimize its cost for a given level of production.
• Existence of Firms: Firms exist to organize production more efficiently than independent individuals.
  • Transaction Costs: If independent individuals mass-produced complex goods like phones, many specialists would have to negotiate and trade, leading to high transaction costs (e.g., lawyer fees).
  • Inefficiency Management: In a firm, inefficient decisions can be identified and eliminated by a centralized authority.
  • Synergies: Firms are structured to find and utilize synergies between different departments or workers.
  • Market Power: Individuals have little power in the market for raw materials, whereas firms can leverage high market power to reduce costs and improve the coordination/quality of outputs.
• The Production Process: This process transforms inputs (factors of production) into outputs.
  • Factors of Production (Inputs):
    • Labor: Includes skilled workers, unskilled workers, and managers.
    • Materials: Raw inputs like steel, paper, water, electricity, etc.
    • Capital: Tangible assets like buildings, machinery, and inventories.
  • Flow Perspective: Production is measured over a specific time period (e.g., outputs and inputs per year or per week).

The Technology of Production

• Production Function: Indicates the highest output ($q$) a firm can produce for any specified combination of inputs, given the state of technology. It is expressed as:
  • $q = F(K, L)$
  • Where $K$ represents Capital and $L$ represents Labor.
• Core Assumptions:
  • Fixed Technology: The production function $F$ remains constant during the analysis.
  • Technical Efficiency: Firms are efficient and do not waste resources; they produce the maximum possible output for their chosen inputs.

Short-Run vs. Long-Run Production

• Short-Run: A time period in which at least one input (a fixed input) cannot be changed. Usually, capital ($K$) is assumed fixed.
  • Production function simplifies to: $q = F(L)$.
• Long-Run: A time period internal to the firm where all inputs (labor, capital, etc.) are variable.
  • Production function is: $q = F(K, L)$.

Production with One Variable Input (Short Run)

• Short-Run Production Data Example (K = 10):
  • Units of Labor ($L$): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
  • Total output ($q$): 0, 10, 30, 60, 80, 95, 108, 112, 112, 108, 100.
  • Average Product of Labor ($AP_{L}$): The output per unit of a particular input.
    • $AP_{L} = \frac{q}{L}$
    • Values from example: 10, 15, 20, 20, 19, 18, 16, 14, 12, 10.
  • Marginal Product of Labor ($MP_{L}$): The additional output produced as an input is increased by one unit.
    • $MP_{L} = \frac{dq}{dL}$
    • Values from example: 10, 20, 30, 20, 15, 13, 4, 0, -4, -8.
• Crucial Relationships in Production Curves:
  • Increasing Phase: By adding labor, total output first increases at an increasing rate, then at a decreasing rate.
  • Maximum Output: Total output is at its maximum when $MP_{L} = 0$ (Point F in the data, where $q = 112$).
  • The Cross-Over Point: When the marginal product is greater than the average product ($MP_{L} > AP_{L}$), the average product is increasing. When the marginal product is less than the average product ($MP_{L} < AP_{L}$), the average product is decreasing. Therefore, $MP_{L} = AP_{L}$ at the point where $AP_{L}$ is at its maximum (Point E/C).

The Law of Diminishing Marginal Returns

• Definition: As more of a variable input is added to fixed inputs, eventually the resulting addition to output (the marginal product) will decrease.
• Logic:
  • At low levels of labor, specialization allows $MP_{L}$ to increase.
  • At high levels of labor, workers may get in each other's way or machines might be over-utilized, causing inefficiencies and a declining $MP_{L}$.
• Note: This law assumes fixed quality of inputs and technology. Technological improvement can shift the entire production function upward, increasing labor productivity even if diminishing returns still apply to the new process.

Production with Two Variable Inputs (Long Run)

• In the long run, firms can vary both labor ($L$) and capital ($K$).
• Assumptions: For any level of capital, output increases with more labor, and for any level of labor, output increases with more capital (non-satiation for firms).
• Isoquants: Curves showing all possible combinations of inputs that yield the same level of output.
  • Isoquant Map: A set of isoquants illustrating a production function.
  • Higher isoquants (farther from the origin) represent higher levels of output.

Marginal Rate of Technical Substitution (MRTS)

• Definition: The amount by which one input can be reduced when one extra unit of another input is added, so that output remains constant.
• Formula: $MRTS = -\frac{dK}{dL}$
• Relationship to Marginal Productivity: Along an isoquant, the total change in output must be zero:
  • $(MP_{L} \times dL) + (MP_{K} \times dK) = 0$
  • This can be rewritten as: $MRTS = -\frac{dK}{dL} = \frac{MP_{L}}{MP_{K}}$
• Convexity: Isoquants are usually convex because of diminishing $MRTS$. As labor increases along an isoquant, the marginal productivity of labor falls, and the marginal productivity of capital rises, meaning the firm is willing to give up less and less capital for each additional unit of labor.

Special Production Function Shapes

• Perfect Substitutes: The $MRTS$ is constant everywhere. The isoquants are linear. Output can be increased by labor or capital alone.
• Fixed Proportions (Leontief): No substitution between inputs is possible. Output requires specific ratios of $L$ and $K$ (e.g., 1 person and 1 typewriter). The isoquants are L-shaped. Adding more of only one input does not increase output.

Economies of Scope

• Production Transformation Curve: Shows all combinations of two outputs (e.g., cars and trucks) possible with a fixed amount of inputs ($L$ and $K$).
• Scope Economies: A firm producing two goods together may produce them more efficiently than two specialized firms producing them separately.
• Scope Diseconomies: Occurs if a single firm producing both goods produces less total output than if production were split between specialized firms.

Understanding Costs

• Accounting Costs: Actual expenditures plus depreciation charges for capital equipment.
• Economic Costs: The cost associated with foregone opportunities (Opportunity Costs).
  • Example: A business owner who manages their own store but doesn't pay themselves a salary still incurs an economic cost (the salary they could have earned elsewhere).
  • House Painting Example: Hiring a painter costs $€75$/hour ($€600$ total for 8 hours). If I paint myself but could earn $€100$/hour consulting, the economic cost of painting my house myself is $€800$. The accountant sees $€0$; the economist sees $€800$.
• Sunk Costs: Expenditures that have been made and cannot be recovered.
  • Rule: Sunk costs should be ignored in future decision-making because they cannot be changed.
  • Example: A firm pays $€500,000$ for an option on a building costing $€5\text{ million}$ (Total $€5.5\text{ million}$). If they find another building for $€5.2\text{ million}$, they should still choose the original building because the $€500,000$ is already spent (sunk). Choosing the second building would result in a total cost of $€5.7\text{ million}$.
• Fixed vs. Variable Costs:
  • Fixed Costs (FC): Costs that do not vary with output level (e.g., rent). These can be recovered if the firm closes (unlike sunk costs).
  • Variable Costs (VC): Costs that change as output changes (e.g., wages, raw materials).
  • Total Cost (TC): $TC(q) = FC + VC(q)$.

Short-Run Costs

• Marginal Cost (MC): The change in cost resulting from producing one extra unit of output.
  • $MC = \frac{dTC}{dq} = \frac{dVC}{dq}$
• Average Total Cost (ATC): Total cost per unit of output.
  • $ATC = \frac{TC}{q} = AFC + AVC$
• Average Fixed Cost (AFC): $AFC = \frac{FC}{q}$
• Average Variable Cost (AVC): $AVC = \frac{VC}{q}$
• Graphical Relationships:
  • $MC$ crosses $AVC$ and $ATC$ at their respective minimum points.
  • If $MC < AVC$, the average variable cost is decreasing.
  • If $MC > AVC$, the average variable cost is increasing.
• Cost and Productivity: Marginal cost is inversely related to marginal product of labor:
  • $MC = \frac{w}{MP_{L}}$
  • Decreasing marginal returns ($MP_{L}$ falling) lead to increasing marginal costs.

Long-Run Costs and Cost Minimization

• User Cost of Capital: The sum of economic depreciation and the interest that could have been earned elsewhere.
  • $\text{User Cost} = (\text{Depreciation Rate} + \text{Interest Rate}) \times \text{Capital}$
  • The sum $(\text{Depreciation Rate} + \text{Interest Rate})$ is combined into the Rental Rate of Capital ($r$).
• Isocost Line: A line showing all combinations of $L$ and $K$ that can be purchased for a specific total cost ($C$).
  • $C = wL + rK$
  • Rearranged for graphing: $K = \frac{C}{r} - (\frac{w}{r})L$
  • Slope: $-\frac{w}{r}$. It represents the rate at which labor can be substituted for capital in the market.
• Cost Minimization Rule: To produce a given output ($q_{1}$) at the lowest cost, find the point where an isocost line is tangent to the isoquant.
  • Slope of Isoquant ($MRTS$) = Slope of Isocost line ($\frac{w}{r}$).
  • $\frac{MP_{L}}{MP_{K}} = \frac{w}{r}$
  • Alternatively: $\frac{MP_{L}}{w} = \frac{MP_{K}}{r}$. This means the marginal production per unit of currency spent must be equal across all inputs.

Expansion Path and Long-Run Scale

• Expansion Path: A curve showing the cost-minimizing combinations of labor and capital for every level of output.
• Long-Run Cost Curve: Derived from the expansion path, showing the relationship between total cost and output when all inputs are flexible.
• Returns to Scale: The rate at which output increases as inputs are increased proportionately.
  • Constant Returns to Scale: Inputs double, output doubles. Isoquants are equidistant.
  • Increasing Returns to Scale: Inputs double, output more than doubles. Isoquants get closer together. Common in mass production (e.g., cars); can lead to monopolies.
  • Decreasing Returns to Scale: Inputs double, output less than doubles. Isoquants get farther apart. Often due to communication or monitoring problems in large organizations.
• Economies vs. Returns to Scale:
  • Economies of Scale: Doubling output costs less than double. This is a cost-based concept.
  • Diseconomies of Scale: Doubling output costs more than double.
• Long-Run Average Cost (LAC):
  • The LAC curve is the "envelope" of the firm's Short-Run Average Cost (SAC) curves.
  • A firm will choose the plant size (represented by a specific SAC curve) that minimizes cost for their planned output level.
  • The long-run AC curve is the lower boundary of all possible short-run AC curves.