Impact of the production function, costs of production and revenue and profit on a firm's price and output decisions

Welcome to the Engine Room of Business Economics!

In the previous chapters, we looked at how consumers behave. Now, we shift our focus to the "Supply Side" – the firms. We are going to explore how companies decide how much to produce and what price to charge. Why does this matter? Because as an actuary, understanding the cost structures and profit motives of companies is vital for risk assessment, pricing insurance products, and valuing businesses.

Don't worry if these graphs and formulas seem a bit like a bowl of spaghetti at first! We will untangle them one strand at a time. By the end of these notes, you'll see how production, costs, and revenue all dance together to determine a firm's success.

1. The Production Function: Turning Inputs into Outputs

Before a firm can sell anything, it has to make it. The production function is simply the relationship between physical inputs (like labour and capital) and the maximum resulting output over a given time period.

Short Run vs. Long Run

This is a crucial distinction in economics. It is not defined by a calendar timeframe, but by flexibility:

1. The Short Run: At least one factor of production is fixed (usually capital, like a factory building or heavy machinery). You can hire more workers, but you cannot expand the factory building overnight.
2. The Long Run: All factors of production are variable. The firm has enough time to build new facilities, purchase new machinery, or alter its production technology completely.

The Law of Diminishing Returns

Imagine a small pizza kitchen with only one oven. If you hire one chef, they can make pizzas. If you hire a second, they can prep ingredients, and output increases significantly. But what if you hire 20 chefs? They will be bumping into each other, fighting over the single oven, and eventually, adding another chef might actually make production slower!

Key Term: Marginal Physical Product (MPP) – The extra output produced by adding one more unit of a variable input (e.g. an extra worker).
The Law of Diminishing Marginal Returns: As successive units of a variable factor are added to a fixed factor, a point will eventually be reached where the addition to total output (MPP) begins to decline.

The Three Measures of Physical Production

Let \( L \) represent the quantity of the variable input (labour) and \( TPP \) (or \( Q \)) represent total output:

1. Total Physical Product (TPP): The total quantity of output produced with a given amount of inputs.
2. Average Physical Product (APP): Total output divided by the amount of the variable factor employed: \( APP = \frac{TPP}{L} \).
3. Marginal Physical Product (MPP): The change in total output resulting from employing one additional unit of the variable factor: \( MPP = \frac{\Delta TPP}{\Delta L} \).

Key Takeaway: In the short run, you cannot simply keep adding workers to increase output indefinitely. Because capital is fixed, diminishing returns inevitably set in, pulling productivity down and unit costs up.

2. Short-Run Costs: The Price of Producing

Costs reflect the underlying technology and input productivity. When the productivity of variable inputs declines due to diminishing returns, costs per unit of additional output inevitably rise.

Fixed vs. Variable Costs

Total Fixed Costs (TFC): Costs that do not change with the level of output (e.g. factory rent, equipment leases). TFC must be paid even if output is zero.
Total Variable Costs (TVC): Costs that vary directly with output (e.g. raw materials, hourly wages, electricity used in production).
Total Cost (TC): The overall expenditure: \( TC = TFC + TVC \).

Average and Marginal Costs

Firms evaluate per-unit costs to make pricing and output decisions:

1. Average Fixed Cost (AFC): Fixed cost per unit of output: \( AFC = \frac{TFC}{Q} \). As output \( Q \) expands, AFC continuously declines (spreading overheads).
2. Average Variable Cost (AVC): Variable cost per unit of output: \( AVC = \frac{TVC}{Q} \).
3. Average Total Cost (ATC): Total cost divided by output: \( ATC = \frac{TC}{Q} = AFC + AVC \). This is your unit cost.
4. Marginal Cost (MC): The addition to total cost from producing one more unit of output: \( MC = \frac{\Delta TC}{\Delta Q} = \frac{\Delta TVC}{\Delta Q} \).

Important Relationship: The Marginal Cost (MC) curve always intersects the Average Variable Cost (AVC) and Average Total Cost (ATC) curves at their minimum points. When \( MC < ATC \), average cost is pulled down; when \( MC > ATC \), average cost is pulled up.

Key Takeaway: Short-run cost curves are U-shaped because of the Law of Diminishing Returns.

3. Long-Run Costs and Input Efficiency

In the long run, all inputs are variable. The firm can choose any combination of labour and capital, leading to the selection of optimal plant sizes and the consideration of Economies of Scale.

How a Business Selects Inputs to Achieve Efficiency

To minimise the cost of producing any given level of output (productive efficiency), a firm balances its inputs using the least-cost combination rule (the equi-marginal principle of production):

\( \frac{MPP_L}{P_L} = \frac{MPP_K}{P_K} \)

Where \( P_L \) is the wage rate of labour and \( P_K \) is the rental price of capital.
- If \( \frac{MPP_L}{P_L} > \frac{MPP_K}{P_K} \), the firm obtains more additional output per pound spent on labour than on capital. It should substitute labour for capital.
- Efficiency is achieved only when the marginal physical product per pound spent is equalised across all factors.

Economies of Scale

This occurs when an increase in the scale of all inputs leads to a lower long-run average cost (LRAC):
- Specialisation and division of labour: Staff specialise in specific tasks, improving speed and precision.
- Indivisibilities: Large, highly productive machinery can only be justified at high volumes.
- Bulk buying (commercial economies): Larger discounts on bulk purchasing of raw materials.
- Financial economies: Large firms can access capital markets and loans at lower interest rates.

Diseconomies of Scale

If a firm grows beyond its optimal scale, average costs may begin to rise:
- Coordination and communication breakdown: Layers of bureaucracy delay decisions and create misunderstandings.
- Alienation and principal-agent friction: Workers and middle managers feel detached from organisational goals, lowering morale and productivity.

Key Takeaway: The Long-Run Average Cost (LRAC) curve acts as an "envelope curve" that touches each short-run average cost (SRAC) curve at the tangency point representing the lowest unit cost for that scale.

4. Revenue: The Money Coming In

Revenue describes the proceeds generated from selling output.

1. Total Revenue (TR): The total receipts: \( TR = P \times Q \).
2. Average Revenue (AR): Revenue per unit sold: \( AR = \frac{TR}{Q} = \frac{P \times Q}{Q} = P \). The AR curve is identical to the demand curve facing the firm.
3. Marginal Revenue (MR): The change in total revenue from selling an additional unit: \( MR = \frac{\Delta TR}{\Delta Q} \).

Market Structure Note: If a firm is a price taker (perfect competition), price is constant and \( MR = AR = P \). If the firm faces a downward-sloping demand curve (imperfect competition), it must cut price on all units to sell more, causing \( MR < AR \).

5. Measuring Profit and the Profit-Maximising Rule

Economists measure total profit as the difference between total revenue and total economic costs:

Economic Profit (\( \Pi \)) = Total Revenue - Total Economic Cost

Economic cost includes both explicit accounting expenses and implicit opportunity costs (such as the return the business owner could have earned on their capital and time elsewhere).

Normal vs. Supernormal Profit

1. Normal Profit: The minimum return required to keep the firm's factors in their current employment. Because it covers owner opportunity costs, normal profit is treated as part of total economic cost (when \( TR = TC \) or \( \Pi = 0 \)).
2. Supernormal (Abnormal) Profit: Profit earned in excess of normal profit (when \( TR > TC \) or \( P > ATC \)).

The Profit-Maximising Output Rule

To maximise total profit (or minimise losses), a firm should produce at the output where:

\( MC = MR \) (with \( MC \) cutting \( MR \) from below)

- If \( MR > MC \): Selling another unit adds more to revenue than to cost. The firm should expand output.
- If \( MC > MR \): The last unit cost more to produce than the revenue it generated. The firm should contract output.
- When \( MC = MR \), no further addition to net profit is possible.

6. Price and Output Decisions and the Shut-Down Rule

Once a firm determines its profit-maximising output using \( MC = MR \), its pricing and operational continuation decisions depend on comparing price (\( P \)) with average cost measures in the short run and long run.

The Three-Step Decision Framework

1. Choose Output: Set output \( Q^* \) where \( MC = MR \).
2. Set Price: Trace vertically from \( Q^* \) to the demand curve (\( AR \)) to find price \( P^* \).
3. Assess Profitability: Compare \( P^* \) with \( ATC \) at \( Q^* \):
- If \( P^* > ATC \), the firm earns supernormal profit.
- If \( P^* = ATC \), the firm breaks even, earning exactly normal profit.
- If \( P^* < ATC \), the firm incurs an economic loss.

The Short-Run Shut-Down Decision

In the short run, fixed costs must be paid regardless of whether the firm produces output or not (they are sunk in the immediate term):
- If \( P \ge AVC \): The firm covers all of its variable operating costs and makes a positive contribution towards paying down fixed costs. Operating generates smaller losses than shutting down immediately. The firm stays open.
- If \( P < AVC \): The firm cannot even cover its direct operating costs. Operating would cause losses exceeding fixed costs. The firm must shut down production immediately (output \( Q = 0 \)).
- Short-Run Shut-Down Point: The minimum point of the \( AVC \) curve (where \( P = AVC \)).

The Long-Run Shut-Down (Exit) Decision

In the long run, all inputs are variable and contracts expire. There are no fixed costs:
- If \( P \ge LRAC \): The firm covers all costs including normal profit. It remains in the industry.
- If \( P < LRAC \): The firm is failing to earn normal profit over the long term. The firm will shut down permanently and exit the industry, reallocating its capital elsewhere.
- Long-Run Shut-Down Point: The minimum point of the \( LRAC \) curve (where \( P = LRAC \)).

Final Takeaway: The firm's short-run supply curve is simply its marginal cost curve above the minimum \( AVC \). In the long run, it is the marginal cost curve above the minimum \( LRAC \). Understanding these boundaries is the foundation of firm behaviour across all market structures in CB2.