MMPC-010 Solved Assignment

Question 1

Resource allocation using the equi-marginal rule (explain with an example)

Core meaning: The equi-marginal principle is an optimisation rule used when a decision-maker must distribute a limited resource (money, time, labour, budget, capacity) across multiple competing uses. The rule says you keep allocating to each use until the marginal benefit per unit of cost becomes equal across all uses. If this equality does not hold, you can improve the outcome by shifting resources from the lower-return use to the higher-return use.

Decision rule in words: Allocate the next unit of resource where it yields the highest marginal benefit per unit cost, and continue rebalancing until the marginal benefit-to-cost ratios become equal (given the resource constraint).

Decision rule in symbols (illustrative):

$$ \frac{MB_1}{MC_1} \;=\; \frac{MB_2}{MC_2} \;=\; \cdots \;=\; \frac{MB_n}{MC_n} $$

In a consumer context, marginal benefit is commonly expressed as marginal utility; with prices as the marginal “cost” per unit purchased. In a producer context, the same logic applies to choosing an input-mix so that marginal returns per unit input cost are balanced across inputs.

Why this works: If one activity gives a higher marginal return per unit cost than another, transferring a small amount of resource from the low-ratio activity to the high-ratio activity increases total satisfaction/profit without increasing total cost. The improvement continues until the ratios equalise.

Worked example (practical and student-friendly): Suppose you have a fixed weekly study budget of 10 hours to distribute between two subjects: Subject A and Subject B.

• Cost per hour: each hour “costs” 1 hour of your budget (so marginal cost per hour is the same).
• Marginal benefit: the extra marks/learning you gain from the next hour of study. Because of diminishing returns, the first few hours usually help more than later hours (the marginal benefit tends to fall as you spend more hours on the same subject).

Assume your realistic marginal benefit estimates look like this:

• Subject A: 1st hour = +10 units, 2nd = +9, 3rd = +8, 4th = +7, 5th = +6
• Subject B: 1st hour = +9 units, 2nd = +8, 3rd = +7, 4th = +6, 5th = +5

Applying equi-marginal thinking: You distribute hours so that the marginal benefit of the last hour spent on A is close to the marginal benefit of the last hour spent on B (since each hour has the same “cost”). A balanced allocation here would be:

• 5 hours on A (last hour benefit about +6)
• 5 hours on B (last hour benefit about +5)

If instead you spent 8 hours on A and 2 hours on B, the last hour of A might be giving you a very small gain while the next hour on B might still give a comparatively higher gain. Shifting time from A to B would improve your overall outcome until the marginal gains become comparable. This is exactly the “equal marginal benefit per unit cost” logic stated in the course material.

Question 2

Income elasticity of demand (meaning, calculation idea, and an example)

Meaning: Income elasticity of demand measures how responsive the quantity demanded (sales) of a product is to changes in consumer income, holding other factors constant (ceteris paribus). It is defined as the percentage change in quantity demanded divided by the percentage change in income.

Basic formula:

$$ E_y \;=\; \frac{\%\Delta Q}{\%\Delta Y} $$

where Q is quantity demanded (sales) and Y is income.

Arc and point concepts (as used in the blocks):

• Arc income elasticity uses two observed points (an interval) and treats responsiveness as an average over that range.
• Point income elasticity measures responsiveness at a specific point on the demand function (using the derivative form).

Standard expressions (illustrative):

$$ E_y^{arc} \;=\; \frac{\frac{Q_2-Q_1}{(Q_1+Q_2)/2}}{\frac{Y_2-Y_1}{(Y_1+Y_2)/2}} \qquad\qquad e_y^{point} \;=\; \left(\frac{dQ}{dY}\right)\left(\frac{Y}{Q}\right) $$

How to interpret values:

• Normal goods: income elasticity is positive (E_y > 0).
• Inferior goods: income elasticity is negative (E_y < 0).
• Income elastic: E_y > 1 (sales rise proportionately more than income).
• Income inelastic: E_y < 1 (sales rise proportionately less than income).
• Unitary: E_y = 1.

Example (forecasting sales using income elasticity): Assume your firm expects average income in its target market to rise by 5% next year. Current annual sales are 150,000 units. If market analysis estimates the product’s income elasticity at 1.1, then (keeping other demand shifters constant) expected sales growth is:

$$ \%\Delta Q \;=\; E_y \times \%\Delta Y \;=\; 1.1 \times 5\% \;=\; 5.5\% $$

So forecast sales are approximately:

$$ Q_{next} \;=\; 150{,}000 \times (1.055) \;=\; 158{,}250 \text{ units} $$

Managerial relevance: This measure is useful for demand planning and capacity decisions across business-cycle phases. Products that are more income elastic tend to expand faster when incomes rise, while necessities are typically less sensitive—so they may not surge as much in expansions but also tend to be more stable when conditions weaken.

Question 3

Relationship between Average Product (AP) and Marginal Product (MP), and between Average Variable Cost (AVC) and Marginal Cost (MC) (with diagram-based explanation)

Step 1: Define the productivity measures (single variable input case): Consider labour L as the only variable input.

• Average Product (AP): output per unit of labour.
• Marginal Product (MP): additional output produced by an additional unit of labour.

$$ AP \;=\; \frac{Q}{L} \qquad\qquad MP \;=\; \frac{\Delta Q}{\Delta L} $$

Step 2: Link productivity to costs (wage rate approach): If the wage rate is W, then total variable cost is:

$$ TVC \;=\; W \times L $$

Average Variable Cost (AVC):

$$ AVC \;=\; \frac{TVC}{Q} \;=\; \frac{W L}{Q} \;=\; \frac{W}{Q/L} \;=\; \frac{W}{AP} $$

Marginal Cost (MC): Since variable cost changes with labour, and wage is assumed given, the marginal cost can be expressed as:

$$ MC \;=\; \frac{\Delta TVC}{\Delta Q} \;=\; \frac{W\,\Delta L}{\Delta Q} \;=\; \frac{W}{\Delta Q/\Delta L} \;=\; \frac{W}{MP} $$

These relationships are explicitly developed in the unit linking productivity and cost behaviour.

Diagram-based interpretation (AP–MP):

• When MP is above AP, the average is rising (AP increases).
• When MP is below AP, the average is falling (AP decreases).
• AP reaches its maximum where AP = MP (MP intersects AP at AP’s peak).

Diagram-based interpretation (AVC–MC):

• If MC is below AVC, then AVC falls.
• If MC is above AVC, then AVC rises.
• MC intersects AVC at AVC’s minimum (where MC = AVC).

Connecting the two relationships (the key insight):

• Because AVC = W/AP, when AP is at its maximum, AVC is at its minimum.
• Because MC = W/MP, when MP is at its maximum, MC is at its minimum.
• At the point where AP is maximum, AP = MP; therefore, at the point where AVC is minimum, AVC = MC.
• When MP is rising, MC falls; when MP is falling, MC rises.

Managerial takeaway: These curve relationships help managers understand why improving productivity (especially marginal productivity) is closely linked to lowering marginal cost, which is central for output decisions, pricing, and expansion/contraction choices when compared with marginal revenue.

Question 4

Profit-maximising output of a perfectly competitive firm in the long run (when all inputs and costs are variable)

Long-run setting: In the long run, the firm can adjust all inputs and choose an appropriate plant size (scale). Unlike the short run, there are no fixed costs; all costs are variable. This flexibility allows the firm to select the most suitable scale for the market conditions.

Price-taking and revenue condition: Under perfect competition, the firm takes the market price as given. Because the demand curve faced by the individual firm is perfectly elastic, the market price equals the firm’s marginal revenue:

$$ P \;=\; MR $$

Thus, the output decision is made by comparing this constant MR with the firm’s long-run marginal cost.

Profit-maximisation rule: The firm expands output as long as marginal revenue exceeds marginal cost, and contracts output when marginal cost exceeds marginal revenue. Therefore, profit is maximised where:

$$ MR \;=\; MC \quad\Longrightarrow\quad P \;=\; LMC $$

This is the same optimisation logic as in the short run, but applied with long-run cost curves (LAC and LMC) and with full scale adjustment available.

What the standard long-run diagram shows (how to “read” it):

• A horizontal line at the market price (D = MR = P).
• Long-run marginal cost (LMC) and long-run average cost (LAC) curves.
• The profit-maximising output is the quantity where the price line intersects LMC.
• If at that quantity P > LAC, the firm earns economic (above-normal) profit; if P = LAC, it earns normal profit; if P < LAC, it incurs economic loss.

Industry adjustment and the long-run outcome: The crucial competitive implication is that economic profit is not sustainable in the long run when entry is open. Economic profit attracts new firms; entry increases industry supply, which pushes price down. This process continues until economic profits are eliminated and firms earn only normal profit. Conversely, economic losses lead to exit, reducing industry supply and pushing price up until losses are eliminated.

Long-run equilibrium condition (competitive benchmark):

$$ P \;=\; LAC \quad\text{(normal profit, zero economic profit)} $$

In that equilibrium, firms still choose output where P = LMC, but the prevailing price has adjusted so that the representative firm’s long-run average cost equals price.

Question 5

Short notes

(a) Decision Tree

A decision tree is a structured tool for analysing decisions that unfold in stages (sequential decisions) and involve uncertainty through chance events (states of nature). It represents:

• Decision nodes: where a manager chooses among alternatives (for example, selecting plant size or market-entry mode).
• Chance nodes/events: where outcomes depend on external factors (for example, competitor response or macroeconomic conditions).
• Branches: showing the possible paths and outcomes.

The probability of a specific final outcome is obtained by multiplying the probabilities along the branches leading to that outcome. The practical advantage is that the decision tree allows the manager to isolate each possible chain of events and follow it through to the end, making complex sequential problems more manageable.

(b) Tastes and Preferences as determinants of demand

Tastes and preferences are important “demand shifters” because changes in what consumers like or dislike can shift the demand curve even when price and income are unchanged. Preference formation is not straightforward to observe or model, but the course material highlights several influences:

• Social and peer effects: buying behaviour is often influenced by reference groups such as colleagues, classmates, and social circles.
• Advertising effects: advertising can shape preferences and is sometimes used as a measurable proxy for this determinant.
• Informational advertising: provides concrete product information (attributes, price, availability).
• Transformational advertising: attempts to change the product’s image and influence the satisfaction consumers associate with it; repeated exposure can alter preferences over time.

When preferences shift in favour of a product category (for example, towards fuel efficiency), demand for those products tends to move rightward; when social attitudes shift against a product (for example, reduced acceptance of smoking), demand can shift leftward.

(c) Economic and Technical Efficiency

• Technical efficiency: a firm is technically efficient when it achieves the maximum possible output from a given combination of inputs. Under technical efficiency, you cannot reduce one input and keep output constant without increasing at least one other input. The production function is treated as representing technically efficient production.
• Economic efficiency: a firm is economically efficient when it produces a given level of output at the lowest possible cost, given input prices. A method can be technically efficient but not economically efficient if it uses relatively expensive inputs in a way that raises cost compared to an alternative feasible method.

Managerial implication: Technical efficiency is about “how much can we produce from what we have,” while economic efficiency is about “how cheaply can we produce a target output given what inputs cost.” Which technically efficient method is economically efficient can change when input prices change.

(d) Monopoly

Monopoly is a market structure where a single firm supplies the entire market for a product that has no close substitutes, and entry is prohibited or difficult. Key characteristics include:

• Single seller (seller concentration is extremely high).
• No close substitutes (product differentiation is such that buyers cannot easily switch to an equivalent alternative).
• Strong barriers to entry (entry/exit conditions prevent competitors from entering freely).

Because the monopolist faces the market demand curve, its demand (average revenue) curve slopes downward, and marginal revenue lies below demand. The monopolist is a price-maker in the sense that it chooses an output level and then charges the price indicated by the demand curve for that output. Profit maximisation occurs where MR = MC, and the monopoly price is then read off from the demand curve at that output (typically implying price exceeds marginal cost at the optimum).