a) Equilibrium price and quantity, and price elasticity of demand

Given: (Q_d = 100 - 4P) and (Q_s = 40 + 4P)

At equilibrium, (Q_d = Q_s):

[ 100 - 4P = 40 + 4P ] [ 100 - 40 = 4P + 4P ] [ 60 = 8P ] [ P = \frac{60}{8} = \text{Rs } 7.5 ]

Quantity:

[ Q_d = 100 - 4(7.5) = 100 - 30 = 70 \text{ units} ]

Check:

[ Q_s = 40 + 4(7.5) = 40 + 30 = 70 \text{ units} ]

Equilibrium price = Rs 7.5; equilibrium quantity = 70 units

Price elasticity of demand at equilibrium:

[ E_d = \frac{dQ_d}{dP} \times \frac{P}{Q} = (-4) \times \frac{7.5}{70} = -0.4286 \approx -0.43 ]

Since (|E_d| = 0.43 < 1), demand for Chokobite chocolate is inelastic at the equilibrium price.

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b) Filled table

Equilibrium from the table: At Rs 5 there is a shortage of 20 units and at Rs 10 there is a surplus of 20 units, so equilibrium lies between Rs 5 and Rs 10. Solving (100 - 4P = 40 + 4P) gives (P = \text{Rs } 7.5) and (Q = 70) units — the same equilibrium price and quantity as in part (a).

Consumer's Equilibrium by the Indifference Curve Approach

1. Meaning of Consumer's Equilibrium

Consumer's equilibrium refers to the situation in which a consumer, with his given income and given prices of two goods, maximizes his total satisfaction. Once he reaches this position, he has no tendency to change his purchase plan, because any reallocation of expenditure would reduce his total satisfaction.

The indifference curve approach explains equilibrium through two tools:

1. Indifference Curve (IC) — shows all combinations of two goods (say X and Y) that give the consumer the same level of satisfaction. It slopes downward, is convex to the origin, and a higher IC means a higher level of satisfaction.
2. Budget Line / Price Line — shows all combinations of the two goods that the consumer can buy with his given income at the given prices. Its slope equals the price ratio $P_x/P_y$.

2. Conditions of Equilibrium

The consumer is in equilibrium when the following two conditions are satisfied:

(a) First-order condition (necessary condition): $MRS_{xy} = \frac{P_x}{P_y}$

That is, the marginal rate of substitution of X for Y must equal the ratio of the price of X to the price of Y. In other words, the slope of the indifference curve must equal the slope of the budget line.

(b) Second-order condition (sufficient condition): The indifference curve must be convex to the origin at the point of tangency. This means $MRS_{xy}$ must be diminishing. If the IC is concave, the tangency point would give minimum, not maximum, satisfaction.

3. Explanation with Diagram

Let the consumer's income be Rs M, and prices of goods X and Y be $P_x$ and $P_y$.

The budget line is: $P_x \cdot X + P_y \cdot Y = M$

Its slope is $-P_x/P_y$. The indifference map consists of IC₁, IC₂, IC₃ … where IC₃ > IC₂ > IC₁ in satisfaction.

   Y
   |
 A |---- IC₃  (unattainable)
   |   /
   |  /   IC₂
   | /  ___
   |/ __/  E  ← Equilibrium point
   | /  \     (tangency of IC₂ and budget line)
   |/    \
   |      \
   O-------\------ B   X

Explanation:

• The consumer wants to reach the highest possible indifference curve.
• IC₃ lies entirely above the budget line, so no point on it is affordable.
• IC₁ cuts the budget line at two points, but these give lower satisfaction than IC₂.
• The highest attainable curve is IC₂, which just touches the budget line at point E.
• At E, the budget line is tangent to IC₂, so: $\text{Slope of IC} = \text{Slope of budget line}$ $MRS_{xy} = \frac{P_x}{P_y}$

Point E is therefore the point of consumer's equilibrium. The consumer buys OM of good X and ON of good Y, and obtains maximum satisfaction.

4. Why Other Points Are Not Equilibrium

• Points on IC₁ (e.g., cutting points): The consumer can move to a higher IC by reallocating expenditure, so he is not maximizing satisfaction.
• Points above the budget line (on IC₃): Not affordable with the given income.
• Point E: No further reallocation can raise satisfaction; hence equilibrium.

5. Conclusion

The consumer attains equilibrium at the point of tangency between the budget line and the highest attainable indifference curve, where $MRS_{xy} = P_x/P_y$ and the indifference curve is convex to the origin. At this point the consumer maximizes satisfaction subject to his income constraint.

Laws of Returns to Scale

Returns to scale refer to the behaviour of output when all inputs are increased in the same proportion in the long run. Since all factors are variable in the long run, the firm can change its scale of production. The laws of returns to scale explain how output responds when the scale of all inputs is changed proportionately.

Let the production function be:

$Q = f(L, K)$

If both labour (L) and capital (K) are increased by a constant proportion $k$ ($k > 1$), then:

$f(kL, kK) = hQ$

The relationship between $k$ and $h$ determines the type of returns to scale. There are three laws of returns to scale.

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1. Law of Increasing Returns to Scale

When output increases in a greater proportion than the increase in inputs, it is called increasing returns to scale.

$f(kL, kK) > kQ \quad \text{or} \quad h > k$

Example: If labour and capital are both increased by 10% ($k = 1.1$) and output rises by 20% ($h = 1.2$), then $h > k$, so increasing returns to scale exist.

Causes:

• Internal economies of scale — specialisation and division of labour, indivisibility of factors, use of better machinery, managerial and technical economies.
• External economies — growth of ancillary industries, improved transport and communication, availability of skilled labour.

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2. Law of Constant Returns to Scale

When output increases in the same proportion as the increase in inputs, it is called constant returns to scale.

$f(kL, kK) = kQ \quad \text{or} \quad h = k$

Example: If all inputs are increased by 10% and output also rises by exactly 10%, constant returns to scale exist. This is a homogeneous production function of degree one (linearly homogeneous).

Causes:

• Economies and diseconomies of scale exactly balance each other.
• The firm operates at the optimum scale of production.

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3. Law of Decreasing Returns to Scale

When output increases in a smaller proportion than the increase in inputs, it is called decreasing returns to scale.

$f(kL, kK) < kQ \quad \text{or} \quad h < k$

Example: If all inputs are increased by 20% but output rises by only 10%, then $h < k$, so decreasing returns to scale exist.

Causes:

• Internal diseconomies — managerial inefficiency, difficulty in supervision and control, scarcity of skilled labour, rising cost of raw materials.
• External diseconomies — congestion, higher factor prices, depletion of natural resources.

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Diagrammatic Illustration

Taking inputs on the X-axis and output on the Y-axis, the total product (TP) curve under returns to scale:

• Increasing returns: TP rises at an increasing rate (convex from below).
• Constant returns: TP rises at a constant rate (straight line from origin).
• Decreasing returns: TP rises at a diminishing rate (concave).

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Conclusion

The three laws of returns to scale operate sequentially in the long run. As a firm expands its scale, it first enjoys increasing returns due to internal economies, then constant returns at the optimum scale, and finally decreasing returns when diseconomies outweigh economies. These laws are a long-run concept, apply only when all inputs change proportionately, and are reversible.

Completing the table

Here $TR = P \times Q$, $AR = TR/Q$ and $MR = \Delta TR/\Delta Q$.

The demand curve is the straight line $P = 22 - 2Q$. So $TR = 22Q - 2Q^2$, giving $AR = 22 - 2Q$ and $MR = 22 - 4Q$.

Graph of AR and MR

Take quantity (Q) on the X-axis and price/revenue (Rs) on the Y-axis.

• Plot AR (= demand) through (1, 20), (2, 18), (3, 16), (4, 14), (5, 12), (6, 10), (7, 8). It is a downward-sloping straight line meeting the Y-axis at 22.
• Plot MR through (1, 20), (2, 16), (3, 12), (4, 8), (5, 4), (6, 0), (7, −4). It is a steeper downward-sloping line that cuts the X-axis at Q = 6 and then goes below it.

Relation between AR and MR curves

1. Both curves slope downward when the firm must lower the price to sell more.
2. MR lies below AR at every output after the first unit, because the lower price applies to all units sold and not only to the extra unit.
3. MR falls twice as fast as AR. For a straight-line demand curve, the slope of AR is −2 and the slope of MR is −4.
4. MR bisects the horizontal distance between the Y-axis and the AR curve. At Rs 16, AR is at Q = 3 and MR is at Q = 1.5, which is exactly halfway.
5. When MR is positive, TR rises. When MR = 0 (Q = 6), TR is maximum (Rs 60). When MR is negative, TR falls.
6. AR is the revenue per unit, and MR is the addition to total revenue from one more unit. Both are equal at the first unit (Rs 20).

Cartel: Meaning

A cartel is a formal, explicit agreement among a small group of oligopolistic firms producing a homogeneous (or close substitute) product to act collectively as a single monopolist. The members agree to fix price, divide the market into quotas, and restrict output so that the combined (joint) profit of all member firms is maximized. OPEC is the classic example.

Assumptions of the Joint Profit Maximization Model

1. The product is homogeneous and the firms sell in the same market.
2. There are only a few firms, and they enter into a formal, enforceable agreement.
3. The cartel behaves as a multi-plant monopolist — it controls the whole market supply.
4. Market demand for the product is known and stable.
5. The cost curves of all member firms are known to the central management of the cartel.
6. No new firm can enter the industry, and there is no fear of competition from outside.
7. Members honestly observe the agreed price and quota.

Price and Output Determination

The cartel, being a monopoly, faces the downward-sloping industry demand curve. Its aim is to maximize joint profit, so it follows the monopoly rule:

$\text{Maximise } \pi = TR - TC \quad\Rightarrow\quad MR = MC \text{ (total)}$

Step 1 — Industry output and price. The cartel management adds the marginal cost curves of all member firms horizontally to obtain the aggregate marginal cost curve $\Sigma MC$. It then equates $\Sigma MC$ with the industry marginal revenue $MR$ (derived from the market demand curve $DD$). The point of intersection $E$ gives the profit-maximizing total output $OQ$ and the marginal cost–marginal revenue level $OP_m$. The price $OP$ is then read off the demand curve at output $OQ$ — it lies above $OP_m$.

Step 2 — Allocation of output among members. The total output $OQ$ is distributed among the firms by the equi-marginal principle:

$MC_1 = MC_2 = \dots = MC_n = MR$

Each firm is given the quota at which its own marginal cost equals the cartel's marginal revenue. A low-cost firm gets a large quota, a high-cost firm a small one. This allocation minimizes the total cost of producing $OQ$ and hence maximizes joint profit.

Diagram (description).

• Panel (a): Price on the vertical axis, output on the horizontal. $DD$ is the market demand curve and $MR$ lies below it. The $\Sigma MC$ curve is the horizontal sum of the members' $MC$ curves. $MR$ and $\Sigma MC$ intersect at $E$, giving total output $OQ$ and the marginal level $OP_m$. Price $OP$ is obtained by extending a vertical line from $E$ up to $DD$.
• Panels (b) and (c): For firms $A$ and $B$, a horizontal line drawn at the level of $E$ cuts $MC_A$ and $MC_B$ at the points that fix the quotas $Oq_A$ and $Oq_B$, with $Oq_A + Oq_B = OQ$.

Numerical Illustration

Two firms $A$ and $B$ form a cartel. Market demand: $P = 100 - Q$, where $Q = q_A + q_B$. Marginal costs: $MC_A = \text{Rs }20$, $MC_B = \text{Rs }20$.

Industry $TR = (100 - Q)Q = 100Q - Q^2$, so $MR = 100 - 2Q$.

Cartel equilibrium: $MR = MC$

$100 - 2Q = 20 \;\Rightarrow\; Q = 40 \text{ units}$

Price: $P = 100 - 40 = \text{Rs }60$

Quota: since $MC_A = MC_B = 20$, each firm produces $q_A = q_B = 20$ units.

Joint profit $= (P - MC)\cdot Q = (60 - 20)\times 40 = \text{Rs }1{,}600$.

(Compare: under competition $P = MC = 20$, $Q = 80$, profit $= 0$. The cartel therefore raises price, restricts output and earns monopoly profit.)

Difficulties in Practice

• Cheating: Each member gains by secretly cutting price or selling above its quota, since its own $MR$ exceeds the cartel price at the margin.
• Quota disputes: Low-cost firms demand larger quotas; high-cost firms resist.
• Enforcement cost: Policing and punishing members is costly.
• New entry and substitutes erode the cartel's market power.
• Legal prohibition in many countries.

Conclusion

Under the joint profit maximization goal, the cartel behaves exactly like a monopoly: it determines total output where the aggregate marginal cost equals industry marginal revenue, sets the price from the market demand curve, and then allocates quotas among members so that the marginal cost of every firm is equal. This yields maximum joint profit, but the arrangement is inherently unstable because each member has an incentive to cheat.

Monopoly: Meaning and Determination of Price and Output

1. Meaning of Monopoly [2]

Monopoly is a market structure in which a single seller (firm) sells a product that has no close substitutes, and where the entry of new firms is completely blocked. The word comes from the Greek mono (one) and poly (seller). Since the monopolist is the only producer of the good, the firm is the industry, and it is a price maker — it can set either price or output, but not both.

Its main features are: (i) one seller and many buyers, (ii) no close substitutes, (iii) strong barriers to entry (patents, exclusive control of raw materials, government licence, large economies of scale), (iv) price-making power, and (v) a downward-sloping demand curve facing the firm.

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2. Price and Output Determination under Monopoly [13]

(a) Revenue curves

Because the monopolist faces the entire downward-sloping market demand curve, its AR curve is the demand curve itself. To sell an extra unit it must lower the price on all units, so MR falls faster than AR and lies below it:

$MR = AR\left(\frac{e-1}{e}\right) \quad \text{or} \quad MR = AR\left(1-\frac{1}{|e|}\right)$

where $e$ is price elasticity of demand. Hence MR is positive when $e>1$, zero when $e=1$, and negative when $e<1$. A profit-maximising monopolist therefore always produces on the elastic portion of the demand curve.

(b) Conditions of equilibrium

Profit is $\pi = TR - TC$. Differentiating, $\dfrac{d\pi}{dQ} = MR - MC = 0$. So:

1. Necessary (first-order) condition: $MR = MC$
2. Sufficient (second-order) condition: the MC curve must cut the MR curve from below, i.e. slope of MC > slope of MR at the point of intersection.
3. Shut-down condition (short run): $AR \ge AVC$; if $AR < AVC$, the firm shuts down.

(c) Price determination

The monopolist first decides output where $MR = MC$, and then reads the price from the AR (demand) curve at that output. Thus, unlike perfect competition, output is determined first and price second.

(d) Short-run equilibrium — three cases

(e) Long-run equilibrium

Since barriers to entry prevent new firms from entering, the monopolist is not forced to the minimum-cost point. In the long run the conditions are:

$MR = LMC \quad \text{and} \quad AR \ge LAC$

Normally the monopolist enjoys supernormal profit even in the long run. Only if demand falls so much that $AR < LAC$ will it leave the industry. Note that long-run equilibrium does not require $P = AC$ or $P = MC$, as in perfect competition.

(f) Diagram (description)

Draw MR and MC; they intersect at point $E$, giving equilibrium output $OQ$. Extend $E$ upward to the AR curve to get price $OP$. If AC lies below AR, the rectangle $PABC$ (between $P$ and $AC$ over output $OQ$) is the supernormal profit.

(g) Numerical illustration

Given demand $P = 100 - 2Q$ and cost $TC = 100 + 20Q$:

$TR = 100Q - 2Q^2 \Rightarrow MR = 100 - 4Q, \qquad MC = 20$

Equilibrium: $MR = MC \Rightarrow 100 - 4Q = 20 \Rightarrow Q = 20$ units

$P = 100 - 2(20) = \text{Rs } 60$

$AC = \frac{100}{20} + 20 = \text{Rs } 25$

$\text{Profit} = (P - AC)Q = (60 - 25)\times 20 = \text{Rs } 700$

Second-order condition: slope of MC (0) > slope of MR (−4) ✓

Comparison with perfect competition: $P = MC \Rightarrow 100 - 2Q = 20 \Rightarrow Q = 40$, $P = \text{Rs } 20$. So monopoly sells half the output at three times the price, and the deadweight loss is $\tfrac12 \times (60-20)\times(40-20) = \text{Rs } 400$.

(h) Conclusion

Under monopoly, price is determined by the AR curve at the output where $MR = MC$. Equilibrium price is higher and output lower than under perfect competition, the firm earns supernormal profit in the long run, and there is no separate supply curve (the same MC can give different price–output combinations for different demand curves). Hence monopoly is allocatively and productively inefficient, which is why governments regulate or nationalise such industries.

a. Price elasticity of demand ((E_p))

Formula used:

[ E_p=\frac{\Delta Q}{\Delta P}\times \frac{P}{Q} ]

where (P) and (Q) are the initial price and quantity.

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At income = Rs. 20,000

[ P_1=180,\ Q_1=80;\quad P_2=220,\ Q_2=60 ]

[ \Delta Q=60-80=-20,\qquad \Delta P=220-180=40 ]

[ E_p=\frac{-20}{40}\times \frac{180}{80} ]

[ E_p=-0.5\times 2.25=-1.125 ]

[ \boxed{|E_p|=1.125>1} ]

So at income Rs. 20,000, demand for ice-cream is relatively elastic.

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At income = Rs. 25,000

[ P_1=180,\ Q_1=100;\quad P_2=220,\ Q_2=80 ]

[ \Delta Q=80-100=-20,\qquad \Delta P=220-180=40 ]

[ E_p=\frac{-20}{40}\times \frac{180}{100} ]

[ E_p=-0.5\times 1.8=-0.9 ]

[ \boxed{|E_p|=0.9<1} ]

So at income Rs. 25,000, demand for ice-cream is relatively inelastic.

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b. Income elasticity of demand ((E_y))

Formula used:

[ E_y=\frac{\Delta Q}{\Delta Y}\times \frac{Y}{Q} ]

where (Y) and (Q) are the initial income and quantity.

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At price = Rs. 180

[ Y_1=20{,}000,\ Q_1=80;\quad Y_2=25{,}000,\ Q_2=100 ]

[ \Delta Q=100-80=20,\qquad \Delta Y=25{,}000-20{,}000=5{,}000 ]

[ E_y=\frac{20}{5{,}000}\times \frac{20{,}000}{80} ]

[ E_y=0.004\times 250=1 ]

[ \boxed{E_y=1} ]

So at price Rs. 180, income elasticity is unitary elastic.

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At price = Rs. 220

[ Y_1=20{,}000,\ Q_1=60;\quad Y_2=25{,}000,\ Q_2=80 ]

[ \Delta Q=80-60=20,\qquad \Delta Y=5{,}000 ]

[ E_y=\frac{20}{5{,}000}\times \frac{20{,}000}{60} ]

[ E_y=0.004\times 333.33=1.333 ]

[ \boxed{E_y=1.333>1} ]

So at price Rs. 220, income elasticity is greater than one, i.e. demand is income elastic.

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c. Comparison and interpretation

1. The negative price elasticities show the law of demand: when price rises from Rs. 180 to Rs. 220, quantity demanded falls.

2. At lower income Rs. 20,000, ice-cream demand is elastic ((|E_p|>1)). At higher income Rs. 25,000, it becomes inelastic ((|E_p|<1)). Thus, as income rises, Aaradhya becomes less sensitive to price changes for ice-cream.

3. The positive income elasticities show that ice-cream is a normal good. When income rises from Rs. 20,000 to Rs. 25,000, quantity demanded increases at both prices.

4. At price Rs. 180, income elasticity is exactly 1, meaning demand increases in the same proportion as income. At price Rs. 220, income elasticity is 1.333, meaning demand increases more than proportionately with income. So at a higher price, ice-cream behaves more like a luxury or income-sensitive good.

Conclusion: Ice-cream is a normal good for Aaradhya. Its demand is price-elastic at lower income but price-inelastic at higher income, while it is income-elastic at the higher price of Rs. 220.

Liquidity preference theory of interest

Introduction. J. M. Keynes put forward this theory in The General Theory (1936). It is a monetary theory. The rate of interest is the reward for parting with liquidity, and it is determined by the demand for and supply of money.

Meaning of liquidity preference. Liquidity preference is the desire of people to hold their wealth in cash, the most liquid asset, instead of lending it or investing it. People give up liquidity only if they are paid for it, and that payment is interest.

Demand for money (liquidity preference). Keynes gave three motives for holding money.

1. Transaction motive. Money is held to meet day-to-day spending of households and firms. It depends mainly on income and is interest-inelastic.
2. Precautionary motive. Money is held for unexpected needs such as illness or accident. It also depends mainly on income.
3. Speculative motive. Money is held to gain from changes in bond prices. It depends on the rate of interest. When the interest rate is high, bond prices are low, people expect them to rise and buy bonds, so cash holding is low. When the interest rate is low, people expect bond prices to fall and hold cash. So the speculative demand curve slopes downward.

$M_d = L_1(Y) + L_2(r)$

Here $L_1$ is the transaction and precautionary demand, which depends on income, and $L_2$ is the speculative demand, which depends on the interest rate. At very low interest rates the curve becomes perfectly elastic (the liquidity trap).

Supply of money. The money supply is fixed by the central bank (Nepal Rastra Bank) and the banking system. It does not depend on the interest rate, so the supply curve is vertical.

Determination of the interest rate. The interest rate is set where the demand for money equals the supply of money ($M_d = M_s$).

• If the interest rate is above the equilibrium, supply exceeds demand. People lend more and the rate falls.
• If it is below the equilibrium, demand exceeds supply. People sell bonds to hold cash and the rate rises.
• If the money supply rises while demand stays the same, the interest rate falls. If the money supply falls, the interest rate rises.

Criticisms

1. Indeterminate theory. The demand for money depends on income, and income depends on the interest rate, so the theory does not determine the rate by itself.
2. Ignores real factors. Productivity of capital, thrift and saving are ignored, yet they influence the interest rate.
3. Ignores the saving–investment relation. Interest is also the reward for saving. This theory treats it only as the reward for parting with liquidity.
4. Unrealistic fixed money supply. The money supply is not completely fixed. It can respond to the interest rate through bank credit.
5. Ignores other assets. Choice is not only between cash and bonds. Shares, land, gold and other assets are also held.
6. Liquidity trap. At very low interest rates, people hold any extra money as cash. The interest rate cannot fall further, so the theory fails to explain the rate at that point.
7. Limited use in underdeveloped countries. In countries like Nepal, bond markets are thin and much of the lending is informal. The speculative motive is therefore weak.
8. Ignores the income effect. Changes in money supply also change income and hence money demand, which the theory leaves out.

Conclusion. The theory explains the monetary side of interest and was a significant advance over classical theory, but it is incomplete without the real factors.