Answer the following questions in about 150 words each : 10×5=50 (a) Show that when prices and income increase in the same…

(a) Let utility be u(x₁, x₂, ..., xₙ), where xᵢ is the quantity of commodity i. Marshallian demand xᵢ(p₁, p₂, ..., pₙ, M) is obtained by maximising u subject to the budget constraint p₁x₁ + p₂x₂ + ... + pₙxₙ = M, with all prices and money income positive. Suppose all prices and money income increase in the same proportion t > 0. The new budget constraint becomes tp₁x₁ + tp₂x₂ + ... + tpₙxₙ = tM. Dividing through by t gives p₁x₁ + p₂x₂ + ... + pₙxₙ = M. Hence the feasible set of commodity bundles is exactly the same as before. Since preferences are unchanged, the utility-maximising bundle is also unchanged.

Formally, the Lagrangian is L = u(x₁, ..., xₙ) + λ(M − p₁x₁ − ... − pₙxₙ). The first-order conditions are ∂u/∂xᵢ = λpᵢ for all i, together with the budget constraint. Eliminating λ between any two goods gives (∂u/∂xᵢ)/(∂u/∂xⱼ) = pᵢ/pⱼ. In the new problem, the same ratio pᵢ/pⱼ is obtained, and after cancelling t the budget constraint is identical. Therefore xᵢ(tp₁, ..., tpₙ, tM) = xᵢ(p₁, ..., pₙ, M). Thus Marshallian demand is homogeneous of degree zero in prices and money income; a proportional rise in all prices and income leaves quantity demanded unchanged. The economic reason is that only relative prices and real income matter. Real income, measured by M/P where P is a suitable price index, remains unchanged, so there is no money illusion.

(b) The IS curve is the locus of combinations of income Y and interest rate r at which the goods market is in equilibrium. In a closed economy with fixed fiscal policy, equilibrium requires Y = C(Y − T) + I(r) + G, where C is consumption, T is taxes, I is investment, and G is government spending. Equivalently, private saving S(Y − T) plus taxes minus government spending equals investment: S(Y − T) + T − G = I(r). To find the slope, differentiate the goods-market equilibrium condition with respect to Y, holding T and G constant: dY = C′(Y − T)dY + I′(r)dr. Hence (1 − C′)dY = I′dr, so dr/dY = (1 − C′)/I′. Since 0 < C′ < 1, the numerator is positive. Since investment is inversely related to the interest rate, I′ < 0. Therefore dr/dY < 0. The IS curve is normally negatively sloped.

The slope measures the change in the interest rate required to restore goods-market equilibrium after a one-unit change in income. Its magnitude is smaller when investment is highly interest-sensitive, i.e. when |I′| is large, and when the marginal propensity to consume C′ is high. It is larger when investment is insensitive to interest and when saving rises strongly with income, i.e. when C′ is low. The negative slope arises because a higher income raises saving and creates an excess of planned saving over planned investment at the initial interest rate. To restore equilibrium, the interest rate must fall so that investment rises. Alternatively, a lower interest rate stimulates investment, which through the multiplier raises equilibrium income. Shifts in G, T, expectations, or autonomous spending shift the IS curve itself.

(c) The classical dichotomy is the analytical separation of the economy into a real sector and a nominal sector. Real variables such as output, employment, real wage, relative prices, and the real interest rate are determined solely by real forces: technology, preferences, endowments, and market clearing. Nominal variables such as money supply, price level, nominal wage, and nominal income are determined by monetary factors. In the classical quantity equation MV = PY, with velocity V and real output Y determined in the real sector, money determines only the price level P. Money is thus treated as a veil.

Neutrality of money means that a change in the nominal money supply leaves all real variables unchanged. Under classical assumptions of flexible prices and wages, absence of money illusion, and stable velocity, a doubling of M doubles P and all nominal magnitudes proportionally. Relative prices and real balances M/P remain unchanged, so output and employment do not change.

The two concepts are related but not identical. The classical dichotomy is the stronger proposition that real variables can be solved independently of nominal variables; nominal variables do not appear in real excess-demand functions. Neutrality is the comparative-static result that changes in money do not affect real variables. Dichotomy generally implies neutrality under classical assumptions, but neutrality can be defined more weakly, for example as a long-run property even when short-run frictions make money non-neutral. Thus the classical dichotomy is not exactly the same as neutrality of money; it is a stronger analytical separation, while neutrality is the absence of real effects of monetary changes.

(d) Market failure occurs when the competitive market fails to achieve a Pareto-efficient allocation. Major reasons include: market power, where monopoly or oligopoly restrict output and set price above marginal cost; externalities, where private costs or benefits differ from social costs or benefits, as in pollution or education; public goods, which are non-rival and non-excludable, leading to free-riding and underprovision; common-pool resources, which are rival but non-excludable and tend to be overused; asymmetric information, causing adverse selection, moral hazard, and missing markets; incomplete markets and coordination failures; and natural monopoly due to increasing returns to scale.

The government can correct these failures through regulation, antitrust policy, Pigouvian taxes and subsidies, cap-and-trade permits, provision or financing of public goods, definition and enforcement of property rights, disclosure requirements, social insurance, and stabilisation policy. However, government intervention may itself fail due to information constraints, rent-seeking, bureaucratic inefficiency, regulatory capture, and political distortions. Thus the role of government is corrective and enabling, but it must be evaluated against the possibility of government failure.

(e) Fisher’s equation of exchange is MV = PT, where M is money supply, V is velocity of money, P is price level, and T is volume of transactions. Velocity is V = PT/M. In Fisher’s approach, V is determined mainly by institutional and technological factors: frequency of wage and salary payments, use of credit and book credit, banking and clearing facilities, availability of money substitutes, transportation and communication, population density, degree of monetisation, and social payment habits. Fisher treated V as broadly stable in the short run.

The Cambridge version is M = kPY, where k is the fraction of nominal income that people choose to hold as cash balances. Here velocity is V = PY/M = 1/k. In the Cambridge approach, k depends on individual and portfolio choices: transactions and precautionary motives, wealth and income, interest rate as the opportunity cost of holding money, price expectations, uncertainty, and financial innovation. It is an income velocity concept, whereas Fisher’s V is a transactions velocity concept.

The key difference is that Fisher emphasises the mechanical and institutional turnover of money as a medium of exchange, often assuming V constant. The Cambridge approach emphasises money as a store of value and derives velocity from the demand for cash balances, making V more variable and sensitive to interest rates and expectations. Hence Fisher treats velocity as largely determined by payment institutions, while Cambridge treats it as the reciprocal of a behavioural cash-balance ratio.