Part 1: Steady-state equilibrium at full employment in Harrod's model

Three growth rates.

• Actual growth rate: $G = \dfrac{\Delta Y}{Y} = \dfrac{s}{C}$, where $s$ is the saving ratio and $C$ the actual capital-output ratio.
• Warranted growth rate: $G_w = \dfrac{s}{C_r}$, where $C_r$ is the required capital-output ratio. It is the rate at which entrepreneurs find their investment just adequate, so saving equals desired investment and they are content to keep growing at that rate.
• Natural growth rate: $G_n = n + t$ (labour-force growth plus technical progress). It is the maximum rate the economy can sustain with full employment.

Steady-state condition. Steady growth with full employment (Harrod's "golden age") requires

$G = G_w = G_n$

• $G = G_w$: expectations are fulfilled, $S = I$, capital is fully utilised and there is no reason to change investment plans. This gives steady growth.
• $G_w = G_n$: capital accumulation just matches the growth of labour efficiency, so labour stays fully employed and capital is not idle. This gives full employment.

Harrod stressed that this equilibrium is a knife-edge. $G_w$ depends on saving habits and technology, $G_n$ on population and technical progress, so nothing makes them equal. Moreover, if $G$ departs from $G_w$ the departure is cumulative (see below).

Part 2: Consequences of equality and disparity

(a) $G = G_w$. Entrepreneurs' expectations are realised, so $S = I$ and the capital stock is just adequate. The economy stays on the warranted path (steady growth).

(b) $G > G_w$ (actual above warranted). Here $C < C_r$, so capital falls short of what is needed. Inventories run down and there is excess demand. Entrepreneurs raise investment, which pushes $G$ even further above $G_w$. The result is a cumulative boom with inflationary pressure, until it is stopped by the full-employment ceiling $G_n$.

(c) $G < G_w$ (actual below warranted). Here $C > C_r$, so there is surplus capital. Inventories pile up and demand is deficient. Entrepreneurs cut investment, income falls and $G$ falls further below $G_w$. The result is a cumulative slump, with unemployment and idle capacity.

(d) Warranted versus natural rate.

Part 3: Recession and recovery in Kaldor's business-cycle model

Kaldor (1940) explains the cycle by non-linear saving and investment functions of the level of activity (income), both also depending on the capital stock $K$:

$S = S(Y, K), \qquad I = I(Y, K)$

• Investment is S-shaped (sigmoid) in $Y$. At low output there is much idle capacity, so investment is insensitive to $Y$ (flat). At normal output it responds strongly (steep). At very high output, rising costs, finance limits and shortages make it flat again.
• Saving is also non-linear, flat in the middle range and steeper at low and high output. As a result the two curves cross three times.

Equilibria. The lower point $A$ and upper point $C$ are stable, because at them $S$ cuts $I$ from below (slope of $S$ greater than slope of $I$). The middle point $B$ is unstable: above $B$, $I > S$ and income rises toward $C$; below $B$, $S > I$ and income falls toward $A$.

Role of the capital stock. Net investment changes $K$. A higher $K$ shifts the $I$ curve down (less need for new capital) and the $S$ curve up. A lower $K$ shifts $I$ up and $S$ down.

Recession.

1. In a boom the economy is at $C$ with $I > 0$, so $K$ keeps rising.
2. The $I$ curve shifts down and the $S$ curve shifts up, so $C$ moves toward $B$ and finally merges with it and disappears.
3. With no upper equilibrium, $S > I$ at the old level. Income, output and employment fall cumulatively toward the lower stable point $A$. This is the recession (the downswing).

Recovery.

1. In the slump the economy is near $A$, where net investment is negative or zero, so capital depreciates and $K$ falls.
2. The $I$ curve shifts up and the $S$ curve shifts down, so $A$ moves toward $B$ and finally disappears.
3. Now $I > S$ at the old level. Income rises cumulatively toward the upper point $C$. This is the recovery (the upswing).

The cycle is therefore self-generating: the boom plants the seeds of the slump through capital accumulation, and the slump plants the seeds of recovery through capital decumulation.

$\text{Boom } (C) \xrightarrow{K\uparrow} \text{Recession} \to A \xrightarrow{K\downarrow} \text{Recovery} \to C$

Contrast. Harrod's model is a growth model with a knife-edge, and its instability is the problem. Kaldor's is a cycle model, where the same instability, bounded by non-linear floors and ceilings, produces recurring fluctuations rather than explosions.

Business Cycle and the Kaldor Model of Recession and Recovery

1. Meaning of Business Cycle

The business cycle (also called the trade cycle) refers to the recurrent, more or less regular fluctuations in the level of aggregate economic activity — output, employment, income, prices and investment — around the long-run trend of growth. It is not a random or seasonal movement; it is a wave-like movement with alternating phases of expansion and contraction that repeat over time.

The standard phases are:

1. Expansion (Boom / Upswing) — output, employment, income, investment and prices rise.
2. Peak (Upper Turning Point) — the maximum level of activity; growth stops and reverses.
3. Recession (Contraction / Downswing) — output, employment, income and investment fall; prices may fall.
4. Trough (Lower Turning Point) — the minimum level of activity; the economy begins to revive.
5. Recovery — the economy moves upward again towards the trend, and the cycle repeats.

Thus a business cycle is self-generating and cumulative in nature: once started in one direction, forces are set in motion that carry it further in the same direction until internal limits are reached and the movement reverses.

2. The Kaldor Model of the Business Cycle

Nicholas Kaldor (1940) explained the cycle by combining the multiplier with a non-linear (non-monotonic) investment function. His model is a non-linear trade cycle model in which the cycle is generated endogenously by the interaction of saving and investment, without any external shocks.

2.1 The Investment Function

Kaldor's key innovation is that investment is not a linear function of income. Instead, investment depends on the level of income (or output) in a non-linear way:

• At very low levels of income, investment is low and rises slowly (excess capacity exists, so induced investment is weak).
• At intermediate (normal) levels of income, investment rises rapidly — the marginal propensity to invest is high.
• At very high levels of income, investment again rises slowly or flattens out (capacity is nearly fully utilised, bottlenecks appear, costs rise).

So the investment curve is S-shaped (sigmoid): its slope (marginal propensity to invest) is first low, then high, then low again. Formally:

$I = I(Y), \quad \frac{dI}{dY} > 0, \quad \frac{d^2I}{dY^2} > 0 \text{ at low } Y, \quad \frac{d^2I}{dY^2} < 0 \text{ at high } Y$

2.2 The Saving Function

Saving is taken as a linear function of income with a constant marginal propensity to save:

$S = S(Y), \quad \frac{dS}{dY} = s > 0 \text{ (constant)}$

2.3 Equilibrium and Stability

Equilibrium occurs where S = I, i.e. where the saving line intersects the investment curve. Because the investment curve is S-shaped and the saving line is straight, they may intersect at three points:

• Point A (low income): investment curve is steeper than saving line → stable equilibrium (a floor).
• Point B (middle income): saving line is steeper than investment curve → unstable equilibrium (a knife-edge).
• Point C (high income): investment curve is steeper than saving line → stable equilibrium (a ceiling).

The middle equilibrium B is unstable. If income is slightly above B, investment exceeds saving, income rises further; if income is slightly below B, saving exceeds investment, income falls further. This instability is the engine of the cycle.

3. Generation of Recession in the Kaldor Model

Starting from the upper stable equilibrium C (peak/boom):

1. Suppose income rises slightly above C due to an investment spurt. At this high income, the economy is near full capacity. The marginal propensity to invest becomes low (the investment curve flattens), while the marginal propensity to save remains constant.
2. Hence saving now exceeds investment (S > I). The multiplier works in reverse: a fall in investment causes a multiple fall in income.
3. As income falls, the economy moves down the downswing. The downward movement is cumulative — falling income reduces investment, which reduces income further, and so on.
4. The recession continues until income falls to the unstable point B. Below B, the investment curve again becomes steeper than the saving line, so investment exceeds saving and the fall is checked.
5. The economy thus has a floor at the lower stable equilibrium A. At A, investment again exceeds saving, so income cannot fall further — the recession ends at the trough.

Key point: Recession is generated because at high income the marginal propensity to invest falls below the marginal propensity to save, making S > I and setting off a downward multiplier process.

4. Generation of Recovery in the Kaldor Model

Starting from the lower stable equilibrium A (trough):

1. At this low income, there is large excess capacity. The marginal propensity to invest is high (investment curve is steep) while the marginal propensity to save is constant and low.
2. Therefore investment exceeds saving (I > S). The multiplier works upward: an increase in investment causes a multiple increase in income.
3. As income rises, the economy moves up the upswing. The upward movement is cumulative — rising income induces more investment, which raises income further.
4. The expansion continues until income reaches the unstable point B. Above B, the saving line becomes steeper than the investment curve, so S > I and the rise is checked.
5. The economy then moves toward the upper stable equilibrium C (peak). At C, investment again falls short of saving, and the next recession begins.

Key point: Recovery is generated because at low income the marginal propensity to invest exceeds the marginal propensity to save, making I > S and setting off an upward multiplier process.

5. Summary of the Mechanism

The cycle is thus endogenous and self-perpetuating: the non-linearity of the investment function creates alternating stable and unstable regions, and the economy oscillates between the floor (A) and the ceiling (C), passing through the unstable middle point B in both directions.

6. Conclusion

The Kaldor model is a significant improvement over earlier linear multiplier–accelerator models because it explains the turning points of the business cycle. The recession is generated when, at high income, the marginal propensity to invest falls below the marginal propensity to save, causing S > I and a downward cumulative multiplier process. The recovery is generated when, at low income, the marginal propensity to invest rises above the marginal propensity to save, causing I > S and an upward cumulative multiplier process. The cycle therefore arises from the interaction of a non-linear investment function with a linear saving function, producing stable floor and ceiling levels between which the economy oscillates indefinitely.

Part 1: Derivation of the balance of payments (BP) curve

The BP curve shows all combinations of income $Y$ and interest rate $i$ at which the balance of payments is in equilibrium (overall balance $= 0$), for a given exchange rate.

$BP = \underbrace{X(e, Y^*) - M(e, Y)}_{\text{current account}} + \underbrace{F(i - i^*)}_{\text{capital account}} = 0$

where exports $X$ depend on the exchange rate $e$ and foreign income $Y^*$, imports $M$ rise with domestic income $Y$ ($M_Y = m > 0$), and net capital inflow $F$ rises with the interest differential, $F' = f > 0$.

Derivation.

1. Fix $e$ and $Y^*$, so exports are exogenous.
2. If $Y$ rises, imports rise and the current account worsens, creating a deficit.
3. To offset it, the capital account must improve, which needs a higher $i$ to attract capital inflow.
4. Setting $BP = 0$ gives $i = i^* + \dfrac{M(Y) - X}{f}$, with slope $\dfrac{di}{dY} = \dfrac{m}{f} > 0$.

So the BP curve is upward sloping:

• Points above it are surpluses (high $i$, large capital inflow).
• Points below it are deficits.
• It is flatter the more mobile capital is (large $f$). With perfect capital mobility it is horizontal at $i = i^*$. With no capital mobility it is vertical.
• A depreciation or devaluation (higher $e$) shifts it right.

Part 2: Effect of a rise in the money supply under a fixed exchange rate

Start at the equilibrium of $IS$, $LM$ and $BP$ (the Mundell-Fleming set-up).

Short run.

1. The rise in $M$ shifts $LM$ right to $LM_1$.
2. The interest rate falls and income rises (investment and consumption increase).
3. The new point lies below $BP$. Higher $Y$ raises imports and the lower $i$ causes capital outflow, so the BoP shows a deficit.

Adjustment. The central bank must defend the fixed rate by selling foreign exchange and buying domestic currency. Foreign reserves fall, and so does the domestic money supply (assuming no sterilisation). $LM_1$ therefore drifts back left.

Final outcome. The process continues until the BoP deficit is eliminated, which is only at the original equilibrium. $LM$ returns to its original position, so:

Case of perfect capital mobility. $BP$ is horizontal at $i = i^*$. The monetary expansion pushes $i$ below $i^*$, causing immediate heavy capital outflow, so the money supply is bid back almost at once. Monetary policy is completely ineffective in changing output.

Conclusion. Under fixed exchange rates, the money supply is endogenous and monetary policy cannot change $Y$ or $i$ except temporarily. The only lasting effect is a loss of foreign exchange reserves. For Nepal, whose rupee is pegged to the Indian rupee at NRs 1.6 per IRs 1, this implies that monetary independence is limited and that fiscal policy is the more effective stabilisation tool.

Basic Propositions of Monetarism and the Relative Effectiveness of Monetary and Fiscal Policy (IS–LM Analysis)

1. Basic Propositions of Monetarism

Monetarism, associated mainly with Milton Friedman and the Chicago School, rests on the following basic propositions:

(a) Quantity Theory of Money. Money is the primary determinant of nominal income. The equation of exchange ( MV = PY ) is restated as a demand-for-money function:

$M^d = kPY \quad \text{or} \quad \frac{M^d}{P} = f(Y, r)$

Money demand is a stable function of a few variables (real income, wealth, expected inflation, interest rate), so changes in money supply produce predictable changes in nominal income.

(b) Money Matters Most. Changes in the money supply are the dominant cause of changes in nominal income, output, and the price level. "Inflation is always and everywhere a monetary phenomenon."

(c) Stability of Velocity. Velocity of circulation ( V = PY/M ) is not constant but is stable and predictable in the short run; it does not move erratically as Keynesians claim.

(d) Long-run Neutrality of Money. In the long run, real output is determined by real factors (labour, capital, technology), so an increase in money supply raises only the price level, not real output. Money is neutral in the long run.

(e) Short-run Non-neutrality. In the short run, unanticipated changes in money supply affect real output and employment because prices and wages are sticky and expectations adjust with a lag.

(f) Rejection of Discretionary Fiscal Policy. Fiscal policy is weak and ineffective because of crowding out; it mainly changes the composition of output rather than its level. Monetary policy should follow a rule (e.g., a fixed growth rate of money) rather than discretionary action.

(g) Natural Rate Hypothesis / Expectations-Augmented Phillips Curve. There is a natural rate of unemployment; attempts to push unemployment below it through expansionary policy only raise inflation in the long run.

(h) Crowding-out Effect. Government borrowing to finance deficits raises interest rates and reduces private investment, offsetting the fiscal stimulus.

2. Relative Effectiveness of Monetary and Fiscal Policy under Monetarism (IS–LM Framework)

The Monetarist IS–LM Setting

Monetarists characterise the economy by:

• A relatively steep LM curve — because money demand is interest-inelastic (Friedman's stable demand-for-money function depends mainly on permanent income, not on the rate of interest), so a given change in money supply requires a large change in the interest rate to restore equilibrium.
• A relatively flat IS curve — because investment is highly interest-elastic, so even a small change in the interest rate produces a large change in investment and hence in income.

(a) Effectiveness of Monetary Policy

With a steep LM curve, an increase in money supply shifts LM to the right (LM₀ → LM₁). Because LM is steep, the interest rate must fall sharply (r₀ → r₁) to induce the public to hold the additional money balances. The large fall in the interest rate, combined with a highly interest-elastic IS curve, produces a large increase in income (Y₀ → Y₁).

$\text{Monetary policy} \Rightarrow \text{large } \Delta Y \text{ (via large fall in } r)$

Thus, monetary policy is highly effective under Monetarism. This is the cornerstone of the monetarist policy prescription: control the money supply to control nominal income.

(b) Effectiveness of Fiscal Policy

An expansionary fiscal policy (increase in government expenditure or cut in taxes) shifts the IS curve to the right (IS₀ → IS₁). Because the LM curve is steep, the rise in income is accompanied by a sharp rise in the interest rate (r₀ → r₁). The higher interest rate crowds out a large amount of private investment (since investment is highly interest-elastic), so the net increase in income (Y₀ → Y₁) is small.

$\text{Fiscal policy} \Rightarrow \text{small } \Delta Y \text{ (large crowding out)}$

Thus, fiscal policy is relatively ineffective under Monetarism. The larger the crowding-out effect, the weaker the fiscal multiplier. In the extreme monetarist case — a vertical LM curve — fiscal policy has no effect at all on income (complete crowding out), while monetary policy is fully effective.

Diagrammatic Summary

Conclusion

Under Monetarism, the LM curve is steep (possibly vertical) and the IS curve is flat. Therefore:

• Monetary policy is relatively more effective — a change in money supply produces a large change in income through a large change in the interest rate.
• Fiscal policy is relatively ineffective — government spending mostly crowds out private investment, leaving income little changed.

This is why monetarists advocate a monetary rule (steady growth of money supply) rather than active fiscal policy to stabilise the economy.

Transitional Dynamics

Transitional dynamics refers to the adjustment path of an economy from its current capital stock toward the steady state in the Solow–Swan growth model. It explains how output per worker, capital per worker, and consumption per worker behave during the period before the steady state is reached — as opposed to the steady state itself, which describes the long-run equilibrium.

The Basic Mechanism

The fundamental equation of the Solow model is:

$\dot{k} = sf(k) - (n + \delta)k$

where $k = K/L$ is capital per worker, $s$ is the saving rate, $f(k)$ is output per worker, $n$ is population growth, and $\delta$ is depreciation.

• Actual investment per worker: $sf(k)$
• Break-even investment per worker: $(n + \delta)k$

Transitional dynamics arise because $sf(k)$ and $(n + \delta)k$ are not equal when the economy is away from $k^*$.

Two Cases

1. $k < k^*$ (below steady state): Since $sf(k) > (n + \delta)k$, we have $\dot{k} > 0$. Capital per worker rises over time, and so does output per worker. The economy is said to be in transitional growth.

2. $k > k^*$ (above steady state): Since $sf(k) < (n + \delta)k$, we have $\dot{k} < 0$. Capital per worker falls, and output per worker declines toward the steady-state level.

In both cases, the economy converges monotonically to $k^*$, where $\dot{k} = 0$.

Key Implications

• Conditional convergence: Poorer economies (lower $k$) grow faster than richer ones, provided they share the same $s$, $n$, $\delta$, and technology. This is the model's central testable prediction.
• Growth without technological progress is temporary: Once $k = k^*$, growth in output per worker stops. Sustained long-run growth requires exogenous technological progress.
• Speed of convergence: Near the steady state, the convergence rate is approximately $\lambda = (1-\alpha)(n + \delta)$, where $\alpha$ is capital's share. A higher $\alpha$ implies slower convergence.
• Policy relevance: A rise in $s$ raises $k^*$; during the transition, growth accelerates temporarily before settling at the new steady state.

Diagrammatic Representation

Plotting $sf(k)$ and $(n + \delta)k$ against $k$: the vertical gap between the two curves equals $\dot{k}$. To the left of $k^*$ the gap is positive; to the right it is negative. The arrows on the horizontal axis point toward $k^*$, confirming stability.

Conclusion

Transitional dynamics bridge the short run and the long run in the Solow model. They show that capital accumulation drives growth only along the path to the steady state, and that convergence — not perpetual growth — is the model's outcome in the absence of technological change.

Rational Expectations

Meaning. The rational expectations hypothesis (REH) was developed by John Muth (1961) and popularised by Robert Lucas, Thomas Sargent and Robert Barro. It states that economic agents form their expectations about future variables by using all available information and the correct economic model, so that their subjective expectations equal the true conditional mathematical expectation:

$P_t^e = E(P_t \mid I_{t-1})$

where $P_t^e$ is the expected value of the variable, $I_{t-1}$ is the information set available at time $t-1$, and $E$ is the mathematical expectation operator. Expectations are therefore "rational" not because agents never err, but because errors are random and unpredictable — they do not repeat systematically.

Key features.

1. Full use of information. Agents use all relevant, costlessly available information, including past data and knowledge of government policy rules.
2. Correct model of the economy. Agents know the true structure of the economy and use it to forecast.
3. Errors are random. Forecast errors have zero mean and are uncorrelated with the information set: $E(P_t - P_t^e \mid I_{t-1}) = 0$. Only genuinely new "news" or unanticipated shocks cause errors.
4. No systematic bias. Unlike adaptive expectations, where errors persist and expectations adjust only gradually, rational expectations adjust immediately to new information.

Policy implication (Lucas critique and policy ineffectiveness). Because agents anticipate systematic policy, only unanticipated policy changes affect real variables such as output and employment. If the central bank announces an expansionary monetary policy and the public rationally expects it, wages and prices rise immediately, so output and employment remain at the natural rate. This is the policy ineffectiveness proposition of Sargent and Wallace. It also implies the Lucas critique: econometric models estimated on past data cannot reliably predict the effects of a new policy regime, since expectations and behaviour change with the regime.

Significance and limitations. REH provides microfoundations for macroeconomics, explains stagflation better than adaptive expectations, and supports credibility and rule-based policymaking. However, it is criticised for assuming unrealistically strong information and computational ability, ignoring uncertainty and learning costs, and failing to explain observed persistence in output and inflation (as in the real world, where anticipated money does affect output in the short run).