Equilibrium in the Goods Market

Part of: Macro-Economics Lecture 06 — Macro-Economics Key concepts: Real Interest Rate, Savings-Investment Equilibrium, Supply Shock, Crowding Out, Partial vs General Equilibrium

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Where This Fits

We now have all three building blocks:

• Supply ($Y$): from the production model
• Consumption demand ($C$): from Lec 02
• Investment demand ($I$): from Lec_05-Investment

This lecture combines them to find equilibrium: the price that clears the goods market and what happens when the economy is hit by a shock.

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The Equilibrium Condition

In a closed economy with exogenous government spending $G$, the goods market clears when:

But what is the price that does the clearing? In a standard supply/demand model, it's the price of the good. Here, we've normalised $P=1$, so the relevant price is the ==real interest rate $r$==.

Rewriting as S = I

Rearrange the equilibrium condition:

The left-hand side is ==national savings $S$==: income not consumed by households or government.

This is equivalent to goods market equilibrium. The real interest rate $r$ adjusts until savings equals investment.

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How r Clears the Market

Effect of r on Savings

From the Lec 02 consumption/saving model:

• Borrowers (current period spending > income): both income and substitution effects push toward more saving when $r$ rises → $S \uparrow$
• Lenders (current saving > zero): effects partially offset — the income effect makes them richer (good), the substitution effect pushes toward more saving. Key assumption: substitution effect dominates.

Result: The savings curve $S(r)$ slopes upward — higher $r$ → more savings.

Effect of r on Investment

From Lec_05-Investment: higher $r$ raises the user cost of capital → optimal $K^f$ falls → less investment.

Result: The investment curve $I(r)$ slopes downward — higher $r$ → less investment.

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The S-I Diagram

The equilibrium real interest rate $r^*$ is where $S = I$. The left panel below shows the baseline; the other two preview the shock analysis that follows.

type: macro-shocks
figure: si-shocks

Left: $S$ slopes up and $I$ slopes down in $r$; their crossing sets $r^*$. Middle: a transitory rise in current $A$ shifts $S$ right → $r$ falls. Right: a rise in future $A^f$ shifts $I$ right and $S$ left → $r$ rises unambiguously.

Every point on the $S$ curve satisfies the household's Euler equation (optimal given $r$). Every point on the $I$ curve satisfies the firm's optimality condition (MPK = user cost). The crossing point $r^*$ is the general equilibrium — both sides optimise simultaneously.

What sets Y in this model?

Output $Y$ is determined entirely by the production function (supply side): $Y = AF(\bar{K}, N)$. The goods market equilibrium determines the split of that $Y$ between $C$, $I$, and $G$ — and the real interest rate $r^*$ that achieves this split. Demand alone cannot change output (see below).

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Analysing Shocks

A ==shock== is an unexpected change in an exogenous variable. The procedure for any shock is:

1. Does the I curve shift? If so, which direction and why?
2. Does the S curve shift? If so, which direction and why?
3. What is the new equilibrium $r^*$, $S^* = I^*$?
4. What else changes ($Y$, $C$)?

We hold $K$ fixed in the current period (it takes time to build capital) and $N$ fixed for now (labor market in Lec 07).

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Example 1 — Transitory Shock to Current $A$

Scenario: Current TFP $A$ jumps up unexpectedly, but is expected to return to normal next period.

Equilibrium result:

• $r^*$ falls (supply of savings exceeds demand for investment at old $r^*$)
• $S$ and $I$ both increase (investment rises because $r$ fell — even though the $I$ curve itself didn't move)
• $Y$, $C$, and $I$ all increase

Partial vs. general equilibrium — this is a key exam distinction

At first glance, a rise in current $A$ seems to have no direct effect on investment (the $I$ curve doesn't shift). But in general equilibrium, $r$ falls, and that lower $r$ induces more investment. This is called the general equilibrium effect — you must trace through the market-clearing mechanism, not just the direct effect.

"Supply shock" terminology

This is called a supply shock: $K$ and $N$ are unchanged, $A$ increases → the economy simply produces more. In short-run macroeconomics, supply shocks move $Y$, $C$, $I$, and $r$ all in predictable directions. This is a building block of Real Business Cycle theory.

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Example 2 — Transitory Shock to Future $A^f$

Scenario: Agents learn today that $A^f$ will be higher next period (then returns to normal).

Equilibrium result:

• $r^*$ rises unambiguously (both shifts push $r$ up)
• Effects on $I$, $S$, and $C$ are ambiguous: the direct effects (more $I$, less $S$) may be partially reversed by the higher $r$ (which discourages investment and encourages saving)

Two possible outcomes

Option 1: Investment increases net — the rightward shift of $I$ outweighs the rise in $r$. Option 2: Investment falls net — the rise in $r$ dominates and pushes back against the original investment increase. Both are consistent with the model. The outcome depends on the elasticities of $S$ and $I$ with respect to $r$.

$I$ and $C$ move in opposite directions here

The future boom creates a scramble for current resources: firms want more capital (↑$I$), households want more current consumption (↓$S$). Since $Y$ is fixed, the higher $r$ rations between these competing demands. This is a taste of the crowding out mechanism.

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Crowding Out and Government Spending

The analysis has an important implication for fiscal policy:

• Output $Y$ is determined by the production function. The goods market equilibrium determines how $Y$ is allocated between $C$, $I$, and $G$.
• If the government increases $G$ without changing $Y$, something else must give: $C + I$ must fall.
• The mechanism: $G$ ↑ → $S = Y - C - G$ falls → $S$ curve shifts left → $r$ ↑ → $I$ falls.

This is ==crowding out==: government spending "crowds out" private investment by raising the real interest rate.

Demand can't change output (in this model)

Since $Y$ is supply-determined, changes in $G$, $C$-preferences, or investor confidence alone cannot change GDP — they only change the composition of spending and the equilibrium $r$. This changes in later models when we allow $N$ to vary (Lec 07) or introduce nominal rigidities (IS-LM).

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Key Takeaways from the Shock Analysis

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Summary

1. Goods market equilibrium $Y = C + I + G$ is equivalent to $S = I$. The real interest rate $r$ is the price that clears the market.
2. $S$ slopes upward in $r$ (substitution effect dominates). $I$ slopes downward in $r$ (user cost rises).
3. Output is supply-determined: $Y = AF(\bar{K}, N)$. The goods market only determines how $Y$ is split across uses and what $r$ achieves that split.
4. Transitory current $A$ shock: shifts $S$ right → $r$ falls, all expenditure components rise. A textbook supply shock.
5. Transitory future $A^f$ shock: shifts both $I$ right and $S$ left → $r$ unambiguously rises; other effects are ambiguous.
6. Crowding out: ↑ $G$ → ↑ $r$ → ↓ $I$. Private investment is squeezed out by public spending.
7. Next step: allowing $N$ to vary (Lec 07) adds another margin of adjustment and lets output respond to more shocks.

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Related Notes

• Built on: Lec_02-Consumption and Saving, Lec_04-Production, Lec_05-Investment
• Next: Lec_07-Labor Market — adds the labour market to complete the short-run general equilibrium
• Future: Fiscal Policy — full analysis of government spending and crowding out
• Future: Real Business Cycles — supply shocks as the driver of fluctuations