Expectations, Consumption & Investment

Investment with Expectations

coreExam · medium

A firm invests when the present value of expected future profits exceeds the cost of the machine. V(Πᵉₜ) = Πᵉₜ₊₁/(1+rₜ) + (1−δ)Πᵉₜ₊₂/[(1+rₜ)(1+rₜ₊₁)] + … with depreciation δ. Aggregate investment depends on expected profits, interest rates, and δ. With constant expectations: V = Π/(r + δ), giving a simple user-cost formula.

Derivation

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The Firm's Investment Decision

A firm considering a new machine computes its present value:

V(\Pi^e_t) = \frac{\Pi^e_{t+1}}{1 + r_t} + \frac{(1-\delta) \Pi^e_{t+2}}{(1+r_t)(1+r_{t+1})} + \cdots

The factor $(1-\delta)^k$ accounts for depreciation — each year the machine loses a fraction $\delta$ of its value.

The Simple Formula

With constant expected profits $\Pi$ and constant $r$ , the series collapses:

V = \frac{\Pi}{r + \delta}

This is the workhorse expression for the value of a marginal unit of capital. $r + \delta$ is the user cost of capital — the right discount rate because capital both earns interest elsewhere and wears out.

Two Channels for Investment

| Channel | Mechanism | |---------|-----------| | Discount rate | $\uparrow r \Rightarrow \uparrow (r+\delta) \Rightarrow \downarrow V \Rightarrow \downarrow I$ | | Expected profits | $\uparrow \Pi^e \Rightarrow \uparrow V \Rightarrow \uparrow I$ |

Tobin's q

The stock market gives us information about $V$ : $q =$ (market value) / (replacement cost). Investment happens when $q > 1$ . Rising stock prices raise $q$ → capex rises. Falling markets depress $q$ → investment collapses.

Why Expectations Shift IS

The IS relation uses the shortcut $I = I(Y, r + \delta)$ . But underneath, a change in expected future profits also moves $I$ — even at unchanged current $Y$ and $r$ . This is the animal-spirits channel:

Recession expectations → $\downarrow \Pi^e$ → $\downarrow V$ → IS shifts left.
Optimism → $\uparrow \Pi^e$ → $\uparrow V$ → IS shifts right.

Worked Example

Expected profits per unit capital Π^e = 15 per year forever. r = 5%, δ = 10%. Machine price = 1.

V = Π/(r + δ) = 15/(0.05 + 0.10) = 15/0.15 = 100.
V = 100 ≫ 1 → strong incentive to invest. Firm invests until marginal Π falls to make V = 1.
If r rises to 10%: V = 15/0.20 = 75. Still V > 1 but I falls. If r = 14%: V = 15/0.24 ≈ 62.5.

V = 100 initially. Rate rise to 10% → V = 75. Rate rise to 14% → V ≈ 62.5. Higher r compresses V through the user-cost channel; investment falls as expected.

Common Mistakes

—Using r instead of r + δ — depreciation is a separate cost in every period.
—Applying V = Π/r (perpetuity without depreciation) instead of V = Π/(r + δ).
—Forgetting that investment depends on expected future profits — current Π is only a proxy when expectations are static.
—Confusing Tobin's q with the average return — q is a ratio of values, with investment happening at the margin where q = 1.

Exam Cues

→Formula: V = Π/(r + δ) under constant expectations.
→Firm invests iff V > price of capital (normalised to 1).
→Two channels for I: discount rate r (↑r → ↓V → ↓I) and expected profits Π^e (↑Π^e → ↑V → ↑I).
→IS shift: shock to Π^e (e.g. recession expectations) shifts IS left at unchanged r — the animal-spirits channel.