The Labor Market

Part of: Macro-Economics Lecture 07 — Macro-Economics Key concepts: Labor Demand, Labor Supply, Static FOC, Income Effect, Substitution Effect, PVLR, Marginal Product of Labor, Real Wage

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Where This Fits

So far in the short-run model, labor $N$ has been held fixed. This lecture endogenises $N$: firms and workers both optimise, and the labor market clears to determine the equilibrium real wage $w^*$ and employment $N^*$.

Adding this completes the short-run general equilibrium:

• Goods market: $S = I$ → equilibrium $r^*$
• Labor market: $N^S = N^D$ → equilibrium $w^*$, $N^*$
• Production function: $Y = A F(K, N^*)$ → equilibrium output $Y^*$

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Model Assumptions

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Labor Demand

The Firm's Problem

Given technology $A$, capital $\bar{K}$, and wage $w$, the firm chooses $N$ to maximise profits:

First-order condition:

Firms hire workers until the ==marginal product of labor== equals the ==real wage==.

Why MPN = w is the optimum

• If $\text{MPN} > w$: the last worker produces more than he costs → hire more
• If $\text{MPN} < w$: the last worker costs more than he produces → fire him
• Only at $\text{MPN} = w$ is there no profitable deviation

The Labor Demand Curve

The labor demand curve is the MPN curve drawn against the real wage $w$. It slopes downward because of diminishing marginal returns to labor: as you hire more workers (holding $\bar{K}$ fixed), the extra output from each additional worker falls.

Shifts vs. Movements Along

The real wage never shifts the labor demand curve

The real wage $w$ is endogenous — it is determined in equilibrium. If $w$ changes, the firm moves along the demand curve to a new $N$. The curve only shifts when the underlying $\text{MPN}$ changes for a given $N$.

What does shift the $N^D$ curve (i.e., changes MPN at every level of $N$)?

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Labor Supply

Preferences

Workers derive utility from two things:

• ==Consumption $C$==: more is better; diminishing marginal utility ($U_C > 0$, $U_{CC} < 0$)
• ==Leisure $\ell = 1 - N$==: more is better; diminishing marginal utility
• Equivalently, ==labor $N$== is a bad: more is worse ($U_N < 0$); and the last unit is more painful than the one before ($U_{NN} < 0$)

The utility function summarises preferences:

Key derivatives:

• $U_C = \frac{\partial U}{\partial C} > 0$ (more consumption is good)
• $U_{CC} < 0$ (diminishing marginal utility)
• $U_N = \frac{\partial U}{\partial N} < 0$ (more work is bad)
• $U_{NN} < 0$ (each extra hour of work is more painful)

The Budget Constraint

Workers can only consume what they earn. In a one-period model with initial wealth $B$:

• $w$ is the real wage (price of each hour of labour in terms of consumption goods)
• $B$ is non-labour wealth (savings, transfers, lottery winnings)
• The constraint holds with equality: there's no reason to throw away money

Graphically: in the (Leisure, $C$) space, the budget line has a slope of $-w$. The horizontal intercept is 1 (all leisure, no work); the vertical intercept is $w + B$ (work every hour).

type: consumption-choice
figure: leisure

Consumption–leisure choice: the budget line has slope $-w$ (the price of an hour of leisure) and vertical intercept set by non-labour wealth $B$. The optimum is the tangency where $MRS = w$ — the static FOC.

The Optimisation Problem

Choose $C$ and $N$ to maximise utility subject to the budget constraint:

Solution using Lagrangian (or substitution):

Divide the second by the first:

This is called the ==static first-order condition (static FOC)==.

Interpreting the Static FOC

The condition says: marginal cost of work = marginal benefit of work.

• Left side $(-U_N)$: the "marginal pain" of working one more hour. Since $U_N < 0$, this is positive — it represents how much utility you lose from the extra labour.
• Right side ($w \cdot U_C$): the marginal benefit of working one more hour. Working an extra hour earns $w$ additional units of consumption; multiplying by $U_C$ converts this into utility terms.

Tangency condition

This is the same as saying the indifference curve is tangent to the budget line: $\text{MRS}_{C,\ell} = w$ The MRS (how much consumption you'd give up for an extra unit of leisure) equals the opportunity cost of leisure (the real wage).

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Income and Substitution Effects

Pure Income Effect (Wealth Shock)

Suppose $B$ increases (you win the lottery), with $w$ unchanged.

The budget line shifts out parallel (same slope, higher intercept). The worker is richer:

• Buys more consumption $C$
• Takes more leisure $\ell$ (i.e., works less)

Why work less when richer?

Leisure is a normal good. If you can afford more of everything, you'll take more leisure — just as you'd buy more of any good when your income rises.

Substitution Effect (Wage Change)

Suppose $w$ rises. The budget line rotates: steeper slope, same intercept at $\ell = 1$, $C = B$.

• Substitution effect alone: leisure just got more expensive (costs $w$ per hour). Work more, take less leisure.
• Income effect: higher $w$ makes you richer → work less, take more leisure.

The two effects work in opposite directions, so the net effect of $w$ on $N^S$ is ambiguous.

Our assumption: substitution dominates

We assume the substitution effect is larger, so the labour supply curve slopes upward: higher real wages lead to more hours worked. This rules out the "backward-bending" labour supply curve region seen in some empirical work on very high earners.

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Labor Supply Curve Shifters

Changes in the current real wage cause a movement along the $N^S$ curve — not a shift. What shifts $N^S$?

What is PVLR?

The Present Value of Lifetime Resources (PVLR) is the total wealth a worker has — including the present value of all future labour income and non-labour income. It's what determines consumption-smoothing. Anything that raises PVLR (other than the current wage) shifts the labour supply curve left.

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Equilibrium in the Labor Market

An equilibrium requires three conditions to hold simultaneously:

1. Households optimise: static FOC $-U_N = w \cdot U_C$
2. Firms optimise: $\text{MPN} = w$
3. Market clearing: $N^S = N^D$

The equilibrium $(N^*, w^*)$ is the crossing point of the supply and demand curves.

type: labor-market
scenario: equilibrium

Labor-market equilibrium: upward-sloping $N^S$ (substitution effect dominates) meets the downward-sloping $N^D$ (= MPN) at $(N^*, w^*)$.

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Shock Analysis

Example 1 — Negative Temporary TFP Shock ($A \downarrow$)

Scenario: TFP falls this period, expected to return to normal next period.

• $N^D$ shifts: Yes, left/down — MPN falls for every $N$ when $A$ falls.
• $N^S$ shifts: No — labor supply depends on the current wage and PVLR. The current wage is endogenous (it will fall, but that's a movement along the curve). PVLR barely changes for a temporary shock.

New equilibrium: lower $N^*$, lower $w^*$.

Output: $Y = AF(\bar{K}, N^*)$ → falls through both channels (lower $A$ and lower $N$).

type: labor-market
scenario: lecture-shocks

Left: a negative TFP shock shifts $N^D$ left (MPN falls) → lower $N$ and lower $w$. Right: a permanent population increase shifts $N^S$ right (short run) → higher $N$ but lower $w$.

Recession interpretation

This is a stylised model of a recession: a negative productivity shock reduces employment, wages, and output. This motivates RBC models where business cycles are driven by TFP fluctuations.

Example 2 — Permanent Population Increase

Short run (K fixed):

• $N^S$ shifts: Yes, right — more workers available at any wage.
  • Additionally: the permanent increase also lowers expected future wages (more labour supply forever), lowering PVLR, which further shifts $N^S$ right.
• $N^D$ shifts: No — MPK and $A$ are unchanged.

Short-run equilibrium: higher $N^*$, lower $w^*$.

Output: $Y = AF(\bar{K}, N^*)$ → rises (more workers). But labour productivity $Y/N$ falls (we slide down the diminishing-returns MPN curve — each worker produces less).

Long run (K can adjust):

More workers in the future means future MPK is higher (complementarity!). Firms will therefore invest more, raising $K$. As $K$ rises:

• $N^D$ shifts right (higher $K$ raises MPN for every $N$)
• The wage rises back toward its original level

Long-run equilibrium: higher $N$, higher $K$, higher $Y$ — but $K/N$ and $Y/N$ return to initial levels (under CRS). Growth in per-capita output requires TFP growth, not just more inputs.

Key long-run result

Capital and labour are complements. A permanent increase in labour supply eventually induces more capital accumulation, which in turn raises labour demand. The economy scales up, but living standards (output per worker) are unchanged unless TFP grows. This foreshadows growth theory.

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Summary

1. Labor demand $N^D$: the MPN curve. Firms hire until $\text{MPN} = w$. Shifts right with higher $A$ or $\bar{K}$.
2. Labor supply $N^S$: workers maximise utility from $C$ and leisure. The static FOC is $-U_N = w \cdot U_C$.
3. Wage effects on $N^S$: ambiguous — substitution (work more) vs. income (work less) effects. We assume substitution dominates → upward-sloping $N^S$.
4. $N^S$ shifters: population, PVLR (from non-wage sources), future wages. Current wage only causes movement along.
5. Equilibrium: $N^S = N^D$ at $(N^*, w^*)$. Then use $Y = AF(\bar{K}, N^*)$.
6. Negative TFP shock: $N^D$ shifts left → lower $N$, $w$, and $Y$.
7. Permanent population increase (short run): $N^S$ shifts right → higher $N$, lower $w$, higher $Y$, lower $Y/N$.
8. Long-run adjustment: higher $N^f$ → higher MPK$^f$ → more investment → higher $K$ → $N^D$ shifts right → wages recover. CRS means $Y/N$ returns to baseline; TFP growth needed to raise living standards.

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Related Notes

• Built on: Lec_04-Production — MPN from the production function
• Built on: Lec_05-Investment — firm optimisation, complementarity of K and N
• Built on: Lec_06-Equilibrium in the Goods Market — the goods market sets $r$; this sets $w$ and $N$
• Future: Growth Models — long-run equilibrium with capital and labour jointly determined
• Future: Unemployment — relaxing the no-frictions assumption