👷 Factor Markets & Distribution Theory

In factor markets, households supply primary productive inputs (labor $L$, capital $K$, land $T$) and firms demand them to produce final goods. Input demand is a derived demand, driven directly by consumer demand for the finished output.

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1. 💼 Competitive Factor Demand

A price-taking firm in both the output market ($P=\overline{P}$) and the labor market ($w=\overline{w}$) maximizes profit:

$$\underset{L}{max}\pi =P\cdot f(L,\overline{K})-wL$$$$\frac{d\pi }{dL}=P\cdot \frac{\partial f}{\partial L}-w=0⟹VM{P}_L=w$$

• Value of Marginal Product of Labor ($VM{P}_L$): $VM{P}_L\equiv P\times M{P}_L$.
• For an imperfectly competitive firm with market power in output ($P=P(Q)$):$$MR{P}_L\equiv MR(Q)\times M{P}_L=P(1-\frac{1}{|{ϵ}_d|})\times M{P}_L=w$$Since $MR<P$ under monopoly, $MR{P}_L<VM{P}_L$, leading to lower factor employment.

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2. 🏛️ Monopsony in Labor Markets

A monopsonist is a sole buyer of labor in a local or specialized labor market. Because the firm faces the upward-sloping market labor supply curve $w(L)$, hiring an additional worker forces the firm to raise wages for all inframarginal workers:

$$TotalLaborCost:TLC(L)=w(L)\cdot L$$$$MarginalFactorCost:MF{C}_L=\frac{d(TLC)}{dL}=w(L)+L\frac{dw}{dL}=w(L)[1+\frac{1}{{ϵ}_s^L}]>w(L)$$

  Wage ($)
    ▲
    │                MFC_L
    │               /
    │              /   Labor Supply w(L)
    │             /   /
w_pc│────────────┼───•
    │           / \ / \
    │          •   /   \
w_m │─────────/───•     \ MRP_L
    └────────┴───┴───────┴────────► Labor (L)
            L_m L_pc

2.1 Monopsony Equilibrium

1. Choose employment $L_m$ where $MF{C}_L=MR{P}_L$.
2. Pay wage $w_m=w(L_m)$ from the labor supply curve.
3. Monopsonistic Exploitation: $w_m<MF{C}_L=MR{P}_L$.

The Minimum Wage Monopsony Paradox

Under perfect competition, a binding minimum wage ($w_{min}>w_{pc}$) always reduces employment. Under monopsony, imposing a moderate minimum wage ($w_m<w_{min}\le w_{pc}$) makes the firm a price-taker on labor up to the supply curve, turning $MF{C}_L$ horizontal and increasing both wages and employment simultaneously!

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3. 🏞️ Economic Rent & Factor Earnings

• Transfer Earnings: The minimum payment required to keep a factor in its current employment (opportunity cost).
• Economic Rent: The excess payment received by a factor over and above its transfer earnings:$$\text{Economic Rent}=\text{Total Factor Earnings}-\text{Transfer Earnings}$$
• For perfectly inelastic factor supply (e.g. land, superstar talent), transfer earnings are zero and 100% of income is pure economic rent.

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Monopsony Labor Market Optimization

Problem: A mining company is the sole employer in a town. Labor supply is $w(L)=10+2L$. The firm's production function is $Q=20L-L^2$, and it sells output in a competitive market at $P=\mathrm{\$}5/\text{unit}$.

1. Derive the $MR{P}_L$ and $MF{C}_L$ equations.
2. Calculate the monopsony profit-maximizing employment $L_m^{\ast }$, wage $w_m^{\ast }$, and deadweight loss.

Step-by-Step Rigorous Solution:

1. Derive $MR{P}_L$ and $MF{C}_L$:

   $$M{P}_L=\frac{dQ}{dL}=20-2L⟹MR{P}_L=P\cdot M{P}_L=5(20-2L)=100-10L$$$$TLC(L)=w(L)\cdot L=(10+2L)L=10L+2L^2⟹MF{C}_L=\frac{dTLC}{dL}=10+4L$$

2. Equate $MF{C}_L=MR{P}_L$:

   $$10+4L=100-10L⟹14L=90⟹L_m^{\ast }=\frac{90}{14}\approx 6.43\text{ workers}$$

3. Determine Monopsony Wage:

   $$w_m^{\ast }=10+2(6.43)=\mathrm{\$}22.86/\text{worker}$$

4. Compare with Competitive Benchmark ($MR{P}_L=w(L)$):

   $$100-10L=10+2L⟹12L=90⟹L_{pc}=7.5,w_{pc}=10+2(7.5)=\mathrm{\$}25.00$$

   Monopsonistic markdown: Firm hires fewer workers ($6.43<7.5$) and pays lower wages ($\mathrm{\$}22.86<\mathrm{\$}25.00$).