Pricing of Factors of Production and Income Distribution Study Notes

Introduction to Income Distribution

• Definition of Income Distribution Theory: This theory focuses on studying how the shares of different factors of production in the total output produced within an economy are determined over a specific time period.

• Simple Case with Two Factors: When considering an economy with only Labour and Capital, the total value of output (denoted as $X$) is distributed as:

  • Share of Labour: $\frac{w \times L}{X}$

  • Share of Capital: $\frac{r \times K}{X}$

  • Where:

    • $w$ = wage rate

    • $L$ = quantity of labour

    • $r$ = rental rate of capital

    • $K$ = quantity of capital

    • $X$ = total value of output produced

Determinants of Factor Shares and Selection

• Key Determinants: The shares attributed to factors of production depend on several variables:

  1. State of Technology: Defined by the production function.

  2. Relative Factor Prices: Represented by the ratio of the wage rate to the rental rate ($\frac{w}{r}$).

  3. Technological Progress: How advancements change the productivity of factors over time.

• Factor Intensity: This is a crucial concept measured by the capital-labour ratio ($K/L$).

• Elasticity of Substitution ($\sigma$):

  • Definition: Measures the responsiveness of factor substitution to changes in relative factor prices.

  • Formula: $\sigma = \frac{\% \Delta (K/L)}{\% \Delta (MRTS_{L,K})} = \frac{d(K/L) / (K/L)}{d(MRTS) / (MRTS)}$

  • Market Equilibrium: In perfectly competitive markets, $MRTS_{L,K} = \frac{w}{r}$.

  • Range: Values for $\sigma$ range from $0$ (no substitution possible) to $\infty$ (perfect substitutes).

• Firm’s Factor Choice (Profit Maximization): The firm identifies the technically efficient factor combination where the slope of the isoquant is exactly equal to the slope of the isocost line. This is expressed as the condition:

  • $MRTS_{L,K} = \frac{w}{r}$

Factor Pricing: General Framework

• Basic Insight: Determining the price of factors follows the same fundamental mechanisms as determining commodity prices; they are dictated by market forces of demand and supply.

• Historical Factor-Price Classifications:

  • Land: Earns Rent.

  • Labour: Earns Wages.

  • Capital: Earns Interest.

  • Entrepreneurship: Earns Profit.

• Variable vs. Fixed Factors:

  • Variable Factors: Factors like labour or raw materials where the supply can change in direct response to price fluctuations.

  • Fixed Factors: Factors such as land or unique natural resources where supply remains constant in the short and sometimes long run.

Marginal Productivity Theory of Distribution in Perfect Competition

• Core Proposition: In perfectly competitive product and input markets, factors are remunerated based on the Value of their Marginal Physical Product ($VMP$).

• The Case of a Single Variable Factor (Labour):

  • Assumptions:

    1. Single commodity ($X$) produced in a perfectly competitive market; price ($P_x$) is given.

    2. Firm’s objective is profit maximization.

    3. Labour market is perfectly competitive; wage ($w$) is fixed.

    4. Technology is constant; the law of variable proportions (diminishing $MPP_L$) applies.

  • Value of Marginal Product ($VMP_L$):

    • Definition: $VMP_L = MPP_L \times P_x$

    • The curve declines as employment increases because the Marginal Physical Product of Labour ($MPP_L$) declines.

  • Profit Maximization Theorem: A firm hires labour until the Marginal Cost of Labour ($MC_L$) equals $VMP_L$. Since the market is competitive and $MC_L = w$, the condition is:

    • $w = VMP_L$

  • Hiring Decisions:

    • If VMP_L > w: The firm hires more labour as the additional revenue exceeds the cost.

    • If VMP_L < w: The firm reduces labour as the additional cost exceeds the revenue.

  • Formal Derivation: Profit ($\Pi$) is $\Pi = P_x \times f(L) - (wL + F)$. The first-order condition ($\frac{d\Pi}{dL} = 0$) yields:

    • $P_x \times \frac{dX}{dL} - w = 0 \rightarrow P_x \times MPP_L = w \rightarrow VMP_L = w$

  • Demand Curve: The $VMP_L$ curve acts as the firm’s demand curve for labour when only one factor is variable.

Factor Demand with Several Variable Factors

• The Caveat: When multiple factors are variable, the $VMP$ curve is not the demand curve. A change in the price of one factor alters the employment of others, which in turn shifts the $MPP$ (and thus the $VMP$) of the original factor.

• Effects of a Wage Fall:

  1. Substitution Effect: As labour becomes cheaper, the firm substitutes labour for capital, moving along the existing isoquant.

  2. Output Effect: Lower costs allow for higher production levels, leading the firm to use more of both factors.

  3. Profit-Maximizing Effect: The firm expands total expenditure to reach a new maximum profit level.

• Net Result: The output and profit-maximizing effects typically offset the substitution effect, causing the $MPP_L$ and $VMP_L$ curves to shift rightward. The long-run demand for labour is the locus of equilibrium points on these shifting $VMP$ curves, maintaining a negative slope.

• Summary of Demand Determinants:

  1. Price of the input (inverse relationship).

  2. Marginal physical product (determined by the production function).

  3. Price of the commodity ($VMP = MPP \times P_x$).

  4. Amount of collaborating factors (e.g., more capital shifts $MPP_L$ right).

  5. Prices of other factors.

  6. Technological progress.

• Market Demand for Labour: This is not a simple horizontal summation of individual firm demand curves. If all firms hire more labour as wages fall, market supply of the commodity increases, driving down $P_x$. This cause each firm's $VMP_L$ curve to shift downward, making the market demand curve more inelastic than the simple sum of individual curves.

Supply of Labour

• Determinants of Market Supply:

  1. Wage rate.

  2. Tastes (work-leisure trade-off).

  3. Population size.

  4. Labour-force participation rate.

  5. Occupational, educational, and geographic distribution.

• Indifference Curve Analysis: An individual maximizes utility where the Marginal Rate of Substitution between leisure and income ($MRS_{leisure, income}$) equals the hourly wage ($w$).

• The Backward-Bending Supply Curve:

  • Up to a threshold wage ($w_3$), a wage increase leads to more hours worked because the substitution effect (leisure is more expensive) outweighs the income effect.

  • Beyond $w_3$, further wage increases result in fewer hours worked because the income effect (demand for leisure rises with wealth) dominates.

• Aggregate Supply Consensus: While individual curves may bend backward, the aggregate market supply is generally considered to have a positive slope, especially in the long run as workers move to higher-paying industries.

Factor Pricing in Imperfect Markets

Model A: Monopoly in Product Market, Perfect Factor Market

• Conditions: Demand for the commodity is downward sloping, meaning Marginal Revenue ($MR_x$) is less than Price ($P_x$).

• Marginal Revenue Product ($MRP_L$):

  • $MRP_L = MPP_L \times MR_x$

  • Since MR_x < P_x, then MRP_L < VMP_L.

• Equilibrium: The monopolist hires labour until $MRP_L = MC_L = w$. The $MRP_L$ curve serves as the demand curve for labour.

• Mathematical Expression: $(MPP_L) \times P_x \times (1 - \frac{1}{e_p}) = w$, where $e_p$ is price elasticity of demand.

• Monopolistic Exploitation (Joan Robinson): Occurs when a factor is paid less than its $VMP$. Since $w = MRP_L$ and MRP_L < VMP_L, the factor is technically exploited.

Model B: Monopsony in Factor Market

• Conditions: A single buyer of labour where the labour supply curve is positively sloped.

• Marginal Expense ($ME_L$): The cost of hiring an additional unit is greater than the wage because the wage must be increased for all existing workers.

  • $ME_L = \frac{d(TE)}{dL} = w + L \times \frac{dw}{dL}$

  • The $ME$ curve lies above the supply curve.

• Equilibrium: The monopsonist hires where $ME_L = MRP_L$. The wage is then determined by the supply curve at that employment level.

• Dual Exploitation:

  1. Monopolistic: w = MRP_L < VMP_L

  2. Monopsonistic: w < MRP_L

• Least-Cost Condition: $\frac{MPP_L}{ME_L} = \frac{MPP_K}{ME_K}$ or $MRTS_{L,K} = \frac{ME_L}{ME_K}$.

Model C: Bilateral Monopoly

• Definition: A single seller (Trade Union/Monopoly) faces a single buyer (Monopsonist Firm).

• Indeterminacy: Economic analysis provides bounds but not a single solution. The monopsonist wants a low wage ($w_F$) and the union wants a high wage ($w_U$). The final wage depends on bargaining skills, political power, and the ability to endure strikes or lock-outs.

Model D: Competitive Buyer vs. Monopoly Union

• Union Goals:

  1. Maximize Employment: Set $w$ where $D_L = S_L$ (Competitive level).

  2. Maximize Total Wage Bill: Set $w$ where Union's Marginal Revenue ($MRS$) equals $0$.

  3. Maximize Total Gains to Union: Set $w$ where $MRS = MC_{Supply}$.

• Union Benefits: In a monopsony, unions can increase both wages and employment by eliminating monopsonistic exploitation. In a competitive buyer scenario, raising wages may lead to unemployment, depending on the price elasticity of labour demand.

Elasticity of Substitution and Income Distribution

• Relative Shares: The ratio of Labour's share to Capital's share is $\frac{wL}{rK} = (\frac{w}{r}) / (\frac{K}{L})$.

• Impact of Change in $\frac{w}{r}$ on Labour's Share:

  • If \sigma < 1 (Inelastic substitution): Labour's share increases.

  • If $\sigma = 1$ (Cobb-Douglas): Share remains constant.

  • If \sigma > 1 (Elastic substitution): Labour's share decreases.

• Hicks’ Classification of Technological Progress:

  1. Neutral: $MRTS$ is unchanged at constant $K/L$; shares stay the same.

  2. Capital-deepening: $MRTS$ declines; labour share decreases.

  3. Labour-deepening: $MRTS$ increases; labour share increases.

• Stylized Fact: Factor shares in developed economies have remained stable despite rising $K/L$ because $\frac{w}{r}$ increased proportionally.

Economic Rent and Quasi-Rent

• Economic Rent: Payment to a factor in excess of its opportunity cost (the amount needed to keep it in its current use).

  • Rent = Actual Payment – Opportunity Cost.

  • Pure Rent: Occurs when supply is perfectly inelastic; price is determined entirely by demand.

  • Distinction: To the economy, rent is price-determined; to the firm, rent is a cost.

• Quasi-Rent: Payment to inputs that are fixed in the short run but variable in the long run.

  • Quasi-Rent = Total Revenue (TR) – Total Variable Cost (TVC).

  • It equals Total Fixed Costs (TFC) plus excess profits. It disappears in the long run.

Wage Differentials

• Causes:

  1. Compensating (Equalizing) Differentials: Offsetting differences in job nature (training costs, risk, unpleasantness, career span, location, cost of living).

  2. Non-Compensating Differentials: Differences in innate biological or acquired abilities (e.g., surgeons, professional athletes).

  3. Product Price Differences: Based on the value of what the labour generates.

  4. Market Imperfections: Immobility, discrimination, minimum wages, and restricted entry by unions.

• Persistence: Differentials tend to narrow over time as labour moves toward high-paying sectors, unless entry is artificially restricted.

Product Exhaustion (The Adding-Up Problem)

• Euler’s Theorem: For a production function homogeneous of degree $\nu$, $L \times MPP_L + K \times MPP_K = \nu Q$. Under Constant Returns to Scale ($\nu = 1$), paying factors their marginal product exactly exhausts the total output: $Q = L \times MPP_L + K \times MPP_K$.

• Clark-Wicksteed-Walras Theorem: Product exhaustion does not require constant returns to scale throughout; it holds for any production function in long-run competitive equilibrium where firms produce at the minimum of the Long-run Average Cost ($LAC$) curve, where output elasticity is exactly $1$.

Questions & Discussion

1. Why is the market demand for labour not simply the horizontal sum of individual firm demand curves?

2. Explain why the $VMP_L$ curve is not the demand curve for labour when there are multiple variable factors.

3. Compare the equilibrium wage and employment levels under: (a) perfect competition, (b) monopoly, (c) monopsony, (d) bilateral monopoly.

4. Under what conditions can a labour union increase both wages and employment simultaneously?

5. Using the concept of elasticity of substitution, explain why factor shares have remained relatively stable despite significant capital-deepening.

6. Distinguish between economic rent and quasi-rent. Provide real-world examples of each.