Economia Politica – Corso Avanzato

Advanced Economics

Chapter 5

The marginalist theory of distribution

Saverio M. Fratini

5.1 The “economic trinity” (1)

In the marginalist theory, the production process is seen as a one-way

avenue from the factors of production to the final output.

But what are the factors of production?

LABOUR

CAPITAL LAND

COMMODITIES

5.1 The “economic trinity” (2)

In capital-profit, or still better capital-interest, land-rent,

labour-wages, in this economic trinity represented as

the connection between the component parts of value

and wealth in general and its sources, we have the

complete mystification of the capitalist mode of

production, the conversion of social relations into

things, the direct coalescence of the material production

relations with their historical and social determination.

It is an enchanted, perverted, topsy-turvy world, in

which Monsieur le Capital and Madame la Terre do their

ghost-walking as social characters and at the same time

directly as mere things. (Marx, Capital III, p. 830)

5.1 The “economic trinity” (3)

In the classical/Sraffian approach

There are 3 social classes: workers, capitalists and landowners

☞ hence there are 3 kinds of income: wage, profit and rent

In the neoclassical/marginalist approach

There are 3 kinds of income: wage, interest and rent

☞ hence there must be 3 factors of production: labour, capital and

land

5.1 The “economic trinity” (4)

In the marginalist theory of distribution:

• production processes employ factors of production. For a commodity

m, with m = 1, 2, …, M, we have: 𝑌𝑚 = 𝐹𝑚 𝐿𝑚, 𝐾𝑚, 𝑁𝑚

• wage rate, interest rate and rent rate are the prices of the factors of

production

• incomes from capital are not residual: they are determined by 𝑟 ∙ 𝐾𝑚

and are part of the costs (costs = 𝑤 ∙ 𝐿𝑚 + 𝑟 ∙ 𝐾𝑚 + 𝜌 ∙ 𝑁𝑚)

• social classes disappears and their place is taken by the economic

agents: households and firms

5.2 The household-firm model (1)

Households:

• decide the consumption plans by utility maximization

• are the owners of the factors of production (endowments)

• sell the use of the factors to the firms (income)

• buy consumption goods from the firms (expenditure)

Firms:

• decide the production plans by profit maximization

• hire the production factors from the households (costs)

• sell the consumption goods to the households (revenues)

5.2 The household-firm model (2)

Households Firms

factor market

commodity market

commodities

revenues

commodities

expenditure

incomes

factors

costs

factors

5.2 The household-firm model (3)

Let us consider a firm that produces a quantity Ym of commodity m,

employing Lm, Km and Nm amounts of factors.

Revenues : 𝑝𝑚 ∙ 𝑌𝑚

Costs : 𝑤 ∙ 𝐿𝑚 + 𝑟 ∙ 𝐾𝑚 + 𝜌 ∙ 𝑁𝑚

Profit : 𝑝𝑚 ∙ 𝑌𝑚 − (𝑤 ∙ 𝐿𝑚 + 𝑟 ∙ 𝐾𝑚 + 𝜌 ∙ 𝑁𝑚)

On the one hand: • the amount of profit is not proportional to the employment of capital

• profit is not an income from capital

On the other hand: • interest on capital is part of the costs

• incomes from capital are not a residual, but the result of a price×quantity

multiplication

5.3 Market equilibrium (1)

The marginalist theory is based on the following items:

a) households’ preferences or tastes

b) technical conditions of production (the set of feasible production

plans)

c) endowments of production factors

Given these items, supply and demand functions for commodities and

factors are built as a result of utility and profit maximizations.

Relative commodity and factor prices are determined in terms of

equilibrium between supply and demand functions.

5.3 Market equilibrium (2)

Market dynamics

In each market:

• supply and demand depend on the price system (commodity and

factor prices)

• the price tends to rise when the demand exceeds the supply and to

fall in the opposite case

Let us assume there are N markets. Let 𝐩𝑡 ∈ ℝ+𝑁 be the price system in

period t. We have: 𝑑𝑝𝑛,𝑡

𝑑𝑡= ℎ𝑛 𝑥𝑛 𝐩𝑡 − 𝑦𝑛 𝐩𝑡 ∀𝑛 = 1, 2, … , 𝑁

5.3 Market equilibrium (3)

Given and initial price vector 𝐩0 ∈ ℝ+𝑁, and the system of differential

equations, the dynamic of prices can be determined: 𝐩 = 𝐩 𝑡, 𝐩0 .

Market equilibrium

An equilibrium is a price vector 𝐩∗ ∈ ℝ+𝑁 such that 𝑑𝑝𝑛,𝑡 𝑑𝑡 = 0, ∀𝑛 =

1, 2, … , 𝑁. In other terms: 𝐩∗ = 𝐩 𝑡, 𝐩∗ .

Accordingly, a price vector 𝐩∗ ∈ ℝ+𝑁 is an equilibrium if and only if:

𝑥𝑛 𝐩∗ − 𝑦𝑛 𝐩∗ = 0 ∀𝑛 = 1, 2, … , 𝑁

5.3 Market equilibrium (4)

The equilibrium p* is asymptotically stable if and only if 𝐩 𝑡, 𝐩0 → 𝐩∗

as 𝑡 → ∞, for every possible initial state 𝐩0 in a certain set.

5.3 Market equilibrium (5)

The most complicated part of this reasoning concerns the conception of

supply and demand as functions of the price system.

These functions come from households’ utility maximization and firms’

profit maximization.

In particular, the demand functions for factors of production are based

on the substitutability between factors and the principle of

diminishing marginal productivity.

5.4 Land-labour model (1)

Let us consider a firm that produces a commodity (corn) employing two

factors: land and labour.

There are many (infinite) methods of production, each employing land

and labour in a different proportion.

These methods are collected into a function of production:

𝑌 = 𝐹(𝑁, 𝐿)

Y : output (corn)

N : employment of land

L : employment of labour

5.4 Land-labour model (2)

A production plan is a vector (−𝑁,−𝐿, 𝑌).

Each firm chooses its production plan, among the set of technically

feasible production plans {(−𝑁,−𝐿, 𝑌) ∈ ℝ3: 𝑌 = 𝐹(𝑁, 𝐿)}, in order to

maximize its profit:

𝑚𝑎𝑥 𝑝𝑌 − (𝜌𝑁 + 𝑤𝐿)𝑠. 𝑡. : 𝑌 = 𝐹(𝑁, 𝐿)

p : output price

𝜌 : rent rate

w : wage rate

5.4 Land-labour model (3)

Let us assume that:

• Corn is the numéraire commodity—i.e. p = 1

• Production has constant returns to scale

Let us introduce the following notation:

• 𝑦 = 𝑌 𝐿

• 𝑛 = 𝑁 𝐿

• 𝑓(𝑛) ≡ 𝐹(𝑛, 1)

Because of constant returns to scale: 𝑦 = 𝑓(𝑛) and 𝐹𝑁′ = 𝑓′(𝑛).

Principle of diminishing marginal product: 𝑓′ 𝑛 > 0 and 𝑓′′ 𝑛 < 0.

BOX: Constant returns to scale

Let us start from an initial the employment of factors in quantities N0

and L0. Hence, the output obtained is Y0 = F(N0, L0).

Now, let us assume a proportional increase of both the factors. That is

N1 = N0 (1 + δ) and L1 = L0 (1 + δ)

We say that the production function exhibits constant returns to scale if

and only if

Y1 = F(N1, L1) = Y0 (1 + δ)

BOX: Marginal products and partial derivatives

Once we have the production function 𝑌 = 𝐹(𝑁, 𝐿), we know that a change in the output obtained depends on the change in the inputs employed by the following rule (total differential):

Δ𝑌 = 𝐹𝑁′ Δ𝑁 + 𝐹𝐿

′Δ𝐿 By definition, the marginal product of land is the increase in the output due to the employment of a further unit of land, with a given employment of labour. Accordingly, if Δ𝑁 = 1 and Δ𝐿 = 0, then Δ𝑌 = 𝑀𝑃𝑁. Therefore:

𝑀𝑃𝑁 = 𝐹𝑁′

Similarly: 𝑀𝑃𝐿 = 𝐹𝐿

′.

5.4 Land-labour model (4)

The production plan (−𝑛, −1, 𝑓(𝑛)) is technically feasible.

In order to choose the optimal plan, each firm must choose the optima

proportion n:

max 𝑓(𝑛) − (𝜌𝑛 + 𝑤)

First order condition:

𝑓 𝑛 − 𝜌 = 0

Because of the concavity of the function 𝑓 𝑛 —i.e. 𝑓′′ 𝑛 < 0—the f.o.c.

is necessary and sufficient to determine the optimal employment of land

per unit of labour, given the rent rate ρ.

5.4 Land-labour model (5)

Because of the decreasing marginal productivity

of land, the f.oc. implies that n moves

in the opposite direction to ρ.

n

MPN, ρ

r

f ’(n)

0

r

n’ n”

5.4 Land-labour model (6)

The f.o.c. – i.e. 𝑓 𝑛 − 𝜌 = 0 – allows us to express the demand for land

(per unit of labour) as a function of the rent rate ρ:

𝑛 = 𝑛𝑑(𝜌)

If all the firms have the same technological knowledge, then the function

of demand for land (per unit of labour) is the same for all the firms.

Let 𝑁 and 𝐿 be the quantities of land and labour available in the

economy, 𝑛 = 𝑁 𝐿 is the endowment of land per unit of labour.

According to the usual market mechanism, the market rent rate tends to

fall whenever 𝑛 > 𝑛, while it tends to rise if 𝑛 < 𝑛.

5.4 Land-labour model (7)

As a consequence, ρ* is

the equilibrium rent rate level

if and only if: 𝑛𝑑 𝜌∗ = 𝑛 .

n

MPN, ρ

r*

f ’(n)

0 𝑛

5.4 Land-labour model (8)

Uniqueness and Stability

𝜌 =𝑑𝜌

𝑑𝑡= ℎ[𝑛𝑑 𝜌 − 𝑛 ]

n

MPN, ρ

r*

f ’(n)

0 𝑛

r

r

𝜌 < 0

𝜌 > 0

BOX: Constant returns to scale (bis)

Proposition: if the production function Y = F(N, L) exhibits constant

returns to scale, then: 𝐹 𝑁, 𝐿 = 𝐹𝑁′𝑁 + 𝐹𝐿

′𝐿.

Proof: Let us start from Y0 = F(N0, L0) and assume to increase just the

employment of land, keeping the employment of labour fixed. In

particular, we assume to employ N1 = N0 (1 + δ) and L0.

The output obtained is: 𝐹 𝑁1, 𝐿0 = 𝑌0 + 𝐹𝑁′𝑁0𝛿.

Then, let us increase the employment of labour, with L1 = L0 (1 + δ),

keeping the employment of land fixed at the previous level.

The output obtained is: 𝐹 𝑁1, 𝐿1 = 𝑌0 + 𝐹𝑁′𝑁0𝛿 + 𝐹𝐿

′𝐿0𝛿.

Since the function has constant returns to scale, we know that

𝐹 𝑁1, 𝐿1 = 𝑌0(1 + 𝛿) . Therefore: 𝑌0 1 + 𝛿 = 𝑌0 + 𝐹𝑁′𝑁0𝛿 + 𝐹𝐿

′𝐿0𝛿 ,

which implies: 𝑌0 = 𝐹𝑁′𝑁0 + 𝐹𝐿

′𝐿0. QED

5.4 Land-labour model (9)

Once the equilibrium rent rate is determined, the wage rate w* is

determined as well.

Wages are the part of the total output that is not devoted to the payment

of rents:

𝑤∗𝐿 = 𝐹 𝑁 , 𝐿 − 𝜌∗𝑁 or 𝑤∗ = 𝑓 𝑛 − 𝜌∗𝑛

Given that Y = F(N, L) has constant returns to scale, then: 𝐹 𝑁, 𝐿 =

𝐹𝑁′𝑁 + 𝐹𝐿

′𝐿.

From equations above we get:

𝑤∗ − 𝐹𝐿′ 𝐿 = 𝜌∗ − 𝐹𝑁

′ 𝑁

Hence, 𝐹𝑁′ = 𝜌∗ ⇔ 𝐹𝐿

′ = 𝑤∗.

5.4 Substitutability between factors (1)

Factors of production are on the same ground. They are substitute

among themselves.

A firm chooses its production plan (−𝑁,−𝐿, 𝑌) in order to maximize its

profit:

𝑚𝑎𝑥 𝑌 − (𝜌𝑁 + 𝑤𝐿)𝑠. 𝑡. : 𝑌 = 𝐹(𝑁, 𝐿)

F.O.C.:

𝑝𝐹𝐿

′ −𝑤 = 0

𝑝𝐹𝑁′ − 𝜌 = 0

⟹ 𝑀𝑅𝑇𝑆 = 𝑤

𝜌

𝑌 = 𝐹(𝑁, 𝐿)

5.4 Substitutability between factors (2)

Let us assume 𝑌 = 𝑌

labour

land

𝑌

𝑁′

𝐿′

𝑁′′

𝐿′′

slope green line: 𝑤′

𝜌′

slope red line: 𝑤′′

𝜌′′

𝑤′

𝜌′>𝑤′′

𝜌′′

5.4 Substitutability between factors (3)

Let us assume 𝐿 = 𝐿 Take 3 different rent rate levels: 𝜌′ < 𝜌′′ < 𝜌′′′ Then: 𝑁′ > 𝑁′′ > 𝑁′′′ and 𝑌′ > 𝑌′′ > 𝑌′′′

The neoclassical parable!

labour

land

𝑌′′′

𝑁′

𝑁′′′

𝑁′′

𝑌′′

𝑌′

𝐿

5.4 Substitutability between factors (4)

The neoclassical parable

If the price of a factor decreases (and the employment of the other factors is fixed), then the quantity employed of this factor increases. If the quantity of a factor increases, with given employments of the other factors, then the quantity produced increases as well. Marginal product of a factor of production : increase in the product obtained due to the use of one unit more that factor, for the same employments of all the other factors