## Ricardian Models: Profits

Notes on The Principles of Political Economy and Taxation

10 Jul 2019
modified: 28 Sep 2019

Under Ricardo’s economic framework, income is the component of valuation remaining after rent, wages and any other consumed capital is deducted from revenue. In considering the distribution of a wealth over the participants in a nation’s economy, profits constitute that portion of production retained by capitalists and entrepreneurs.

### Context and assumptions

Ricardo assumes simple relationships between the amount of food produced, the labor required to produce that food, and the price of food.

Define the following terms:

• $Q_F^{*}$ : quantity of food or raw produce generated with a fixed quantity of capital and labor.
• $Q_L^{*}$ : quantity of labor required to generate a fixed quantity of food on land that pays no rent
• $v_p$ : price for a fixed quantity of food or raw produce

Then Ricardo maintains that the price of a fixed basket of necessaries/provisions at the margin is proportional to the quantity of labor required to produce it; and inversely related the amount of food produce with a fixed quantity of labor:

$\begin{eqnarray} v_p &\propto& Q_L^{*} \\[2em] v_p &\propto& \frac{1}{Q_F^{*}} \end{eqnarray}$

For Ricardo, a worker’s annual wage will depend on the price of necessaries modified by two additional factors:

1. a factor $k$ scales the price of food to a level sufficient to sustain a laborer and family over the course of an annual period.
2. a premium or decrement that accounts for fluctuations in supply and demand of labor $f\left(\frac{c_L}{P} \right)$, where $c_L$ is the capital allocated to labor and $P$ is the population.

worker wages

$w = kv_p + f\left(\frac{c_L}{P} \right)$

The total annual wages paid for production simply scales with the number of workers employed during production of commodities:

total wages

$\begin{eqnarray} W &=& w \times Q_L \\ &=& \left(kv_p + f\left(\frac{c_L}{P} \right) \right) \times Q_L \end{eqnarray}$

Then the annual income achieved amounts to the value of the commodity produced with a given quantity of labor scaled by the total labor utilized during production.

income

$M = Q_F \times v - W$

And the annual rate of profits is the income less any additional consumption of capital divided by the sum of wages and additional capital consumption.

profits

$r = \frac{Q_F \times v_p - W - c}{W + c}$