Ricardian Models: Valuation

Notes on The Principles of Political Economy and Taxation

05 Jul 2019
modified: 05 Jul 2019

David Ricardo begins The Principles of Political Economy with a discussion on valuation and exchange rates between two commodities: eg., the quantity of cotton that exchanges for a specific quantity of corn; or the quantity of corn that exchanges for a specific quantity of silver specie (coin money). Generalized in this way, valuation extends to barter as well as currency-based economies. For any two commodities A and B, the exchange rate is expressed as \(\frac{v_B}{v_A}\), a ratio representing the quantity of commodity A that exchanges for a unit quantity of commodity B. For example, if commodity B has twice the value in exchange as commodity A, \(v_B = 2v_A\), then two units of commodity A will exchange for one unit of commodity B.

Context and assumptions

Importantly, Ricardo develops his model for valuation under a critically important assumption: perfect competition prevails in both capital and labor markets:

  • workers are mobile and incur no transition costs in migrating to alternative employment opportunities.
  • capital is liquid, and incurs no transition cost in migrating to alternative venture opportunities.
  • both labor (ie., workers) and capital are highly responsive to changes in wages and profits.

Under this assumption, labor and capital quickly crowd into relatively favorable opportunities for wages or profits, respectively; and just as quickly abandon relatively less favorable employments and ventures. As a consequence, long run returns in all competitive ventures achieve a common rate of profit.

Scarcity, utility, and labor

Ricardo identifies three primary drivers for valuation: utility, scarcity, and the quantity of labor required for production of commodities. Utility refers to the value derived from consumption of commodities; utility induces demand for commodities. Scarcity refers to the availability of commodities and captures both natural and artificial barriers to obtaining commodities; scarcity reflects the supply of commodities. Quantity of labor accounts for labor requirements necessary for production of commodities; quantity of labor accounts for duration and intensity of work as well as other abstract features of labor like unpleasantness, erudition, and developed skill. For Ricardo, this quantity includes both immediate labor consumed during production, and consumption of previous labor expenditures towards fixed capital inputs (buildings, tools, technology, etc) or intermediate products.

Ricardo proposes his theory of valuation on an assumption that ventures survive under long-run market conditions by producing commodities adequately possessed of utility (demand) and scarcity (supply). Commodities must possess both qualities in order that entrepreneurs trouble themselves with the effort of bringing them to market. However, Ricardo generally ignores utility and scarcity in developing his model of valuation and focuses nearly exclusively on quantity of labor instead. The rationale for this narrow focus is justified by limiting the scope of his model to commodities unrestricted by monopoly, and from his assumption of perfect competition of labor and capital.

Perfect competition
Prices dynamically vary with changes in scarcity and utility of commodities. And while price increase may temporarily increase profits or price decrease may lower them, labor and capital mobilize in response to this variation. Thus, market supply is appropriately apportioned to meet market demand, and any temporary advantages soon diminish with competition; that is, commodities brought to market will simply return the common rate of profit over the long term.

The effects of perfect competition are illustrated in the diagram below. The three panels each reflect a scenario in which revenues from capital expenditure \(c_0\) generates a rate of return less than, equal to, or in excess of the common rate of return. In the left panel, the rate of return falls short of the common rate of profits; capital will be withdrawn from this venture until the supply of products in this venture bring in the common rate of return \(\left(v(r_0)\right)\). The middle panel shows a venture in which capital generates revenues that match the common rate of return. Capital investments will neither increase or decrease for this venture. The right panel reflects a case where capital generates revenues that exceed the common rate of return. Capital will flow towards this venture to compete for these high profits until competition drives down revenues towards the common rate of profits.

Scarcity and monopoly
Ricardo limits his discussion of valuation to commodities unrestricted by monopolistic barriers to production. He cites rare wines as an example of a commodity facing such restrictions since production is limited by the ideosyncratic soil and terrain uniquely situated within certain vinyards. No amount of labor can increase production of such commodities. For commodities unrestricted by monopoly, changes to valuation ultimately redound to the changes in labor or capital required to produce these commodities.

Labor theory of value

Since long run pricing on commodities reflects only a commmon rate of profits, the only measure for relative valuation between commodities is the quantity of labor expended during production of each commodity. As previously suggested, quantity of labor consumed in the production of commodities encompasses both immediate and intermediate labor expenditures. Immediate consumption of labor can be measured by the wages paid during production of commodities over a fixed period of time. Intermediate consumption of labor includes the consumption of capital, and expenditures towards investments and intermediate products prior to the period in which commodities are brought to market. These factors are outlined in the following diagram and discussed in further detail below.

Additional factors for exchange rates
In addition to the immediate quantity of labor in production of commodities, Ricardo identifies two intermediate factors that influence the exchange rates between commodities.

  1. Consumption of labor as fixed capital expenditures.
    • fixed capital expenditures are invested with an expectation of at least a common rate of return. When these expenditures occur over time periods prior to revenue realization, opportunity costs are incurred as the expected rate of return on this expenditure. Commodities requiring more fixed capital expenditure prior to revenue realization come to market with higher valuations than those with less capital expenditure.
    • changes in wages induce changes in the common rate of profits, so that wages can interact with and alter the expected opportunity costs associated with fixed capital expenditures. Rates of profit differing from those prevailing during periods of fixed capital expenditure will modify the opportunity costs incurred prior to revenue realization and alter final valuation of commodities.
    • as fixed capital depreciates over time, consumption of these investments accrues towards the valuation of commodities.
  2. Consumption of labor as working capital expenditure
    • Frequency of circulation (time from investment to market sales) in working capital can also influence valuation of commodities; since capital expenditures require at minimum a common rate of return, commodites with production periods of longer duration will exhibit higher valution than those brought to market after a shorter duration.
      Differences between any of these additional factors for different ventures lead to changes in exchange rates, even when the different ventures require exactly the same quantity of capital and labor for production during the period of revenue realization. These additional factors are each discussed below.

Influence of fixed capital investment prior to revenue realization on valuation

Ricardo shows two examples that illustrate how employment of fixed capital influences exchange rates between commodities. In his first example, he supposes the production of two commodities over the course of two years. Venture A requires \(\$5000\) capital investment toward manufacturing labor during each year, and commodities are brought to market with realized revenues at the end of each year. For each year, the expected rate of profit is \(10\%\).

\[A\left\lbrace \overbrace{$5000}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}} \times (1 + 0.1) = $5000 + $500 \right.\]

A second venture B requires capital investment \(\$5000\) toward construction of a machine over the first year (fixed capital), with no production of commodities during this period; in the second year, production of commodities commences by employing the machine and an additional capital investment \(\$5000\) toward manufacturing labor.

\[B\left\lbrace\begin{eqnarray} \overbrace{$5000}^{\overset{\text{fixed}}{\scriptsize{\text{capital}}}} \times (1 + 0.1) &=& $5000 + \overbrace{$500}^{\overset{\text{unrealized}}{\scriptsize{\text{profit}}}} &\nonumber& &\text{year} \ 1& \\ ($500 + \overbrace{$5000}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}}) \times (1 + 0.1) &=& $5500 + $550 &\nonumber& &\text{year} \ 2&\\ \end{eqnarray}\right.\]

For venture B, the expected, but unrealized, profit on the capital expenditure during the first year is \(\$500\). Since the capitalist expects a return on investment, the capitalist treats this unrealized profit as additional capital expenditure during the second year; ie., the total capital invested is counted as \(\$5500\) during the second year.

Ricardo’s example shows that each venture requires the same quantity of labor - measured by capital investments for the manufacture of commodities (ie., \(\$5000\)), but the exchange rate between commodities A and B is not unity. During the second year, both ventures require a capital investment of \(\$5000\), but the commodites produced from venture B will exchange for more than the commodities produced from venture A.

\[\frac{v_B}{v_A} = \frac{$6050}{$5500}\]

The unrealized profits from previous fixed capital investments are added to capital expenditures during the current period, and lead to an exchange rate greater than unity.

The effect of fixed capital on exchange rates generalizes for commodities that require multiple periods before realization of revenues. For commodity B, assume

  1. An initial investment of \(c_{F_1}\) towards fixed capital.
  2. A constant rate of expected profit \(r\) over each period.
  3. An elapsed time of \(T \geq 2\) periods between the expenditure towards fixed capital and the period in which commodities are brought to market.
  4. A capital investment of \(c_{L_T}\) towards manufacturing labor during the final period \(T\).

For commodity A, assume the same investment of \(c_{L_T}\) towards manufactures brought to market at time \(T\). The exchange rate of B to A is given as:

\[\begin{eqnarray} \frac{v_B}{v_A} &=& \frac{c_{F_1} r (1 + r)^{T - 1} + c_{L_T}(1 + r)}{c_{L_T}(1 + r)} \\ &\nonumber& \\ &=& 1 + \frac{c_{F_1} r (1 + r)^{T - 1}}{c_{L_T}(1 + r)} \end{eqnarray} \label{a}\tag{1}\]

Interactions between fixed capital investment and wages.

In a second example, Ricardo illustrates the way in wages of labor interact with costs of fixed capital to influence valuation of commodities. Ricardo supposes an increase in the wages of labor, with a consequent decrease in the rate of profits (from \(10\%\) to \(9\%\))

For venture A, the commodities brought to market require an increase in capital investment to account for the rise in wages:

\[A\left\lbrace \overbrace{$5045.87}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}} \times (1 + 0.09) = $5045.87 + $454.13 \right.\]

And for venture B, the same wage increase affects manufacturing capital investment.

\[B\left\lbrace\begin{eqnarray} \overbrace{$5045.87}^{\overset{\text{fixed}}{\scriptsize{\text{capital}}}} \times (1 + 0.09) &=& $5045.87 + \overbrace{$454.13}^{\overset{\text{unrealized}}{\scriptsize{\text{profit}}}} &\nonumber& &\text{year} \ 1& \\ ($454.13 + \overbrace{$5045.87}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}}) \times (1 + 0.09) &=& $5500 + $495 &\nonumber& &\text{year} \ 2&\\ \end{eqnarray}\right.\]

The change in the rate of profits alters the exchange rate between commodities B and A as

\[\frac{v_B}{v_A} = \frac{$5995}{$5500}\]

Similar to equation \(\ref{a}\), we can generalize a change in the rate of profits over multiple periods. For venture B, assume

  1. An initial investment of \(c_{F_1}\) towards fixed capital.
  2. A variable rate of expected profit \(r_{_i}\) over each period \(i\).
  3. An elapsed time of \(T \geq 2\) periods before commodities are brought to market.
  4. A capital investment of \(c_{L_T}\) towards manufactures during the final period \(T\).

For venture A, we assume the same simple investment of \(c_{L_T}\) towards manufactures brought to market at time \(T\). These assumptions give exchange rates between commodities B and A as

\[\begin{eqnarray} \frac{v_B}{v_A} &=& \frac{c_{F_1} r_{{}_1} \prod_{i=2}^{T} (1 + r_{{}_i}) + c_{L_T}(1 + r_{_T})}{c_{L_T}(1 + r_{{}_T})} \\ &\nonumber& \\ &=& 1 + \frac{c_{F_1} r_{{}_1} \prod_{i=2}^{T} (1 + r_{{}_i})}{c_{L_T}(1 + r_{_T})} \end{eqnarray} \label{b}\tag{2}\]

Interactions between fixed capital investment and quantity of labor.
Ricardo points out that the effect of changing wages for exchange rates is insignificant compared to changes in the quantity of labor. He supposes the latter factor exhibits a much more frequent and more significant effect. By way of illustration, he imagines a \(20\%\) decline in the cost of manufacturing required for production of both commodities. For venture A, this reduction in manufacturing expense results in a valuation proportional to:

\[A\left\lbrace \overbrace{$4000}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}} \times (1 + 0.1) = $4000 + $400 \right.\]

Whereas for venture B, we have

\[B\left\lbrace\begin{eqnarray} \overbrace{$5000}^{\overset{\text{fixed}}{\scriptsize{\text{capital}}}} \times (1 + 0.1) &=& $5000 + \overbrace{$500}^{\overset{\text{unrealized}}{\scriptsize{\text{profit}}}} &\nonumber& &\text{year} \ 1& \\ ($500 + \overbrace{$4000}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}}) \times (1 + 0.1) &=& $4500 + $450 &\nonumber& &\text{year} \ 2&\\ \end{eqnarray}\right.\]

Altogether, \(20\%\) reduction in manufacturing labor leaves us with an exchange rate between commodities B and A as

\[\frac{v_B}{v_A} = \frac{$4950}{$4400}\]

The examples above show that the supposed reduction in required labor has more than twice the absolute effect on exchange rates than that resulting from a change in wages of labor; Ricardo himself acknowledges the influence of wages, but cautions his readers against readily concerning themselves over this relatively insignificant factor interacting with fixed capital investments.

Influence of frequency of circulation for working capital in exchange rates

The frequency of circulation of working capital also influences exchange rates between commodities. Commodities from venture B will exchange for a greater quantity of commodities from venture A when venture B requires more time before revenues are realized than for venture A.

Ricardo supposes that venture A requires a capital investment towards manufacturing of \(\$2000\), a rate of profit at \(10\%\), and a realization of those profits after a single year.

\[A\left\lbrace \overbrace{$2000}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}} \times (1 + 0.1) = $2000 + $200 \right.\]

For venture B, suppose a capital investment towards manufacturing during a first year of \(\$1000\) to work up raw materials into intermediate products, and a second investment of \(\$1000\) towards manufactures during a second year to complete production of commodities.

\[B\left\lbrace\begin{eqnarray} \overbrace{$1000}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}} \times (1 + 0.1) &=& \overbrace{$1000}^{\overset{\text{intermediate}}{\scriptsize{\text{materials}}}} + \overbrace{$100}^{\overset{\text{unrealized}}{\scriptsize{\text{profit}}}} &\nonumber& &\text{year} \ 1& \\ ($1000 + $100 + \overbrace{$1000}^{\overset{\text{manufacturing}}{\scriptsize{\text{capital}}}}) \times (1 + 0.1) &=& $2100 + $210 &\nonumber& &\text{year} \ 2&\\ \end{eqnarray}\right.\]

When commodites produced from both ventures are brought to market during the second period, the exchange rate is given as

\[\frac{v_B}{v_A} = \frac{$2310}{$2200}\]

When both costs \(c_L\) and rates of profit \(r\) vary over multiple periods \(j\), an exchange rate between a multi-period manufacturing venture B and a single-period manufacturing venture A sharing the same manufacturing investment \(c_L\) over the final period \(T\) will generalize to

\[\begin{eqnarray} \frac{v_B}{v_A} &=& \frac{\sum_{j=1}^{T} c_{L_j} \prod_{k=j}^{T} (1 + r_{_k})}{c_{L_T}(1 + r_{_T})} \end{eqnarray} \tag{3}\]

Influence of duration of fixed capital in exchange rates

Finally, Ricardo considers how variation in durability of fixed capital influences exchange rates between commodities. A fixed capital investment \(c_F\) that is entirely consumed over \(S\) periods will factor into the value of commodities brought to market. Assuming a fixed capital invested during the first period of production is consumed linearly during production of commodities that are brought to market after \(T\) periods, we would add \(\frac{T}{S}c_{F_1}\) to the value of the commodities.

Generalizing Ricardian Valuation

Generalizing factors relating to variation in fixed capital, rates of profit, frequency of circulating capital, and durability of fixed capital, we can formulate the value of commodities brought to market after \(T \geq 2\) periods under the following assumptions:

  1. An initial investment towards fixed capital \(c_{F_1}\).
  2. A variable rate of wages (and profits) \(r\) over period \(j\).
  3. A variable investment in manufacturing costs \(c_L\) over period \(j\).
  4. A total consumption of fixed capital after \(S\) periods.

Commodities satisfying these assumptions come to market with a value expressed as

\[v \propto \overbrace{c_{F_1} r_{_1} \prod_{i=2}^{T} (1 + r_{_i})}^{\overset{\text{expected return}}{\scriptsize{\text{on fixed capital}}}} + \overbrace{\sum_{j=1}^{T} c_{L_j} \prod_{k=j}^{T} (1 + r_{_k})}^{\overset{\text{cost and expected returns}}{\scriptsize{\text{on manufacturing capital}}}} + \overbrace{\frac{T}{S}c_{F_1}}^{\overset{\text{consumption of}}{\scriptsize{\text{fixed capital}}}} \label{c} \tag{4}\]

For multiple capital investments, denote fixed capital expenditure during period \(j\) as \(c_{F_j}\), with a total consumption of this capital occuring after a period \(S_{_j}\); then equation \(\ref{c}\) generalizes to

\[v \propto \sum_{j=1}^{T} \overbrace{c_{F_j} r_{_j} \prod_{i=j+1}^{T} (1 + r_{_i})}^{\overset{\text{expected return}}{\scriptsize{\text{on fixed capital}}}} + \overbrace{c_{L_j} \prod_{k=j}^{T} (1 + r_{_k})}^{\overset{\text{cost and expected returns}}{\scriptsize{\text{on manufacturing capital}}}} + \overbrace{\frac{T-j+1}{S_j}c_{F_j}}^{\overset{\text{consumption of}}{\scriptsize{\text{fixed capital}}}} \tag{5}\]

Labor inputs vs labor at command

Breaking with his predecessor Adam Smith, Ricardo distinguishes between the quantity of labor to produce a commodity and the quantity of labor that a commodity will command (or purchase). Adam Smith expressed the value in exchange for a commodity as both the quantity of labor employed for its production, as well as the quantity of labor that commodity could command. Ricardo notes that the quantity of labor commanded by a commodity varies with the supply and demand of labor, and thus cannot represent an invariable standard for the exchangeable value for that commodity.