Negative Price Wicksell Effect, Positive Real Wicksell Effect

1.0 Introduction
I have previously suggested a taxonomy of Wicksell effects. This post presents an example with:

• The cost-minimizing technique varying continuously along the so-called factor-price frontier
• Negative price Wicksell effects
• Positive real Wicksell effects
• Price Wicksell effects greater in magnitude than real Wicksell effects.

This example is due to Saverio M. Fratini ("Reswitching and Decreasing Demand for Capital").

2.0 Technology
Suppose technology consists of a continuum of techniques indexed by the parameter θ, where θ is a real number restricted to the interval [0, 1]:

0 ≤ θ ≤ 1

Each technique consists of the three Constant-Returns-to-Scale processes in Table 1. No commodity is basic, in Sraffa's sense, in any technique in this technology. In the first process in a technique, θ-grade iron is produced directly from unassisted labor. In the second process, labor transforms the θ-grade iron into θ-grade steel. Finally, in the third process, labor transforms θ-grade steel into corn, the consumption good in the model. All processes take a year to complete, and all processes totally use up their input.

Table 1: The Technique Indexed by θ

Inputs	Industry Sector
	θ-Grade Iron	θ-Grade Steel	Corn
Labor (Person-Yrs)	1/(1 + θ)	θ	3/(1 + θ)
Iron (Tons)	0	1	0
Steel (Tons)	0	0	1
Corn (Bushels)	0	0	0
Output	1 Ton	1 Ton	1 Bushel

Capital goods are specific in their uses in this example. θ₁-grade steel cannot be made out of θ₂-grade iron when θ₁ ≠ θ₂.

3.0 Stationary-State Quantity Flows
Suppose in Table 1 that:

• The first process is used to produce (1 + θ)/(4 + θ + θ²) tons of θ-grade iron
• The second process is used to produce (1 + θ)/(4 + θ + θ²) tons of θ-grade steel
• The third process is used to produce (1 + θ)/(4 + θ + θ²) bushels corn

Then one person-year would be employed over these three processes. Capital goods would consist of (1 + θ)/(4 + θ + θ²) tons of θ-grade iron and (1 + θ)/(4 + θ + θ²) tons of θ-grade steel. The capital goods would be used up throughout the latter two sectors, but reproduced at the end of the year. Net output would consist of (1 + θ)/(4 + θ + θ²) bushels corn per person-year.

4.0 Prices
Given the technique, stationary state prices must satisfy the following three equations:

[1/(1 + θ)] w = p₁

p₁(1 + r) + θ w = p₂

p₂(1 + r) + [3/(1 + θ)] w = 1

where:

• p₁ is the price of a ton θ-grade iron;
• p₂ is the price of a ton θ-grade steel;
• w is the wage;
• r is the rate of profits.

A bushel corn is the numeraire. The above equations embody the assumption that labor is paid at the end of the year.

The above is a system of three equations in four unknowns, given the technique. It is a linear system, given the rate of profits. The solution in terms of the rate of profits is easily found. The so-called factor-price curve for a technique is:

w(r, θ) = (1 + θ)/[3 + θ(1 + θ)(1 + r) + (1 + r)²]

The price of a ton θ-grade iron is:

p₁(r, θ) = 1/[3 + θ(1 + θ)(1 + r) + (1 + r)²]

The price of a ton θ-grade steel is:

p₂(r, θ) = [(1 + r) + θ(1 + θ)]/[3 + θ(1 + θ)(1 + r) + (1 + r)²]

Given the technique and the rate of profits, these prices can be used to evaluate the value of the capital goods used in a stationary state.

5.0 The Cost-Minimizing Technique
The optimal technique to use at any given rate of profits maximizes the wage. The first-order condition for such maximization is found by equating the derivative of the factor-price curve to zero:

dw/dθ = 0

Or:

3 + θ(1 + θ)(1 + r) + (1 + r)² - (1 + θ)(1 + 2θ)(1 + r) = 0

For 0 ≤ r ≤ 2, the cost-minimizing technique is then:

θ(r) = {[3 + (1 + r)²]/(1 + r)}^1/2 - 1

For r > 2, a corner solution is found:

θ(r) = 1

Figure 1 illustrates the cost-minimizing technique.

Figure 1: The Choice of Technique

The graph in Figure 1 reaches a minimum at a rate of profits of (3^1/2 - 1). For (12^1/4 - 1) < θ < 1, two rate of profits have the corresponding cost-minimizing technique indexed by the given value of θ. In other words, this is an example of reswitching.

The index for the cost-minimizing technique can be plugged into the factor price curve for the technique to which it corresponds at a given rate of profits. Figure 2 displays the resulting so-called factor price frontier. The index θ varies continuously for 0 ≤ r ≤ 200% in Figure 2. As the rate of profits increases without bound, the frontier approaches a wage of zero.

Figure 2: The Factor-Price Frontier

6.0 Capital and Labor "Markets"
Fratini’s notes that this is a reswitching example in which the capital market initially appears to be in accord with out-dated neoclassical intuition. The above analysis has shown how to find physical quantities of capital goods per worker, how to evaluate them at equilibrium prices, and how to find net output per worker. Figure 3 shows the resulting plot of the value of capital per unit output. Fratini looks at the value of capital per worker instead. Either curve is continuous and downward-sloping. The regions above and below the rate of profits of (3^1/2 - 1) appear qualitatively similar and visually indistinguishable. This curve might be said to be a downward-sloping demand function for capital.

Figure 3: The Capital Market

The analogous curve looks very different for the labor market (Figure 4). The region with a positive Wicksell effect is a region with a high rate of profits and thus a low real wage. The demand function for labor might be said to be upward-sloping in the region with a positive real Wicksell effect.

Figure 4: The Labor Market

7.0 Conclusion
The example makes Fratini’s point. The shape of the relationship between the value of capital, either per worker or per unit output, and the rate of profits is not necessarily a good indicator of the presence of reswitching or reverse capital-deepening.

A Taxonomy Of The Effects Of Wicksell Effects

Consider a typical circulating capital model in which commodities are produced from commodities and labor. The technique in use is described by a square Leontief input-output matrix and a vector of labor coefficients. In a long run equilibrium, in which prices are stationary, the technique is selected from a set of techniques to minimize production costs at a given interest rate. That set is known as the technology.

Suppose the composition and quantity is taken as given, along with the interest rate and the technology. The difference in the value of the capital goods at two different interest rates is the sum of the price Wicksell and real Wicksell effects. The price Wicksell effect is the sum of the differences in prices among the capital goods for a given technique. But the cost-minimizing technique might not be the same at two interest rate. The real Wicksell effect is difference in the value of the capital goods for two techniques, given the price system at a one interest rate.

I want to compare the relative magnitude of price and real Wicksell effects at a given interest rate. Thus, I want to consider derivatives at a given interest rate. Therefore, suppose the technology consists of a continuum of techniques that might be eligible along the so-called factor price frontier. Table 1 shows all combinations of price and real Wicksell effect. A Wicksell effect is negative when the equilibrium at the higher interest rate has a lower value of capital, from the effects of price and quantity changes, respectively.

Table 1: Possibilities

	Technology Property		Labor Market Response	Capital Market Response
	Price Wicksell Effects	Real Wicksell Effects	Higher Wage	Lower Interest Rate
A	Negative	Negative	Less Employment	Increased Value of Capital
B	Negative	Positive	More Employment	Indeterminate
C	Positive	Negative	Less Employment	Indeterminate
D	Positive	Positive	More Employment	Decreased Value of Capital
E	Zero	Negative	Less Employment	Increased Value of Capital
F?	Zero	Positive	More Employment	Decreased Value of Capital
G	Negative	Zero	Unchanged Employment	Increased Value of Capital
H	Positive	Zero	Unchanged Employment	Decreased Value of Capital
I	Zero	Zero	Unchanged Employment	Unchanged Value of Capital

Row A in Table 1 conforms to the outdated neoclassical intuition of equilibrium prices as indices of relative scarcity. But, as Edwin Burmeister has noted, nobody knows what special case assumptions need to be imposed on technology to ensure that Wicksell effects happen to fall in any given direction.

I have the response in the capital market shown as indeterminate for rows B and C. The claim is that, for the case of a technology representable by a continuum of techniques, the price Wicksell effect can, but need not, swamp the real Wicksell effect. It is essential for this swamping to occur at a single interest that the technology be continuous. Pierangelo Garegnani, Heinz D. Kurz & Neri Salvadori, and Saverio M. Fratini have examples that illustrate some possibilities with a continuum of techniques.

Row E is the case of Samuelson's Surrogate Production Function. Price Wicksell effects are zero when the factor price curve for a given technique is a straight line. The question mark after the label for Row F reflects my belief that this row catalogs an impossibility. If factor price curves are straight lines along their entire length, capital-reversing cannot arise.

Rows G, H, and I are cases in which the real Wicksell effect is zero. The real Wicksell effect is zero in the discrete case when the factor price curves are tangent at a switch point. I'm not sure how this extends to the continuous case, in which all points along the factor price frontier are non-switching points. If the Row I case is possible, the technique is not determined by the location of the corresponding factor price curve. I think this may be so for non-straight line factor price curves, but I'm unsure about this case.

These remarks suggest a research program. First, demonstrate that no possibilities exist that are not listed in Table 1. This would seem to be obvious. But I don't understand Andreu Mas-Colell's paper "Capital Theory Paradoxes: Anything Goes" (in Joan Robinson and Modern Economic Theory (ed. by G. Feiwel) (1989)). He shows some multi-valued relations where I would expect functions. Second, for those rows that are impossible in Table 1, demonstrate this impossibility. Third, for each possible row, construct a numeric example. For rows B and C, one should construct at least two examples, one for each direction of the capital market response. I suppose a third example, in which price and real Wicksell effects are exactly matched in magnitude would be amusing. Much of this research would be non-original; many components are in the literature.

Process Recurrence

1.0 Introduction
This post presents another anomaly for neoclassical economics established in Sraffa's framework. It does not present a numeric example, but only outlines the possibility of one. Maybe Kurz and Salvadori (1995) or Woods (1990) contain specific numeric values in which a process can recur along the so-called factor price frontier without the reswitching or recurrence of techniques also arising. Han and Schefold found process recurrence empirically to be more common than reswitching. They say:

"In fact, it is often observed that returns of processes are connected with reverse capital deepening within the scope of our investigation (but it can easily be seen that either phenomenon can also occur without the other)." - Han and Schefold (2006)

2.0 The Model
Consider a simple economy in which workers produce two commodities, apples and bananas from inputs, also composed of apples and bananas. Two processes are known for producing apples, and these processes are labeled I_A and II_A. Two processes, I_B and II_B are also known for producing bananas. This a circulating capital model; all inputs are used up in each production in producing the outputs.

A technique consists of two processes, one for producing apples and one for producing bananas. The claim is that processes can be such that techniques are cost-minimizing at the rates of profits indicated in Table 1. Notice that in this table, each technique is cost-minimizing only in one interval for the rate of profits. Techniques do not reswitch or recur. Yet the first process for producing apples does recur. That process is part of the cost-minimizing technique in both the first and the last interval for the rate of profits. The reswitching of techniques is sufficient for processes to recur, but is not necessary.

Table 1: Cost-Minimizing Techniques

Rate of Profit	Technique
0 ≤ r ≤ r1	IA, IB
r1 ≤ r ≤ r2	IIA, IB
r3 ≤ r ≤ r1	IIA, IIB
r3 ≤ r ≤ rmax	IA, IIB

Rates of profit at which more than one technique is cost minimizing are called switch points. r₁, r₂, r₃, and r₄ are switch points in the example. Capital reversing is the phenomenon in which around a switch point the cost-minimizing technique at the higher rate of profit also has a higher ratio of the value of capital goods to the value of a physically-specified output. In exploded neoclassical intuition, equilibrium prices are scarcity indices. A higher price was thought to indicate that a commodity is more scarce and to lead producers to subsitute other inputs for the more scarce commodity. If the interest rate were the price of capital, a higher interest rate would lead producers to adopt less capital-intensive techniques, in some sense, from a known book of blueprints for techniques. Capital reversing is paradoxical from this perspective and shows that neoclassical heuristics are logically invalid.

Process recurrence is also paradoxical from the exploded neoclassical perspective. An ill-trained neoclassical economist might expect the principle of substitution to hold for individual industries, as well as at the level of the entire economy. In the simple sort of model from which the outlined example is drawn, a higher rate of profits is associated with a lower wage. The specification of a process includes the specification of the how many person-years of labor is needed per unit output in the given industry. Figure 1 shows this normalized direct labor input as a function of the wage. Process recurrence is incompatible with the supposed neoclassical principle of substitution.

Figure 1: Labor Intensity Versus Wage In A Single Industry

References

• Z. Han and B. Schefold (2006) "An Empirical Investigation of Paradoxes: Reswitching and Reverse Capital Deepening in Capital Theory", Cambridge Journal of Economics, V. 30
• Heinz D. Kurz and Neri Salvadori (1995) Theory of Production: A Long-Period Analysis, Cambridge University Press
• J. E. Woods (1990) The Production of Commodities: An Introduction to Sraffa, Humanities Press

Now, Judge, I Had Debts No Honest Man Could Pay/The Bank Was Holding My Mortage And They Were Gonna Take My House Away

1.0 Introduction
This example illustrates one aspect of how Sraffa analyzed natural resources. In this case, natural resources consist of land of various qualities or grades. The quantity of each grade is given; more land cannot be produced. Land is not destroyed either. Appropriate production processes yield as much land as a joint output as given as input. So this sort of model does not incorporate a natural resource like oil that is used up in production.

This example demonstrates that owners of less efficient (productive) natural resources can receive a greater rent.

By the way, Sraffa introduces a distinction between basic and non-basic commodities. Lands with positive rent are non-basic, and therefore their rent is a candidate for taxation.

2.0 Technology
This is a simple economy in which only corn is produced. Table 1 shows the available processes that have corn as an output. Each process requires the use of one grade of land, and no more than one process is known for each grade of land. (This is an example of a model of extensive rent.) Suppose this economy has available 150 acres of land of grade I, 162 acres of grade II land, and 210 acres of grade III land.

Table 1: Technology

Inputs	Process
	Alpha	Beta	Gamma
Labor (Person-Years)	2/5	1	2/7
Grade I Land (Acres)	1	0	0
Grade II Land (Acres)	0	3/2	0
Grade III Land (Acres)	0	0	1
Corn (Bushels)	2/5	1/6	4/7
Output (Bushels)	1	1	1

3.0 Prices
The question to be addressed is what prices and distribution of income are compatible with a long-period position, given the technology. The amount of corn required for net output is a parameter that must be known to answer this question.

3.1 When Only One Grade Of Land Is Cultivated
Suppose the requirements for use for corn in this economy can be satisfied by cultivating any one of the three grades of land. Two grades, and maybe some of the third grade, can lie fallow. So at most 90 bushels are produced each year, after replacinging up the seed corn.

And suppose that the wage is 1/2 bushels per person-year. The wage is paid out at the end of the year. These assumptions are enough to derive the factor-price curves shown in Figure 1. These curves are drawn under the assumption that all grades of land paid no rent.

Figure 2: Factor Price Curves

Since the beta factor-price curve is rightmost (on the outer frontier) at the given wage, all corn is produced on land of the second grade, and this land pays no rent. The wage and the rate of profits must satisfy the following equation:

(1/6)(1 + r) + w = 1

where a bushel corn is the numeraire. For a wage of 1/2 bushels per person-year, the rate of profits is 200%.

Under the given assumptions, the cost of producing a bushel corn on the first grade of land, even if that land were to pay no rent, is 7/5 bushels. Since this cost exceeds the revenues from selling a bushel of land, no capitalist would want to produce on the first grade of land. The reader can check that the cost of producing a bushel corn on the third grade of land also exceeds unity.

3.2 When Two Grades Of Land Are Cultivated
Now suppose the requirements for use are such that two grades of land must be cultivated. The net output of this economy is between 90 and 180 bushels corn. The wage remains 1/2 bushels per person-year. In this case, the first and second grades are cultivated, with the second grade paying rent. The price equations are:

(2/5)(1 + r) + (2/5)w + ρ₁ = 1

(1/6)(1 + r) + w + (3/2)ρ₂ = 1

ρ₁ ρ₂ = 0

ρ₁, ρ₂ ≥ 0

The equations specify that no land can have a negative rent and that at least one grade of land must have a rent of zero. The rate of profits is 100%, when the wage is 1/2 bushels per person-year. Land of grade I pays no rent, and the rent on land of grade II is 1/9 bushels per acre.

The cost of producing a bushel corn on the third grade of land, accounting just for outlays of seed corn and labor, is 9/7 bushels. Capitalists will not want to cultivate the third grade of land, even if it is free.

3.3 When Three Grades Of Land Are Cultivated
Finally, suppose the requirements for use for use require all three grades of land to be cultivated. The price equations are:

(2/5)(1 + r) + (2/5)w + ρ₁ = 1

(1/6)(1 + r) + w + (3/2)ρ₂ = 1

(4/7)(1 + r) + (2/7)w + ρ₃ = 1

ρ₁ ρ₂, ρ₃ = 0

ρ₁, ρ₂, ρ₃ ≥ 0

The rate of profits is 50%, when the wage is 1/2 bushels per person-year. The rent on land of grade I is 1/5 bushels per acre. The rent on grade II land is 1/6 bushels per acre

3.4 Orders Of Efficiency And Rentability
The above analysis has identified a definite order in which different grades of land will be cultivated as greater quantities of output are required to be produced. This is the order of efficiency. In this example, the order of efficiency, from most efficient to least efficient, is: Grade II, Grade I, Grade III.

One can also rank the grades of land from high rent to low rent, when all three grades of land are cultivated and the wage is 1/2 bushels per person-year. The order of rentability, from highest rent to zero rent, is: Grade I, Grade II, Grade III.

The orders of rentability and efficiency differ. It is possible for a less productive, that is, the less efficient, resource to provide its owner with a greater rent than the more efficient resource.

This example is not driven by the existence of switch points.

Reference

• Alberto Quadrio-Curzio, "Rent, Income Distribution, and Orders of Efficiency and Rentability", in Essays on the Theory of Joint Production (Edited by L. L. Pasinetti), Columbia University Press, 1980.

Slope Of "Demand Curve" Varying With Numeraire

1.0 Introduction
As I have previously blogged, Ian Steedman has a number of articles explaining price theory. These articles typically explain the implications, for example, for the slopes of certain functions, but often do not contain graphical illustrations. Here then is an opportunity for me to develop blog posts. For instance, this post works through the example in Section 4 of Arrigo Opocher and Ian Steedman's article, "Input price-input quantity relations and the numéraire" (Cambridge Journal of Economics, V. 33, N. 5 (2009): 937-948).

2.0 Technology
Consider a simple economy in which three commodities - iron, steel, and corn - are produced. Corn is the only commodity people consume, and corn is not used as an input in the production of any commodity. The problem considered here is how much of each input into the production of corn will be demanded by the firms in the corn-production industry.

Iron is produced by unassisted labor. The production function for iron is:

q₁ = f₁(l₁, x_1,1, x_2,1, x_3,1) = l₁

where:

• q_i; i = 1, 2, 3; is the quantity of the ith commodity produced in the period under consideration.
• f_i( ); i = 1, 2, 3; is the production function for the ith commodity.
• l_j; j = 1, 2, 3; is the number of person-years hired in the period under consideration to produce the jth commodity.
• x_i,j; i = 1, 2, 3; j = 1, 2, 3; is the quantity of the ith commodity used in the production of the jth commodity in the period under consideration.

Steel is produced by labor from iron. Steel production is modeled by a Cobb-Douglas production function:

q₂ = f₂(l₂, x_1,2, x_2,2, x_3,2) = (l₂)^D (x_1,2)^E/(D^D E^E)

where D and E are positive parameters and D + E = 1.

Corn is produced by labor from inputs of iron and steel. The production function for corn is:

q₃ = f₃(l₃, x_1,3, x_2,3, x_3,3) = (l₃)^d (x_1,3)^e (x_2,3)^f/(d^d e^ef^f)

where d, e, and f are positive parameters and d + e + f = 1.

All production functions exhibit Constant Returns to Scale (CRS). All capital goods are totally used up in production, and all production processes require the same amount of time.

3.0 Cost Functions and Coefficients of Production
Consider competitive firms producing a commodity who want to adopt a cost-minimizing technique. For definitiveness, consider a firm producing steel. Let w be the wage (paid at the beginning of the period) and p_j; j = 1, 2, 3; be the spot price of the jth commodity. The unit cost function, c₂( ), for producing steel is the value of the objective function in the solution to the following mathematical programming problem:

Given w, p₁, p₂, p₃

Choose l₂, x_1,2, x_2,2, x_3,2

To Minimize wl₂ + p₁x_1,2 + p₂x_2,2 + p₃x_3,2

Such that:

f₂(l₂, x_1,2, x_2,2, x_3,2) = 1

l₂ ≥ 0; x_i,2 ≥ 0, i = 1, 2, 3.

Solving this programming problem, one finds the unit cost function for producing steel is:

c₂(w, p₁, p₂, p₃) = (w)^D (p₁)^E

In working out the cost function, I also figured out how much labor and iron a steel-producing firm would hire to produce one ton of steel. Opocher and Steedman pose the problem with given cost functions, not production functions. They derive the coefficients of production by invoking Shephard’s Lemma.

A similar cost-minimizing problem arises for corn-making firms.
The unit cost function, c₃( ), for producing corn is:

c₃(w, p₁, p₂, p₃) = (w)^d(p₁)^e(p₂)^f

The derivatives of the cost functions are summarized by the coefficients of production, a₀ and A, where:

• a₀ is a three-element row vector such that a_0,j; j = 1, 2, 3; is the person-years of labor hired per unit output of the jth industry.
• A is a 3x3 matrix such that a_i,j; i = 1, 2, 3; j = 1, 2, 3; is the quantity of the ith commodity purchased as an input per unit output in the jth industry.

Table 1 displays coefficients of production.

Table 1: The Cost-Minimizing Technique

	Iron Industry	Steel Industry	Corn Industry
Labor	a0,1 = 1	a0,2 = D (p1/w)E	a0,3 = dwd - 1 (p1)e(p2)f
Iron	a1,1 = 0	a1,2 = E (w/p1)D	a1,3 = ewd (p1)e - 1(p2)f
Steel	a2,1 = 0	a2,2 = 0	a2,3 = fwd (p1)e(p2)f - 1
Corn	a3,1 = 0	a3,2 = 0	a3,3 = 0
Output	1 ton iron	1 ton steel	1 bushel corn

4.0 Long Period Price Equations
The above shows the coefficients of production that perfectly competitive firms choose, given prices. Each column in Table 1 has been derived independently of the others. Firms will continue to produce corn, the consumption good, period after period only if some firms also choose to produce iron and steel. Cost-minimizing firms will not make these choices for any configuration of prices. The long-period condition that firm choices be self-sustaining yields the following system of three equations:

a_0,1 w (1 + r) = p₁

[p₁ a_1,2(w, p₁) + w a_0,2(w, p₁)](1 + r) = p₂

[p₁ a_1,3(w, p₁, p₂) + p₂ a_2,3(w, p₁, p₂) + w a_0,3(w, p₁, p₂)](1 + r) = p₃

where r is the rate of profits. Since production takes time, the rate of profits is generally positive. These equations, when solved, define long-period equilibrium prices for produced commodities as functions of the wage and the rate of profits:

p₁ = w(1 + r)

p₂ = w(1 + r)^{1 + E}

p₃ = w(1 + r)^{1 + e + f + (1 + E)f}

5.0 The Numeraire
The above system of price equations has two degrees of freedom. One degree of freedom is removed when the numeraire is specified. Let the numeraire consist of σ₁ units of iron, σ₂ units of steel, and λ person-years:

σ₁p₁ + σ₂p₂ + λw = 1

Prices of the commodities comprising the numeraire are then:

w = 1/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

p₁ = (1 + r)/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

p₂ = (1 + r)^{1 + E}/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

Figure 1 shows the relationship between the wage and the rate of profits. This relationship is known as the wage-rate of profits frontier. In this example, coefficients of production vary continuously along the frontier.

Figure 1: Wage-Rate Of Profits Frontier

6.0 Quantity Demanded for Inputs
The above derivations allow one to draw various graphs for specific parameter values. I set the parameters for the production functions as follows: D = 3/4; E = 1/4; d = 2/3; e = 1/6; and f = 1/6. And I considered two numeraires. For the first numeriare, σ₁ = 1/3; σ₂ = 1/3; and λ = 1/3. For the second numeriare, σ₁ = 1/3; σ₂ = 1/2; and λ = 1/6.

For each numeraire, the wage and other prices in a long-period position are defined as functions of the rate of profits. And the cost-minimizing coefficients of production are functions of these prices, that is, ultimately of the rate of profits. Figure 2 shows a locus constructed out of these functions. Curves at the top of the figure plot the price of iron against the quantity of iron demanded by the (non-vertically integrated) corn-producing industry. In drawing this figure, the quantity of corn produced is constrained to be unity. The curves are analogous to conditional demand curves in neoclassical economic theory. And one can see that, for certain regions, whether the slope of such a curve is positive or negative depends on the numeraire. But is not a main point of neoclassical economics to argue that demand curves are downward-sloping (for arbitrary numeraires)?

Figure 2: A Locus for Firms in Long Period Equilibrium

7.0 Conclusion
So much for explaining the price of a capital good by well-behaved supply and demand curves in the market for that commodity.

Yet Another Cambridge Controversy (YACC) Example

1.0 Introduction
This is another example of reswitching and capital reversing. In this example, a capital good can only be used with a fixed quantity of labor in producing the single consumption good. If entrepreneurs would like to employ a different quantity of labor per unit output of the consumption good, they must use the services of some other capital good. In other words, this is an example of fixed coefficients, in some sense. But the existence of reswitching and capital-reversing are compatible with variable coefficients and the possibility of marginal adjustments. I think many economists may be confused on this point.

This example can also be seen as an illustration of a generalization of Paul Samuelson's "Surrogate Production Function" model. Samuelson assumed that alternative processes for producing the consumption good each require the use of a different capital good. But Samuelson required the special-case assumption that, for a given capital good, the desired ratio of physical units of the capital good to labor was invariant among processes producing the consumption good and producing the capital good. As Garegnani pointed out, this special case effectively collapses Samuelson's model to a one-good model. And so Samuelson's attempted defense of the aggregate production function fails.

2.0 The Technology and Quantity Flows
Consider a firm whose managers are aware of the technology shown in Table 1. Each column defines a Constant-Returns to Scale process for producing the output indicated by the column heading. The managers know of two processes for producing corn, a good used only in consumption. Steel and labor are each used in producing either more steel or in producing corn. Similarly for tin. Each process requires a year to complete, and the capital good is totally used up in producing the output. In the jargon, this is an example with no fixed capital; all capital is circulating capital.

Table 1: The CRS Technology

Inputs	Steel Industry	Tin Industry	Corn Industry
			Alpha	Beta
Labor (Person-Years):	1/3	1/2	1	3/2
Steel (Tons):	1/6	0	1	0
Tin (Tons):	0	1/4	0	1/4
Corn (Bushels):	0	0	0	0
Output (Various):	1	1	1	1

Two techniques are available for the vertically-integrated firm to produce corn. Each technique consists of two processes operated in parallel. The first technique, called the alpha technique, consists of the steel-producing process and the first of the two corn-producing processes. Suppose these processes are operated at a scale to produce 6/5 tons of steel gross and one bushel corn. Of the output of the steel-producing process, 1/5 tons replaces the steel inputs used up in producing the steel. The remaing ton replaces the steel used up in producing the corn. Thus, at this scale, the net output of the vertically-integrated firm consists of one bushel corn. Since 7/5 person-years are used across both processes, the ratio of the capital-inputs in physical terms to worker is 6/7 tons steel per person-year. Output per worker consists of 5/7 bushels per person-year.

The second technique, called the beta technique, consists of the tin-producing process and the second corn-producing process. When these processes are used to produce 1/3 tons tin gross and one bushel corn, the net output consists of one bushel corn. At this scale of operations, 5/3 person-years are employed. The ratio of the capital-inputs to worker is 1/5 tons tin per person-year. Output per worker is 3/5 bushels per person-year.

The alpha technique produces more output per worker. In the traditional and incorrect neoclassical analysis of the production function, greater output per worker for a known technology is achieved by the use of more capital per worker. Firms choose to use a less labor-intensive process, in this incorrect analysis, when consumers choose to save more and the interest rate (called the rate of profits below) consequently falls. Notice that one cannot tell from the above arithmetic with the example whether the capital-to-labor ratio is higher in the alpha or in the beta technique. The units of comparison so far are incommensurable.

3.0 Prices
The adoption of each technique in a steady-state competitive capitalist economy specifies certain relationships among price variables.

3.1 The Alpha Technique
If the alpha technique is cost-minimizing, the following equations must hold:

(1/6) p_s (1 + r) + (1/3) w = p_s

p_s (1 + r) + w = 1

where p_s is the price of steel in units of bushels corn per ton steel, w is the wage in units of bushels corn per person year, and r is the rate of profits. These equations incorporate the assumption that wages are paid out of the surplus at the end of the year. This system of equations shows that the same rate of profits is earned in producing both steel and corn.

This is a system of two equations in three variables. As Sraffa notes, it has one degree of freedom. Two of the variables can be found in terms of the third. I take the rate of profits as the independent variable. The wage is then:

w_α(r) = (5 - r)/(7 + r)

And the price of steel is:

p_s(r) = 2/(7 + r)

The wage-rate of profits curve is a declining function in the first quadrant of the wage-rate of profits space (Figure 1). Since this is a model of the production of commodities by means of commodities, the wage-rate of profits curve cuts both axes. The maximum rate of profits in the alpha system, found when the workers live on air, is r = 500%. The maximum wage, where the capitalists receive none of the surplus, is 5/7 bushels per person-year.

The price of steel can be used to evaluate the capital goods used per worker in the alpha technique:

I_α(r) = 12/[7 (7 + r)] bushels per person-year

The price Wicksell effect is the variation, given the technique, in the value of capital per worker with the rate of profits.

Figure 1: Wage-Rate Of Profits Curves

3.2 The Beta Technique
The price equations for the beta technique consist of:

(1/4) p_t (1 + r) + (1/2) w = p_t

(1/4) p_t (1 + r) + (3/2) w = 1

where p_t is the price of tin in units of bushels corn per ton tin. The solution of this system of equations is:

w_β(r) = (3 - r)/(5 - r)

p_t(r) = 2/(5 - r)

The maximum rate of profits in the beta system is r = 300%, which is lower than the maximum in the alpha system. The maximum wage in the beta system is 3/5 bushels per person-year, which is also lower than in the alpha system. The value of capital per worker is:

I_β(r) = 2/[5 (5 - r)] bushels per person-year

3.3 Switch Points
The technique with the highest wage, at a given rate of profits, is the cost-minimizing technique at that rate of profits. The wage-rate of profits frontier in models like the example is the outer envelope in Figue 1 of all wage-rate of profits curves. The alpha technique is cost-minimizing both for rates of profits between 0% and 100% and for rates of profits between 200% and 500%. The beta technique is cost minimizing for rates of profits between 100% and 200%.

At a switch point, more than one technique is cost minimizing. In the example, switch points are at (r, w) = (100%, 1/2) and at (200%, 1/3).

3.4 Selected Consequences for Capital and Labor "Markets"
The analysis of the choice of technique allows one to graph the value of capital per worker in a steady state versus the rate of profits (Figure 2). The graph displays a combination of price and real Wicksell effects. Consider the horizontal lines at the rate of profits of 100% and 200% for the switch points. The real Wicksell effect is the variation in the in the value of capital per worker with the technique at a rate of profits for a switch point. The price Wicksell effect is shown by the curves not being vertical between switch points. In Figure 2, the switch point with a positive real Wicksell effect is indicated. The example demonstrates the logical invalidity of traditional neoclassical theory, in which real Wicksell effects are always negative.

Figure 2: Investment Function

The analysis of the choice of technique also allows one to graph the level of employment firms offer at each wage, given the level of net output. Figure 2 indicates the switch point around which firms will attempt to hire more labor if the wage is increased.

Figure 3: Labor Employed per Unit Net Output as a Function of Wages

As an aside, I wonder if the following quotation includes an allusion to the Cambridge Capital Controversy:

"For output in goods-producing industries to increase, the quantity of intermediate goods used to produce the output must also be increased (e.g. cloth to produce a shirt, ham to produce a ham sandwich, etc.). However, the concept of the marginal product of labor (or capital) requires that as the input of labor (or capital) is increased, all other inputs must be held constant. But this is not possible for intermediate inputs in goods-producing industries. Therefore, the concept of the marginal product of labor (or capital) is not possible when there are intermediate goods in the production function. Again, we prefer not to teach such logical errors to our students." -- Bernard Guerrien and Emmanuelle Benicourt (2008)

4.0 Conclusions
So much for the logical validity of typical arguments for improved labor market flexibility and for the belief that minimum wages cause unemployment.

Bibliography

• Bernard Guerrien and Emmanuelle Benicourt (2008) "Is Anything Worth Keeping in Microeconomics?" Review of Radical Political Economics, V. 40, N. 3 (Summer): 317-323
• P. Garegnani (1970) "Heterogeneous Capital, the Production Function and the Theory of Distribution", Review of Economic Studies, V. 37, N. 3 (July): 407-436.
• Paul A. Samuelson (1962) "Parable and Realism in Capital Theory: The Surrogate Production Function", Review of Economic Studies, V. 29, N. 3 (June): 193-206

A Reswitching Example With Fixed Capital

1.0 Introduction
This post presents an example of reswitching with fixed capital. This is Example 5 in Chapter 9 of J. E. Woods, The Production of Commodities: An Introduction To Sraffa (Humanities Press, 1990). Fixed capital is a special case of joint production. I often analyze the choice of technique in circulating capital models by constructing the so-called factor-price frontier as the outer envelope of factor-price curves for each technique. This method of analysis does not generalize to general models of joint production. I confine myself in this post to methods of analysis that apply generally to joint production. This example illustrates the reswitching of techniques. A manifestation of reswitching in this example is the non-monotonic dependence of the economic life of machinery on the rate of profits.

2.0 Technology
The example is of a vertically-integrated industry producing a net output of corn with inputs of corn, machines, and labor. Machines can last for two yearly cycles of production, while corn inputs are entirely used up each year in production. Firms have available the three Constant-Returns-to-Scale (CRS) processes shown in Figure 1. New machines are produced by the first process, while corn is produced by both of the two remaining processes. The third process uses a one-year old machine as an input. The one-year old machine is produced jointly with corn by the second process.

Table 1: The Technology

	Machine Industry	Corn Industry
Inputs
Labor	1/10 Person-Year	43/40 Person-Year	1 Person-Year
Corn	1/16 Bushel	1/16 Bushel	1/4 Bushel
New Machines		1	0
Old Machines		0	1
Outputs
Corn	0 Bushels	1 Bushel	1 Bushel
New Machines	1	0	0
Old Machines	0	1	0

3.0 Quantity Flows
A choice of technique arises over the economic life of the machine. A firm might choose to discard the machine after the first year and never use the third process, whether in conjunction with the other processes or not. (I assume that machines can be freely disposed of after both one and two years of operation.)

The proportions in which processes are operated varies, depending on how many years the machine is used. Table 2 shows the first two process being used to produced a net output of one bushel corn. Notice that when these processes are operated in parallel new machines are simultaneously produced by the first process and used up by the second process. (I suppose I could scale up the processes in Tables 2 and 3 by 91 so that an integral number of machines is used in each technique and the net output of both is the same quantity of bushels of corn.)

Table 2: Use Of Machine For One Year

	Machine Industry	Corn Industry
Inputs
Labor	4/35 Person-Year	43/35 Person-Year
Corn	1/14 Bushel	1/14 Bushel
New Machines	0	8/7
Outputs
Corn	0 Bushels	8/7 Bushel
New Machines	8/7	0

Table 3 shows how all three processes must be scaled when machines are used for two years and the net output is one bushel corn. Tables 2 and 3 can be used to calculate the flow of labor inputs and capital goods required to produce a net output of one bushel corn for each technique. Table 4 exhibits the result of this calculation. If machines are junked after one year, 2/455 person-years more labor is hired per bushel corn net, as compared to machines being junked after two years.

Table 3: Use Of Machine For Two Years

	Machine Industry	Corn Industry
Inputs
Labor	4/65 Person-Year	43/65 Person-Year	8/13 Person-Year
Corn	1/26 Bushel	1/26 Bushel	2/13 Bushel
New Machines	0	8/13	0
Old Machines	0	0	8/13
Outputs
Corn	0 Bushels	8/13 Bushel	8/13 Bushel
New Machines	8/13	0	0
Old Machines	0	8/13	0

Table 4: Inputs Per Net Output of 1 Bushel Corn

	Years Machine Is Operated
	1 Year	2 Years
Labor	47/35 Person-Years	87/65 Person-Years
Capital	1/7 Bushel, 8/7 New Machines	3/13 Bushel, 8/13 New Machines, 8/13 Old Machines

4.0 Prices and the Choice of Technique
Firms choose the technique - that is how long machines are used - based on profitability. Hence, an analysis of the choice of technique requires an analysis of prices.

4.1 Prices When Machine is Operated One Year
Suppose, to begin with, that the machine is operated for one year only. Prices are such that no pure economic profits can be earned in producing new machines or in operating machines for the first year. Furthermore, costs (inclusive of a charge on capital goods advanced) do not exceed revenues in either of these two processes. These assumptions imply two equalities are met:

(1/16) (1 + r) + (1/10) w_α = p_{0, α}

[(1/16) + p_{0, α}](1 + r) + (43/40) w_α = 1,

where a bushel corn is the numeraire, p_{0, α} is the price of a new machine, w_α is the wage, and r is the rate of profits. These equations embody the implicit assumption that wages are paid after the laborers have been working for a year. The price of a year-old machine is zero.

This is a system of two equations in three unknowns. If the rate of profits is taken as given outside this system, the other two variables can be found as a function of the rate of profits:

w_α(r) = (5/2)(14 - 3 r - r²)/(4 r + 47)

p_{0, α}(r) = (1/16)(103 + 39 r)/(4 r + 47)

The larger is the wage, the smaller is the rate of profits. The maximum rate of profits in this system arises for a wage of zero. That maximum is approximately 253.11%.

The above solution can be used to analyze the profitability of operating the machine for another year. Figure 1 shows the costs and revenues (in bushels corn) for producing a bushel of corn by operating the third process in Table 1. Inputs are valued at the prices given by the above solution. Note that revenues would exceed costs if the rate of profits were below approximately 33%. Likewise, revenues would exceed costs if the rate of profits were above 50% and below the maximum. In these regions, firms would choose to operate the machines for a second year, and the prices would not be equilibrium prices.

Figure 1: Costs and Revenues for Operating Machine for Second Year

On the other hand, the above solution gives equilibrium prices if the rate of profits lies between approximately 33% and 50%. In this region, the costs of operating the machine for a second year exceed the revenues. Firms would only operate the machine for one year, consistently with the hypothesis for this case.

4.2 Prices When Machine is Operated Two Years
The other case arises when prices are consistent with the machine being operated for two years. By assumption, pure economic profits cannot be earned in any of the three processes in the technology. Likewise, costs do not exceed revenues for any of the three processes. These assumptions yield a system of three equations:

(1/16) (1 + r) + (1/10) w_β = p_{0, β}

[(1/16) + p_{0, β}](1 + r) + (43/40) w_β = 1 + p_{1, β}

[(1/4) + p_{1, β}](1 + r) + w_β = 1

where p_{0, β} is the price of a new machine, p_{1, β} is the price of a machine after operating for one year, w_β is the wage, and r is once again the exogeneously specified rate of profits.

This system has one more equation than the system in Section 4.1. But it contains one more variable also. Thus, the wage and the prices of the new and one year-old machine can be found in terms of the rate of profits:

w_β(r) = (5/2)(26 + 7r - 4r² - r³)/(87 + 51r + 4 r²)

p_{0, β}(r) = (1/80)(135 - 166 w_β - 86 w_β r + 50 r - 5 r²)/(1 + r)²

p_{1, β}(r) = (1/4)(3 - 4 w_β - r)/(1 + r)

In this case also, the maximum rate of profits arises when the wage is zero. The maximum rate of profits is approximately 258.77%

The analysis of the choice of technique in this case is based on examining the price of the year-old machine (Figure 2). Only non-negative prices are consistent with equilibria. The price of an used machine becoming negative is a sign that it is cheaper to truncate the use of a machine after it is used for one year. That is, for a rate of profits between approximately 33% and 50%, firms will junk the machine after one year of operation.

Figure 2: Price of One-Year Old Machine

4.3 Choice of Years of Operation
The analysis of the choice of technique is consistent, whether one begins with the price system for machines being used for one or two years. When the rate of profits is below approximately 33%, but non-negative, machines are used for two years. When the rate of profits exceeds 50% and below the maximum, machines are also used for two years. When the rate of profits exceed approximately 33% and are below 50%, machines are discarded after one year of operation.

5.0 Conclusion
Suppose one incorrectly accepts pre-Sraffian neoclassical or Austrian intuition. Then one would believe that a low rate of profits reflects capital, in some sense, being relatively less scarce than labor. And one would expect firms to adopt a more capital-intensive technique at a lower rate of profits. In this context, one would expect the economic lives of machines to be longer at a lower rate of profits. But, as shown by Figure 3, such a belief is incorrect, in general.

Figure 3: Economic Life of Machine versus Rate of Profit

If one had such incorrect pre-Sraffian neoclassical or Austrian intuition, one would likewise expect firms to hire more workers at lower wages, also under conditions of perfect competition. Figure 4 demonstrates this belief is false as well. So much for the idea that wages and employment are determined by the interaction of well-behaved supply and demand functions.

Figure 4: Labor Intensity versus Wage

Another Reswitching Example

1.0 Introduction
I have previously presented examples of reswitching and capital reversing. These phenomena were discovered as part of the Cambridge Capital Controversy, and you will occasionally see commentators bring up the CCC on various blogs. This post presents another example. This example has a different structure than my favorite models. In this example, not all commodities are produced with the aid of commodities. Some capital goods not explicitly shown are produced directly with unassisted labor.

2.0 Some Theory And An Example
Consider a firm choosing among a set of techniques for producing a given commodity. Each technique, as in Table 1, exhibits Constant Returns to Scale (CRS) and can be represented as a series of dated labor inputs. The first element in this series is the amount of labor that must be applied in the technique during the current production cycle to produce one unit of the commodity. The second element is the amount of labor that must be applied one cycle ago to produce the capital goods required for use by the first labor input in the series. And so on. This representation of a technique is known as a flow-input, point-output model.

Table 1: The Technology

Year Before Output	Labor Hired for Each Technique
	Alpha	Beta
0	33 Person-Years	0 Person-Years
1	0 Person-Years	52 Person-Years
2	20 Person-Years	0 Person-Years

Consider the the cost of producing a unit of output with a given technique, where the calculation of the cost is performed at the end of the year in which the output becomes available. The cost is the sum of the cost of the labor inputs over previous years, with the cost of each labor input including an interest charge:

L₀ w + L₁ w (1 + r) + L₂ w (1 + r)² + ... + L_n w (1 + r)ⁿ + ...

where w is the wage for a unit of labor, r is the rate of profits, and (L₀, L₁, L₂, ...) is the series of dated labor inputs representing the technique. The wage is paid at the end of the year for the labor expended during that year.

The best rate of profits (also known as the interest rate) available to the firm should be used in finding the cost of producing a unit of output with each technique. Since the only use of financial capital available to the firm here is to produce with one or another of the techniques of production, the best rate of profits is the rate of profits obtainable with cost-minimizing techniques. The following algorithm traverses the techniques to find the rate of profits associated with cost-minimizing techniques:

1. Choose a technique.
2. For the given wage w, find the rate of profits r that, when used to cost up the labor inputs, equates that cost to the price of a unit of the produced commodity.
3. Cost up the labor inuts for all techniques with this wage and rate of profits.
4. Choose a technique with the minimum of the costs found in Step 3.
5. If the technique for which the rate of profits was obtained in Step 2 can be selected in Step 4, stop. You have found a cost-minimizing technique and the rate of profits.
6. Else, go to Step 2 and repeat with the newly selected technique.

Figures 1, 2, and 3 illustrate the application of the algorithm to the example. Figure 1 shows the wage-rate of profits curves associated with each technique. In flow-input, point-output models, these curves never cross the axis for the rate of profits. Instead, the curves approach it asymptotically. Points at which such curves intersect are called switch points. The "perverse" or "paradoxical" switch point is at a wage of 1/78 units of output per person year and a rate of profits of 50%. The higher-wage switch point is at a wage of 5/286 units of output per person-year and a rate of profits of 10%. A switch point is called "paradoxical" merely because it illustrates behavior contradicting mistaken and outdated neoclassical beliefs. The algorithm can be considerably simplified:

Theorem: Consider a technology specified as a choice among techniques, where each technique is represented by a flow-input, point-output model. Let the outer envelope of the wage-rate of profits curves for each technique be constructed by finding the maximum rate of profits for each wage over all wage-rate of profits curves. For each point on the outer envelope, a technique in which the technique’s corresponding wage-rate of profits curve contains that point is selected by the above algorithm as a cost-minimizing technique at that wage.

Figure 1: Rate of Profits by Technique

The theorem is easily applied to the example. By the theorem, the alpha technique is cost-minimizing for wages between the switch points. The beta technique is cost-minimizing for wages below 1/78 units of output per person-year and between wages of 5/286 and the maximum wage of 1/52 units of output per person-year.

Now to show that the algorithm yields the same answers. Suppose one starts by choosing the alpha technique at Step 1. The rate of profits sought in Step 2 is read off the wage-rate of profits curve for the alpha technique in Figure 1, given the wage. The costs found in Step 3 are shown in Figure 2. Figure 1 shows that the alpha technique is, indeed, cost minimizing for intermediate wages, that is, between the switch points. If the wage is either too low or too high, the beta technique is selected in Step 4, as shown in Figure 2. According to Step 6, the flow of control in the algorithm then goes to Step 2 with the beta technique selected. Figure 3 shows the costs of the techniques using the rate of profits for the beta technique. Here too, the algorithm concludes with the same answer as suggested by the theorem.

Figure 2: Cost of Techniques at Alpha Rate of Profits

Figure 3: Cost of Techniques at Beta Rate of Profit

3.0 Conclusion
Once the cost-minimizing technique has been determined, one can consider how much labor firms will want to hire per unit output at any given wage. Figure 4, which is based on the assumption of a stationary-state output, illustrates this calculation for the example. If a firm is operating the alpha technique, for example, to produce an unchanged output over the next three years, workers producing consumption goods available at the end of the current year, producing capital goods to be used to produce consumption goods available at the end of the next year, and producing capital goods to be used to produce capital goods available at the end of the next year will all be working side-by-side. That is, 53 workers will be employed per unit output under the alpha technique for a stationary state. As the graph shows, around the switch point at the lower wage, a higher wage is associated with a cost-minimizing technique in which firms want to employ more workers for a given stationary-state output.

Figure 4: Labor Intensity of Cheapest Technique

So much for the theory that wages and employment are determined by the intersection of well-behaved supply and demand curves in the labor market.

Therefore I Ain't Got No Money To Pay The Rent

1.0 Introduction
A vulgar belief is that competitive factor prices reward factor owners for the physical contributions of their services to output, in some sense. Although David Ricardo was not a vulgar economist - he is one of the paradigms for defining Classical economics - he made a similar mistake. In particular, Ricardo treated land as capable of being ordered from high rent to low rent based on physical data (fertility) alone. He only considered a special case where the order of rentability can be read off of physical data alone, rather than the more general case where the order of rentability can be altered by differences in distribution between wages and profits. Understanding the more general case seems appropriate for one who considers economics a social science.

This post demonstrates the limitations of the Ricardian doctrine. These limitations are shown by means of an example. Since so-called Neoclassical economics is often claimed to have been developed by extending the Ricardian theory of rent, the analysis presented here may be of slightly more contemporary interest as well.

2.0 The Technology
Consider a very simple economy that produces a single consumption good, corn, from inputs of labor and (seed) corn. Corn is grown on two grades of land. All production processes in this example require a year to complete. Only one production process is known for producing corn on each grade of land. These processes require the inputs shown in Table 1 to be available at the beginning of the year for each bushel corn produced and available at the end of the year.

Table 1: Inputs Required Per Bushel Corn Produced

Grade I Land	Grade II Land
1 Acre	1 Acre
1/4 Person-Years	2/3 Person-Years
1/2 Bushels Corn	1/3 Bushels Corn

Notice that the proportions of labor and seed corn differ on the two grades of land. This idea can be generalized to a model of the choice of technique in long period positions, where this model includes both extensive and intensive rent:

"...it is very well possible that a larger output per hectare is obtained by using less (or even nothing) of some input(s) and more of some other input(s) or some positive amount of some input(s) not used at all at the smaller level of production." -- Heinz D. Kurz and Neri Salvadori

Assume that the total corn output required in this economy can be grown on some combination of both grades of land, but exceeds the amount that can be grown on either grade alone. In other words, when enough corn is produced to satisfy this economy's requirement for use, some, but not all, of one grade of land will lie fallow. Thus, rent will almost always be paid on one grade of land, and the other is free (non-scarce).

Here's how Ricardo explains the emergence of rent:

"On the first settling of a country, in which there is an abundance of rich and fertile land, a very small proportion of which is required to be cultivated for the support of the actual population, or indeed can be cultivated with the capital which the population can command, there will be no rent; for no one would pay for the use of land, when there was an abundant quantity not yet appropriated, and, therefore, at the disposal of whosoever might choose to cultivate it...

If all land had the same properties, if it were unlimited in quantity, and uniform in quality, no charge could be made for its use, unless where it possessed peculiar advantages of situation. It is only, then, because land is not unlimited in quantity and uniform in quality, and because in the progress of population, land of an inferior quality, or less advantageously situated, is called into cultivation, that rent is ever paid for the use of it. When in the progress of society, land of the second degree of fertility is taken into cultivation, rent immediately commences on that of the first quality, and the amount of that rent will depend on the difference in the quality of these two portions of land...

Thus suppose land - No. 1, 2, 3, - to yield, with an equal employment of capital and labor, a net produce of 100, 90, and 80 quarters of corn. In a new country, where there is an abundance of fertile land compared with the population, and where therefore it is only necessary to cultivate No. 1, the whole net produce will belong to the cultivator, and will be the profits of the stock which he advances. As soon as population had so far increased as to make it necessary to cultivate No. 2, from which ninety quarters only can be obtained after supporting the labourers, rent would commence on No. 1; for either there must be two rates of profit on agricultural capital, or ten quarters, or the value of ten quarters must be withdrawn from the produce of No. 1, for some other purpose. Whether the proprietor of the land, or any other person, cultivated No. 1, those ten quarters would equally constitute rent; for the cultivator of No. 2 would get the same result with his capital, whether he cultivated No. 1, paying 10 quarters for rent, or continued to cultivate No. 2, paying no rent. In the same manner it might be shown that when No. 3 is brought into cultivation, the rent of No. 2 must be ten quarters, or the value of ten quarters, whilst the rent of No. 1 would rise to twenty quarters; for the cultivator of No. 3 would have have the same profits whether he paid twenty quarters for the rent of No. 1, ten quarters for the rent of No. 2, or cultivated No. 3 free of all rent." -- David Ricardo (1821)

3.0 Prices
Consider stationary prices where seed corn is paid for at the beginning of the year, and wages and rent are paid at the end of the year. Then prices must satisfy the following pair of equations:

( 1/2 )( 1 + r ) + ( 1/4 ) w + ρ(I) = 1

( 1/3 )( 1 + r ) + ( 2/3 ) w + ρ(II) = 1

where

• Corn is the numeraire
• w is the wage
• r is the (common) rate of profits
• ρ(I) and ρ(II) are the rents on the respective grades of land.

The condition that at least one grade of land be free is expressed by a third equation:

ρ(I) ρ(II) = 0

The last condition is that all rents must be nonnegative.

3.1 Case 1
Consider the case when wages are 1/2 bushel per person-year. The rate of profits is then 75%. The rent on grade II land is 1/12 bushel per acre, and grade I land is free.

3.2 Case 2
Consider the case when wages are 5/6 bushel per person-year. The rate of profits is 33.3%. The rent on grade I land is 1/8 bushel per acre, and grade II land is free.

4.0 Conclusions
A figure is useful for visualizing the analysis of rent. Figure 1 shows the wage-rate of profits curves for this example. One commodity is produced in the above example, and that commodity functions both as a capital good and as a consumption good. This leads to the wage-rate of profits curves being straight lines. In models with more goods, these curves, while still decreasing, can be of varying convexity. In models with more than two grades of land, more than two curves exist. The figure can be used to locate the grade of land at the extensive margin in working inward from the outer frontier. With a given real wage, one works from right to left. Can the grade of land associated with the rightmost curve satisfy the requirements for use? If so, all land is free, and that grade is used. Otherwise, look at the grade associated with the next curve inward. If both grades can satisfy the requirements for use, then the first grade pays rent, and the second grade is free. In the general case, the frontier associated with the extensive margin may be neither the outer nor the inner frontier formed by the curves on the graph. Furthermore, the order in which grades of land are introduced to satisfy higher requirements for use need not be the order from high-rent land to lower-rent land. If I fully understood my references, I probably could note other counterintuitive results.

Figure 1: Wage-Rate of Profits Frontier for Rent Analysis

Table 2: Summary

Wage Per Person-Year	Location of Extensive Margin
Between 0 and 2/3 Bushels	Land I Non-Scarce, Land II Scarce
Between 2/3 and 1 Bushels	Land I Scarce, Land II Non-Scarce

Table II summarizes the results from the example. The scarcity of land of the best grade is reflected in the use of both grades of land. But which land is best, and therefore scarce, varies with the distribution of income and the price system. Scarcity, then, generally cannot be simply read off of the technological data. It is determined with the price system.

Ricardo was wrong; the order of rentability may be altered by differences in distribution between wages and profits.

References

• Christian Bidard, Prices, Reproduction, Scarcity, Cambridge University Press (2004).
• Heinz D. Kurz and Neri Salvadori, "A Single Theory or Two? Walras's Critique of Ricardo".
• Alberto Quadri-Curzio, "Rent, Income Distribution, and Orders of Efficiency and Rentability", in Essays on the Theory of Joint Production (ed. by Luigi L. Pasinetti), Columbia University Press (1980).
• David Ricardo, On the Principles of Political Economy and Taxation, 3rd edition (1821).
• Bertram Schefold, Mr. Sraffa on Joint Production and Other Essays, Unwin Hyman (1989).