
Figure 1: WageRate of Profits Curves and their Frontier 
1.0 Introduction
This post is a generalization of a neoclassical onegood model. It advances a comparison of Sraffian analysis of the choice of the costminimizing choice of the technique and neoclassical analyses, correctly understood, of marginal productivity. Accordingly, all production functions are smooth in this example. If substitutability is seen as a technological property of production functions, then the single capital good and labor can be substituted in each of the two industries in this model.
2.0 The Technology
Consider a simple economy in which steel and corn are produced from inputs of steel and labor. The steel used as an input in production is totally used up in yearly cycles, and the outputs become available at the end of the year. In other words, this is a model without fixed capital, and all production processes require a year to complete.
2.1 Production Functions
The production function for steel is:
Q_{1} = F_{1}(X_{1}, L_{1}) = A_{1} X_{1}^{α1} L_{1}^{(1  α1)}
where:
 Q_{1} is (gross) output of steel (in tons).
 X_{1} is steel (tons) used as a capital good in the steel industry.
 L_{1} is labor (personyears) used as an input in the steel industry.
and A_{1} and α_{1} are positive constants such that:
0 < α_{1} < 1
The production function for corn is:
Q_{2} = F_{2}(X_{2}, L_{2}) = A_{2} X_{2}^{α2} L_{2}^{(1  α2)}
where:
 Q_{2} is (gross) output of corn (in bushels).
 X_{2} is steel (tons) used as a capital good in the corn industry.
 L_{2} is labor (personyears) used as an input in the corn industry.
and A_{2} and α_{2} are positive constants such that:
0 < α_{2} < 1
2.2 A Set of Coefficients of Production
An alternative specification of this ConstantReturnstoScale (CRS) technology is as a set of coefficients of production a_{01}(s_{1}), a_{02}(s_{2}), a_{11}(s_{1}), a_{12}(s_{2}) from the set:
{ (a_{01}(s_{1}), a_{02}(s_{2}), a_{11}(s_{1}), a_{12}(s_{2}))  0 < s_{1}, 0 < s_{2}}
where:
a_{01}(s_{1}) = [1/(A_{1}s_{1})]^{[1/(1  α1)]}
a_{02}(s_{2}) = [1/(A_{2}s_{2})]^{[1/(1  α2)]
}
a_{11}(s_{1}) = s_{1}^{(1/α1)}
a_{12}(s_{2}) = s_{2}^{(1/α2)
}
and
 a_{01}(s_{1}) is the labor required, in the steel industry, per ton steel produced.
 a_{02}(s_{2}) is the labor required, in the corn industry, per bushel corn produced produced.
 a_{11}(s_{1}) is the steel input required, in the steel industry. per ton steel produced (gross).
 a_{12}(s_{2}) is the steel input required, in the corn industry, per bushel corn produced.
2.0 Quantity and Price Equations, Given the Technique
Consider a stationary state in which the firms employ one personyear of labor each year, and prices are stationary. For notational convenience below, define the following function:
f(R) = (a_{01}a_{12}  a_{02}a_{11})R + a_{02}
2.1 Quantity Relations
The amount of steel produced each year, measured in tons, is:
q_{1} = a_{12}/f(1)
The amount of corn produced each year, measured in bushels, is:
q_{2} = (1  a_{11})/f(1)
These quantities must satisfy two equalities. First, the amount of labor employed is unity:
1 = a_{01}q_{1} + a_{02}q_{2}
Second, consider the following equation:
q_{1} = a_{11}q_{1} + a_{12}q_{2}
The lefthand side of the above equation denotes the quantity of steel produced each year and available, as output from the steel industry, at the end of each year. The righthand side denotes the sum of steel used as inputs in the steel and corn industries, respectively. These inputs must be available at the start of each year. Hence, the above equation is a necessary condition when the economy is in a selfsustaining, stationary state.
2.2 Price Relations
I take the consumption good, corn, as the numeraire. The price of steel, in units of bushels per ton, is
p = a_{01}/f(1 + r),
where r is the rate of profits. The wage is:
w = [1  a_{11}(1 + r)]/f(1 + r)
The above equation is known as the wagerate of profits curve.
The price of steel, the wage, and the rate of profits must satisfy two equations. The condition that the price of steel just cover the cost of producing steel is:
pa_{11}(1 + r) + a_{01}w = p
The lefthand side of the above equation shows the cost of producing a ton of steel. Costs are inclusive of normal profits, so to speak, on the cost advanced to purchase physical inputs at the start of the year. In this case, those inputs consist of steel, the single capital good in this model. Although labor is hired at the start of the year to work throughout the year, the price equations in this model show labor being paid out of the harvest gathered at the end of the year.
The condition that the price of corn just cover the cost of producing corn yields a similar equation:
pa_{12}(1 + r) + a_{02}w = 1
2.3 The CapitalLabor Ratio
"Capital" is an ambiguous term. It denotes both physicallyexisting means of production. And it denotes the value of those means of production, when embedded in certain social relations. For example, in this model, the distribution of the capital goods over the two industries is assumed to be appropriate to the continued selfreproduction of the economy. In a sense, the plans of entrepreneurs and firms managers are coordinated.
At any rate, the relationships described so far allow one to express the value of capital, in numeraire units, per personyears, given the technique:
k = p q_{1}
k = a_{01}a_{12}/[f(1)f(1 + r)]
The capitallabor ratio (in units of bushels per personyears) does not appear in any legitimate marginal product. Nevertheless, I find it a useful quantity for further analysis in multicommodity models.
3.0 The Chosen Technique
The costminimizing technique differs with the rate of profits. For analytical convenience, I take the rate of profits as exogenous in this model. One could, instead, if one so chose, take the wage as given and find the rate of profits endogenously. At any rate, this model is open, and the distribution of income is not determined in the model. The equations below set out each of the four coefficients of production in this model as functions of the rate of profits:
a_{01} = (1/A_{1})^{[1/(1  α1)]} [(1 + r)/α_{1}]^{[α1/(1  α1)]}
a_{02} = (1/A_{2})
x {(1  α_{2})/[(α_{1})^{[α1/(1  α1)]}(1  α_{1})α_{2}]}^{α2}
x [(1 + r)/A_{1}]^{[α2/(1  α1)]}
a_{11} = α_{1}/(1 + r)
a_{12} = (1/A_{2})
x [(α_{1})^{[α1/(1  α1)]}(1  α_{1})α_{2}/(1  α_{2})]^{(1  α2)}
x [A_{1}/(1 + r)]^{(1  α2)/(1  α1)}
3.1 Steel as a Basic Commodity and the OneGood Case
I have previously set out an analysis of the choice of technique for a onegood model with an aggregate CobbDouglas production function. In the twogood model set out in this post, the coefficients of production for steel, a_{01} and a_{11}, when the costminimizing technique is chosen, are the same as the coefficients of production in that onegood model. This is not surprising.
In the model in this post, steel enters, as an input, into the production of both steel and corn, for all possible techniques. On the other hand, corn never enters as an input into the production of any commodity. In the technical terminology of postSraffian economics, steel is always a basic commodity, and corn is never a basic commodity. Thus, the production of steel can be analyzed, in some sense, prior to the analysis of the production of corn.
3.2 A OneGood Special Case
Consider the special case in which:
α_{1} = α_{2} = α
A_{1} = A_{2} = A
In effect, steel and corn are the same commodity. The coefficients of production, for the costminimizing technique are:
a_{02} = a_{01} = (1/A)^{[1/(1  α)]} [(1 + r)/α]^{[α/(1  α)]}
a_{12} = a_{11} = α/(1 + r)
So this case reduces to the onegood model, as it should. This concludes my analysis of this special case.
4.0 The Chosen Technique on Unit Isoquants and Marginal Productivity Conditions
The coefficients of production are such that the steel industry lies on its unit isoquant:
1 = F_{1}(a_{11}, a_{01})
Likewise, the corn industry lies on its unit isoquant:
1 = F_{2}(a_{12}, a_{02})
Since the coefficients of production in Section 3 above are for the costminimizing technique, all valid marginal productivity relationships must hold. I have chosen to express each marginal productivity condition in numeraire units per unit input. And, the cost of an input and its marginal product are equated here at the end of the year.
Following these conventions, the following display equates the cost of steel to the value of the marginal product of steel in the steel industry:
p(1 + r) = p ∂F_{1}(a_{11}, a_{01})/∂a_{11}
Likewise, the following display equates the cost of steel to the value of the marginal product of steel in the corn industry:
p(1 + r) = ∂F_{2}(a_{12}, a_{02})/∂a_{12}
Since wages are paid out of the harvest, the rate of profits does not appear in my statement of marginal productivity conditions for labor. The following display equates the wage and the value of the marginal product of labor in the steel industry:
w = p ∂F_{1}(a_{11}, a_{01})/∂a_{01}
Likewise, the following display equates the wage and the value of the marginal product of labor in the corn industry:
w = ∂F_{2}(a_{12}, a_{02})/∂a_{02}
I have checked the above equations for the isoquants and the four marginal productivity equations. This is quite tedious.
Above, I have listed six equations, two expressing the condition that the coefficients of production lie upon unit isoquants and four marginal productivity equations. These six equations are sufficient to determine the six unknowns (w, p, a_{01}, a_{02}, a_{11}, and a_{12}) in terms of the model parameters and the externally specified rate of profits. In other words, this model illustrates that marginal productivity is a theory of the choice of technique, not of the (functional) distribution of income.
5.0 The WageRate of Profits Frontier
An alternate analysis of the choice of technique can be based on the wagerate of profits frontier. And this analysis yields the same answer as the above analysis based on marginal productivity.
Recall, from Section 2.2, that a technique can be specified as an ordered pair chosen from the specified index set. The index variables for the costminimizing technique, as a function of the rate of profits are:
s_{1} = [α_{1}/(1 + r)]^{α1}
s_{2} = (1/A_{2})^{α2}
x [(α_{1})^{[α1/(1  α1)]}(1  α_{1})α_{2}/(1  α_{2})]^{[(1  α2)α2]}
x [A_{1}/(1 + r)]^{[(1  α2)α2/(1  α1)]}
I think it of interest to note that both the optimal process for producing steel and the optimal process for producing corn, in a stationary state, vary continuously with the rate of profits. This is not a generic result for a discrete technology. In a discrete technology, the costminimizing techniques at a switch point typically differ in the process used in only one industry; a small variation in the rate of profits thus affects only the specification of a process in one industry.
5.1 First Order Conditions
Since the coefficients of production are functions of the index variables, the wagerate of profits curve for a technique can be viewed as a function of:
 The index variables s_{1} and s_{2},
 The rate of profits r, and
 The model parameters α_{1}, A_{1}, α_{2}, and A_{2}.
A necessary condition for a technique to be costminimizing, at a given rate of profits, is that the wage be a maximum. This maximum is taken from the wage on each wagerate of profits curve, over all techniques. In the current context, with a model with smooth production functions, the first derivative of the wagerate of profits frontier, with respect to each index variable, must be zero at the maximum:
∂w/∂s_{1} = 0
∂w/∂s_{2} = 0
Note that the above is a system of two equations in the two unknown index variables. I did not actually calculate the above derivatives for this model. Perhaps Figure 1 provides some confidence in this mathematics. I deliberate drew three wagerates of profits curves on the frontier and one off of it.
5.2 Second Order Conditions
The FOCs determine a critical point. The calculus is consistent with such a critical point being a local maximum, a local minimum, or a saddle point. The following are sufficient conditions, in this context, for a critical point to be a local maximum:
∂^{2}w/∂s_{1}^{2} < 0
∂^{2}w/∂s_{2}^{2} < 0
D(s_{1}, s_{1}) > 0
where D(s_{1}, s_{1}) is defined by:
D(s_{1}, s_{1}) = [∂^{2}w/∂s_{1}^{2}][∂^{2}w/∂s_{2}^{2}]  [∂^{2}w/∂s_{1}∂s_{2}]^{2}
Of the three SOCs, either the first or the second is redundant.
6.0 Conclusion
I still have some ideas for future work with this model. But I think this is enough for one blog post. I hope the above presentation suggests that marginal productivity is not a theory of distribution, in general. One cannot validly hold, for example, that real wages are determined by the marginal product of labor. Furthermore, the Sraffian analysis of the choice of technique is analytically equivalent to the determination of the choice of technique, given, for example, the rate of profits, by marginal productivity.