Wednesday, January 01, 2020
Welcome
The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.
In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.
I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.
Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.
Friday, November 16, 2018
Pattern Analysis for a Fixed Capital Example
Figure 1: A Pattern Diagram |
In this example, I perturb parameters in an example of Bertram Schefold's. I was disappointed in that, as far as I can see, one can analyze the choice of technique in this example by the construction of the wage-rate of profits frontier. As far as I understand, this is not true for joint production in general. I guess I also need to find an example in which the physical life of a machine is at least three years so as to find a three-technique pattern.
This example does highlight differences in different measures of capital-intensity.
2.0 TechnologyTable 1 presents the technology for this example. Machines and corn are produced in this economy. Corn is the only consumption good. New machines are produced from inputs of labor and corn. Corn is produced from inputs of labor, corn, and machines. A machine can be worked for two years. After the end of the first year of its working life, it is known as an old machine. I assume each process requires a year to complete and exhibits constant returns to scale.
Inputs | Industry | ||
Machine | Corn | ||
Labor | a_{0,1} = (1/10) u(t) | a_{0,2} = (43/40) u(t) | a_{0,3} = u(t) |
Corn | a_{1,1} = (1/16) u(t) | a_{1,2} = (1/16) u(t) | a_{1,3} = (1/4) u(t) |
New Machines | 0 | 1 | 0 |
Old Machines | 0 | 0 | 1 |
Outputs | |||
Corn | 0 | 1 | 1 |
New Machines | 1 | 0 | 0 |
Old Machines | 0 | 1 | 0 |
I model technical progress by constantly decreasing inputs into each process, other than machines:
u(t) = e^{1 - σ t}
When σ t is unity, this is Bertram Schefold's example of restitching, at rates of profits of 1/3 and 1/2.
3.0 Prices of ProductionThe first row in Table 1 can be summarized by a row vector, a_{0}, of labor coefficients. The next three rows are expressed by a square matrix A. The last three rows form the matrix B. Suppose wages are paid out of the surplus product at the end of the year. If the same rate of profits is to be made in all operating processes, prices must satisfy the following system of equations;
p A (1 + r) + w a_{0} = p B
I let corn be the numerator:
p e_{1} = 1
where e_{1} is the first column of the identity matrix.
Given the wage, w, in a range between zero and some maximum, the above system of price equations can be solved for the rate of profits, r, the price of a new machine, p_{2}, and the price of an old machine, p_{3}.
4.0 Choice of TechniqueThe managers of firms need not run the machine for two years. They could discard the machine after only one year. (I assume free disposal.) The managers will be cost-minimizing if they run the machine for only one year if the price of an old machine is negative.
Alternatively, consider the price system when the machine is operated only two years. The matrices A and B are 2x2 square matrices, and a_{0} is a row vector with two elements. With these prices and the price of an old machine of zero, one could calculate the cost of operating the machine for a second year to produce a bushel of corn. When this cost is less than unity (the price of a bushel of corn), it is cost-minimizing to operate the machine for both years.
These two methods of analyzing the choice of technique yield the same answer for this example. Figure 1, above, illustrates the results. Until time reaches the pattern over the axis for the rate of profits, it is cost-minimizing to operate the machine for only one year. In Region 2, the machine is operated for two years when wages are low, and for one year when wages are higher. Region 3 is an example of reswitching. Eventually, it is cost-minimizing to operate the machine for two years, for all feasible wages.
5.0 CapitalIn outdated neoclassical intuition, a higher wage indicates that labor is more scarce, in some sense, and capital is relatively more abundant. One might, wrongly, except the price system to encourage capitalists to adopt less labor-intensive or more capital-intensive techniques, in some sense. And, in a simple example like this one, one might expect the more capital-intensive technique to be one in which the machine is run for both years.
The example confounds these expectations in both Region 2 and Region 3. Around the switch point in Region 2, a higher wage is associated with the adoption of a technique in which the machine is only operated for the first year. The same is true of the same switch point - the one at the lower wage - in Region 3. From this viewpoint, the switch point is "perverse" in both regions.
This result contrasts with the usual analysis based on real Wicksell effects. The real Wicksell effect is negative for the switch point in Region 2. It is positive for the same switch point in Region 3. For a switch point with a negative real Wicksell effect, a higher wage is associated with the adoption of a technique with more net output per person-year employed. And that is so in this case too. The switch point is only 'perverse', from this perspective, in Region 3.
6.0 ConclusionThis post has illustrated that what I am calling pattern analysis can be applied to examples of joint production in which joint production is only manifested in production and use of long-lived machines. It has focused attention on the distinction between different intuitions about the capital-intensity of a technique.
Saturday, November 10, 2018
A Linear Program for Markup Pricing
Figure 1: A Partition of Price-Wage Space for a Two-Commodity Reswitching Example |
This post generalizes my approach in Vienneau (2005). In that article, I present a Linear Programming (LP) problem for the firm. In the case of an economy that produces two commodities, one can present a graphical display that clarifies how Sraffa's equations arise. The dual LP is important in this development. Here, I show how that approach can work for a case in which rates of profits systematically vary among industries.
I was pleased that this approach works out for markup pricing. In a sense, this post derives both a direct and an indirect approach for analyzing the choice of technique, in the context of a model of markup pricing.
2.0 The ModelTo begin with, consider a model of the production of N commodities from labor and these commodities. This is a model with circulating capital and no joint production. Assume that managers of firms know of U_{j} processes for producing the output of that industry.
Each process is defined by:
- a_{0, j}(u), u = 1, 2, ..., U_{j}, the person-years of labor needed to produce one unit of the jth commodity.
- a_{., j}(u) = [a_{1, j}(u) ..., a_{N, j}(u)]^{T}, the inputs of each commodity needed to produce one unit of the jth commodity.
Each process exhibits constant returns to scale (CRS), requires a year to complete, and use up all their inputs. I also take a set of weights for industries, 1/s_{1}, ..., 1/s_{N}, as givens. Let prices be p = [p_{1}, ..., p_{N}]. Also, let e = [e_{1}, ..., e_{N}]^{T} be the numeraire, so that:
p e = 1
I should have some assumptions on coefficients ensuring that the economy can be productive by a suitable choice of technique.
I introduce some variables as abbreviations:
k_{j}(u) = p a_{., j}(u)
c_{j}(u) = p a_{., j}(u) + w a_{0, j}(u)
π_{j}(u) = p_{j} - c_{j}(u)
r_{j}(u) = π_{j}(u)/k_{j}(u)2.1 The Firm's LP
The managers of a firm take the wage, w and prices p as given. Let ω = [ω_{1}, ..., ω_{N}]^{T} be the firm's inventory of each commodity at the start of the year. Let q_{j}(u) be the quantity of the jth commodity that the firm produces with the uth process known for producing that commodity. Let q_{N + 1} be the value of inventory not used for purchasing inputs into production.
Each year the managers of the firm choose how much to produce of each commodity and with which process so as to maximize the weighted increment of value:
(1/s_{1})[π_{1}(1) q_{1}(1) + π_{1}(2) q_{1}(2) + ... + π_{1}(U_{1}) q_{1}(U_{1})]+ (1/s_{2})[π_{2}(1) q_{2}(1) + ... + π_{2}(U_{2}) q_{2}(U_{2})]...+ (1/s_{N})[π_{N}(1) q_{N}(1) + ... + π_{N}(U_{N}) q_{N}(U_{N})]
Such that the firm can purchase all of the inputs into production needed at the beginning of the year:
k_{1}(1) q_{1}(1) + k_{1}(2) q_{1}(2) + ... + k_{1}(U_{1}) q_{1}(U_{1})+ k_{2}(1) q_{2}(1) + k_{2}(2) q_{2}(2) + ... + k_{2}(U_{2}) q_{2}(U_{2})...+ k_{N}(1) q_{N}(1) + k_{N}(2) q_{N}(2) + ... + k_{N}(U_{N}) q_{N}(U_{N}) ≤ p ω
For all j, u:
q_{j}(u) ≥ 0
The weights formalize the concept that managers find some industries more desirable or easier to invest in than others. It works out that an industry that managers are less willing to contest or expand production in has a larger rate of profits, in the system of prices of production.
2.2 The Dual LP
The above LP has a dual problem. It is to choose r to minimize:
p ω r
Such that for all j, u:
p a_{., j}(u) (1 + rs_{j}) + w a_{0, j}(u) ≥ p_{j}
r ≥ 0
When a decision variable is positive in a solution to the primal LP, the corresponding constraint is met with equality in the dual LP. Suppose the solution of the primal LP leads to each commodity being produced by a specific process in each industry. The price system defined by the technique composed of those process will be satisfied. The economy will be on the wage curve for that technique.
3.0 Solution of the Primal LP
The solution to the primal LP is illustrated by Table 1. In a solution, only basis variables are positive The table specifies the value of each basis variable, when only it is positive in the solution, and conditions that must hold for it to be in the basis. The decision variable q_{N + 1} is a slack variable, introduced to convert the inequality constraint in the primal LP into an equality. It represents the value of inventory carried over, without supporting production. The conditions for when a decision variable is in the basis are intuitive. Consider the first row. A given commodity is produced with a given process only if the rate of profits made in other processes producing that commodity do not exceed the rate of profits made in the given process. Furthermore, the marked-up rate of profits in producing other commodities must not exceed the marked-up rate of profits in the given process. Finally, the (undiscounted) cost of producing a the given commodity must not exceed the revenue made from selling iron. (I am aware that there is some redundancy in how I have stated conditions in the table.)
Variable in Basis | Value | When Optimal |
q_{J}(V) | p ω/k_{J}(V) | For u = 1, 2, ...,U_{J} [p_{J} - w a_{0, J}(V)]/k_{J}(V) ≥ [p_{J} - w a_{0, J}(u)]/k_{J}(u) |
For all j, u (1/s_{J})r_{J}(V) ≥ (1/s_{j})r_{j}(u) | ||
c_{J}(V) ≤ p | ||
q_{N + 1} | p ω | For all j, u c_{j}(u) ≥ p |
The solution to the primal LP, in a two-commodity example, is easily visualized. The second commodity is the numeraire, and the price of the first commodity is graphed on the ordinate. Figure 1 partitions the space formed from the price of iron and the wage. A single decision variable enters the basis inside each region in Figure 1. Each region is labeled by that decision variable, in an obvious notation. On the boundaries, a solution to the LP can be formed from a linear combination of decision variables. In the example, both commodities must be produced for the economy to be self-sustaining. Firms are willing to produce both only if prices lie along the heavy locus. The figure shows that this is a reswitching example. One technique is adopted at low and high wages, while the other technique is used at intermediate wages. The figure also illustrates that the wage cannot exceed a maximum.
4.0 ConclusionI have thought about how this LP approach generalizes. In a general joint production framework, it is not immediately obviously how to assign processes to industries. So I do not see how to define the weights. I suppose one could have a weight for each process, instead of for each industry.
Land presents another difficulty. One would like to impose additional constraints in the primal LP to specify that overall production cannot require that more than a given quantity of some inputs cannot be used in production. Then multiple processes would be used, in a model of extensive rent, in certain industries. But should not such constraints be imposed above the level of the firm? That is, if a firm's production meets the constraints, they might still be violated in the economy as a whole.
But, I suppose, this LP approach applies to cases of fixed capital, where joint production is such that firms in an industry can choose to operate multiple processes, each jointly with a machine of a specific age.
Reference- Robert L. Vienneau (2005). On Labour Demand and Equilibria of the Firm. The Manchester School 73 (5): 612-619.
Friday, November 02, 2018
Extending An Example With Markup Pricing
Figure 1: A Two-Dimensional Pattern Diagram |
The example in this working paper is of an economy in which two commodities are produced. Technical progress is modeled as decreasing the coefficients of production in one of the processes for producing corn. They decrease at a rate of σ of ten percent.
Figure 2 shows how the pattern of switch points vary with technical progress. Initially, the Beta technique is cost-minimizing. Then it becomes a reswitching example. Around the switch point at the lower rate of profits, a higher wage is associated with more labor being hired, per unit of net output. Also, a higher wage is associated with the adoption of more direct labor being hired in corn production, per bushel corn produced gross. This is called a reverse substitution of labor. The other switch point disappears over the wage axis with more technical progress. The remaining switch point still exhibits a reverse substitution of labor. Eventually, that switch point no longer exhibits such a reverse substitution. Finally, it disappears entirely.
Figure 2: A Pattern Diagram |
I have been exploring how this example behaves with full cost pricing. I let the rate of profits in the iron industry be s_{1} r, and the rate of profits in the corn industry be s_{2} r. Figure 1 illustrates how these modeling choices for technical progress and markup pricing interact when s_{2} = 1.
Figure 2 illustrates the characteristics of switch points along a horizontal line, at s_{1}, in Figure 1. The numbered areas in the two figures correspond. Only one switch point exists in the region numbered 6, and it has a positive real Wicksell effect.
The example illustrates that an increase in the markup in a specific industry can result in the creation of a switch point in which higher wages are associated with firms wanting to employ more workers, both per unit net output in the economy as a whole and per unit gross output in a specific industry. Think of a vertical line going through Regions 6, 2, and 1, and, specifically, the partition between Regions 6 and 2. On the other hand, the transition from Region 5 to Region 3 is associated with creation of a switch point that only exhibits a reverse substitution of labor; it still has a negative real Wicksell effect.
Thanks to the comments of Sturai for encouraging me to write this post and for pointing out a paper by Antonio D'Agata that I'll have to read.
Wednesday, October 24, 2018
Structural Economic Dynamics, Real Wicksell Effects, and the Reverse Substitution of Labor
I have uploaded another working paper:
This article presents an example in which technical progress results in variations in the labor market. Around a switch point with a positive real Wicksell effect, a higher wage is associated with firms wanting to employer more labor per unit output of net product. Around a switch point with a reverse substitution of labor, firms in a particular industry want to hire more labor per unit output of gross product. Technical progress can bring about and take away circumstances favorable for workers wanting to press claims for higher wages.
My research approach can generate fluke switch points. I have decided that such flukes are more interesting when placed in a story about the perturbation of parameters.
Saturday, October 20, 2018
A Visualization of the Choice of Technique
Figure 1: Regions for Basis Variables |
I introduced a new way of visualizing the choice of technique for two-commodity models back in 2005. As far as I know, nobody has taken up this idea. I modify my method slightly by having labor advanced; wages are paid out of the surplus at the end of the year. I cite John Roemer in my paper linked previously.
2.0 TechnologyTable 1 specifies the technology I use for illustration. Each row lists the inputs needed to produce one unit (ton or bushel) for the indicated industry. As usual, this is a model of circulating capital.
Input | Industry | ||
Iron | Corn | ||
Alpha | Beta | ||
Labor | a_{0, 1} = 1 | a^{α}_{0, 2} ≈ 0.9364 | a^{β}_{0, 2} ≈ 0.6174 |
Iron | a_{1, 1} = 9/20 | a^{α}_{1, 2} ≈ 0.02602 | a^{β}_{1, 2} ≈ 0.001518 |
Corn | a_{2, 1} = 2 | a^{α}_{2, 2} ≈ 0.1041 | a^{β}_{2, 2} ≈ 0.4636 |
For this economy to be reproducible, both iron and corn must be produced. The iron-producing process can be combined with either of the corn-producing processes. Thus, there are two possible techniques, the Alpha and Beta techniques, each of which include the corn-producing process with the corresponding label. (The approach in this post can be extended to include any number of available processes in either industry.)
3.0 A Linear Program for the FirmConsider a firm that starts the year with an inventory of ω_{1} tons iron and ω_{2} bushels corn. I take corn as the numeraire. The firm faces a price for iron of p bushels per ton and a wage of w bushels per person years. The managers of the firm must set the value of the following decision variables:
- q_{1}: The tons iron produced with the iron-producing process.
- q^{α}_{2}: The bushels corn produced with the Alpha corn-producing process.
- q^{β}_{2}: The bushels corn produced with the Beta corn-producing process.
- q_{3}: The value of inventory that the firm carries over unused to the next year.
The firm is constrained by the value of its inventory. Its level of production cannot require it to advance more than the value of its inventory.
The managers of the firm attempt to maximize the increment of value. Their problem can be formulated as a Linear Program (LP). They choose q_{1}, q^{α}_{2}, and q^{β}_{2} to maximize:
z = (p - pa_{1, 1} - a_{2, 1} - a_{0, 1}w)q_{1}+ (1 - pa^{α}_{1, 2} - a^{α}_{2, 2} - a^{α}_{0, 2}w)q^{α}_{2}+ (1 - pa^{β}_{1, 2} - a^{β}_{2, 2} - a^{β}_{0, 2}w)q^{β}_{2}
Such that:
(pa_{1, 1} + a_{2, 1})q_{1}+ (pa^{α}_{1, 2} + a^{α}_{2, 2})q^{α}_{2}+ (pa^{β}_{1, 2} + a^{β}_{2, 2})q^{β}_{2}≤ p ω_{1} + ω_{2}
q_{1} ≥ 0, q^{α}_{2} ≥ 0, q^{β}_{2} ≥ 0
In solving this LP by the simplex method, it is convenient to introduce the slack variable, q_{3}, to convert the constraint to an equality.
4.0 The Dual LPThe above LP has a dual. It is to choose a non-negative rate of profits so as to minimize the capital charge on the inventory. Constraints are such that the cost of each production process, including a charge for capital, does not fall below the revenue from operating that process. Formally, choose r to minimize:
(p ω_{1} + ω_{2}) r
Such that:
(pa_{1, 1} + a_{2, 1})(1 + r) + a_{0, 1}w ≥ p
(pa^{α}_{1, 2} + a^{α}_{2, 2})(1 + r) + a^{α}_{0, 2}w ≥ 1
(pa^{β}_{1, 2} + a^{β}_{2, 2})(1 + r) + a^{β}_{0, 2}w ≥ 1
r ≥ 0
If the primal LP has a solution, so will the dual LP. And the value of the objective functions will be the same, for a solution, for both the primal and dual LP. When a decision variable is positive in a solution to the primal LP, the corresponding constraint is met with equality in the dual LP. Thus, if the solution of the primal LP leads to corn being produced and iron being produced with the Alpha iron-producing process, the economy will be on the wage curve for the Alpha technique. Similar remarks apply to the Beta technique.
Variable in Basis | Value | When Optimal |
q_{1} | (p ω_{1} + ω_{2})/(pa_{1, 1} + a_{2, 1}) | r_{1} ≥ r^{α}_{2} |
r_{1} ≥ r^{β} | ||
c_{1} ≤ p | ||
q^{α}_{2} | (p ω_{1} + ω_{2})/(pa^{α}_{1, 2} + a^{α}_{2, 2}) | r_{1} ≤ r^{α}_{2} |
r^{α}_{2} ≥ r^{β}_{2} | ||
c^{α}_{2} ≤ 1 | ||
q^{β}_{2} | (p ω_{1} + ω_{2})/(pa^{β}_{1, 2} + a^{β}_{2, 2}) | r_{1} ≤ r^{β}_{2} |
r^{α}_{2} ≤ r^{β}_{2} | ||
c^{β}_{2} ≤ 1 | ||
q_{3} | p ω_{1} + ω_{2} | c_{1} ≥ p |
c^{α}_{2} ≥ 1 | ||
c^{β}_{2} ≥ 1 |
5.0 The Solution of the Primal LP
The solution to the primal LP is illustrated by Table 2. In a solution, only basis variables are positive. The table specifies the value of each basis variable, when only it is positive in the solution, and conditions that must hold for it to be in the basis. These conditions are specified in terms of certain variables introduced as abbreviations. The rates of profits in each process are:
r_{1} = (p - a_{0, 1}w)/(pa_{1, 1} + a_{2, 1})
r^{α}_{2} = (1 - a^{α}_{0, 2}w)/(pa^{α}_{1, 2} + a^{α}_{2, 2})
r^{β}_{2} = (1 - a^{β}_{0, 2}w)/(pa^{β}_{1, 2} + a^{β}_{2, 2})
The (undiscounted) costs of each process are:
c_{1} = pa_{1, 1} + a_{2, 1} + a_{0, 1}w
c^{α}_{2} = pa^{α}_{1, 2} + a^{α}_{2, 2} + a^{α}_{0, 2}w
c^{β}_{2} = pa^{β}_{1, 2} + a^{β}_{2, 2} + a^{β}_{0, 2}w
The conditions for when a decision variable is in the basis are intuitive. Consider the first row. Corn is produced only if the rate of profits made in either of the iron-producing processes does not exceed the rate of profits made in the corn producing process. Furthermore, the (undiscounted) cost of producing a bushel corn must not exceed the revenue made from selling corn.
6.0 Visualization
The solution to the primal LP, in a two-commodity example, is easily visualized. Figure 1 partitions the space formed from the price of iron and the wage. A single decision variable enters the basis inside each region in the figure, and that region is labeled by that decision variable. On the boundaries, a solution to the LP can be formed from a linear combination of decision variables. Iron and corn must be both produced for the economy to be self-sustaining. Firms are willing to produce both only if prices lie along the heavy locus. The figure shows that this is a reswitching example. The Beta technique is adopted at low and high wages, while the Alpha technique is used at intermediate wages. The figure also illustrates that the wage cannot exceed a maximum.
7.0 ConclusionIf you think about it, the above is a derivation of the usual method of analyzing the choice of technique by constructing the outer frontier of the wage curves for all available techniques. It is not restricted to a two-commodity example, although the diagram is so restricted. The proof follows from duality theory in linear programming. The graph illustrates that equilibrium prices must vary with the wage.
I remain puzzled about why mainstream economists continue to teach that, under the ideal assumptions of free competition, wages and employment are determined by the interaction of supply and demand in labor markets.
Wednesday, October 17, 2018
William Nordhaus, 2018 "Nobel" Laureate, On Labor Values
Suppose one wants to quantitatively measure the growth in productivity over centuries. And one wants to look at specific commodities that can be said to have existed over such a long time. Think of a lumen of light or a food calorie. How can one do this? The definition of a price index over such a long time period is questionable.
Adam Smith addressed this problem. Some would find his approach common sense. One could ask how long must a common laborer work to be able to afford the commodity in question. More recently, William Nordhaus considered the question. He, too, advocated the use of a Smithian labor-commanded standard to measure technological change. (I haven't read the reference below in decades.)
- William D. Nordhaus (1997). Traditional Productivity Estimates are Asleep at the (Technological) Switch. Economic Journal 107 (444): pp. 1548-1559.