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"Accordingly, to a follower of Menger, the determination of economic equilibrium would not merely involve the determination of the prices of those goods which have prices (as in Walras); it should also involve the determination of which goods are to have prices and which are to be free. The weakness of the Walras-Cassel lies in the implied assumption that the whole amount of each available factor is utilized; once that assumption is dropped, the awkwardness of the construction....can be shown to disappear."
(John Hicks, "Linear Theory", Economic Journal, 1960)
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Contents
(1) Problems in the Walras-Cassel Model
(2) The Schlesinger-Wald Inequalities and Shadow Values
(3) A Simple Graphical Illustration
(4) Existence and Uniqueness of Equilibrium
(5) The Wald System with Capital
[For alternative presentations, consult Kuhn (1956), Dorfman, Samuelson and Solow (1958: Ch. 13), Karlin (1959), Lancaster (1968: Ch.9) and Weintraub (1983)].
(1) Problems in the Walras-Cassel Model
Recall that in the simplest Walras-Cassel system we had the following set of equations:
(i) Output market equilibrium: x = D(p, w)
(ii) Factor market equilibrium: v = B¢ x
(iii) Price-cost equalities: p = Bw
If we have n produced goods and m factors, then have n equations in (i), m equations in (ii) and n equations in (iii), but as we can always remove one equation by Walras's Law, then we have a total of 2n+m-1 equations. The unknowns in this system are x (n unknowns), p (n unknowns) and w (m unknowns), and, by the numeraire, we could omit one of these, thus we would have total of 2n+m-1 unknowns.
The first problem we obtain with Walras's method of solution is that setting the number of equations equal to the number of unknowns is neither necessary nor sufficient for the existence of a solution. It is not sufficient because xy = 3 and x + y = 1 are two equations with two unknowns which has no real solution. Furthermore, it is not necessary because, for instance, x^{2} + y^{2} = 0 has a unique solution, but it is one equation with two unkowns.
Heinrich von Stackelberg (1933) pointed out this difficulty and noted the following: namely, if the number of industries using a particular set of factors is less than the number of factors, it is conceivable that prices and quantities become indeterminate. His objection can be illustrated directly by supposing that we have merely one industry and two factors, so that p_{ }= b_{1}w_{1} + b_{2}w_{2} is the only price-cost equation. Clearly, this has no solution: there are numerous combinations of w_{1} and w_{2} that can yield the same output price p (in terms of our earlier diagram, we would only have one P-locus in our two-dimensional factor price space). This problem would not arise if we had two outputs: as we illustrated with our simple two-sector Walras-Cassel system and outputs in this case are quite determinate.
Thus, Stackelberg indicated, for determinacy, we need at least the condition that n ³ m. More specifically, we need to ensure that if there are m factors, then there must be at least m industries which use all m factors. The reason for the strength of this last statement can be gathered if we have, for instance, four outputs (x_{1} to x_{4}) and four inputs (v_{1} to v_{4}), where industry x_{1} uses factors v_{1} and v_{2} exclusively while industries x_{2}, x_{3} and x_{4} use combinations of the remaining factors v_{3} and v_{4}. Thus, even though n = m as a whole, notice that because factors v_{1} and v_{2} are only used by one industry (x_{1}), the indeterminacy problem remains.
Another problem emerges in this example which was also noticed by Stackelberg: namely, if we have three industries (x_{2}, x_{3} and x_{4}) which use two factors (v_{3} and v_{4}), we cannot guarantee that the Walras-Cassel conditions will be met. Heuristically, imagine three price locii in a two-dimensional factor returns space: it is highly improbable that all three price locii will intersect at a single point. In other words, there is (probably) no combination of the two factor returns where all the price-cost equations are satisfied. Noticing this, Stackelberg suggested that we must allow for some of the equalities to remain unsatisfied.
As was first pointed out by Hans Neisser (1932), the imposition of the equalities (ii) v = B¢ x and (iii) p = Bw yields another troublesome issue. If factor supply must equal factor demand, it is entirely conceivable that this holds at negative prices for some factors. (heuristically, see Figure 1, where for a particular market, demand D(p) and supply S(p) intersect at e, yielding negative equilibrium price, p*). Similarly, if prices must equal cost of production for every commodity, then we are not excluding situations where this condition is met at negative levels of output for some commodities (cf. Figure 1, where D(p) and S¢ (p) intersect at e¢ , yielding negative equilibrium output, q*). Yet while negative prices and negative quantities are entirely possible they are not economically meaningful.
Figure 1 - Neisser's Critique
Friederik Zeuthen (1933) was the first to explicitly suggest that inequalities be introduced into the Walras-Cassel system so that "free goods" could be admitted. As Zeuthen notes, the Walras-Cassel system imposes the arbitrary assumption that all factors are scarce and thus that all have "prices". However, a factor is scarce only if there is more demand for it than is available - hence the price. But it is conceivable, Zeuthen noted, that a factor may be so plentiful or demand for it so minimal that it is not scarce (i.e. supply exceeds demand at any positive price, e.g. S(p) and D(p) in Figure 1) and thus that factor should have a zero price. In any case, one cannot decide a priori which factors are to be free and which are not. As Zeuthen reminded everyone the decision as to which commodities were "economic" and "non-economic", as Carl Menger (1871) had taught long ago, should be result of the equilibrium and thus endogneous to the problem. As he writes, "one does not know at the outset which goods are free goods, so one should insert into the equality the possibility of an unused residual and, at the same time, stipulate among the conditions that either this residual or the price of the resource equals zero." (Zeuthen, 1933).
(2) Schlesinger-Wald Inequalities and Shadow Values
The participants of the Vienna Colloquium, notably Karl Schlesinger (1935) and Abraham Wald (1935, 1936), followed up on the Neisser-Stackelberg-Zeuthen critique of the Walras-Cassel system by introducing inequalities into the factor market clearing equations. Thus, allowing for an inequality in the factor market equilibrium condition, we obtain:
v ³ B¢ x
so that, in equilibrium, for a particular factor j, either the quantity supplied of that factor is equal to the quantity demanded (i.e. v_{j} = B_{j¢ }x) or the quantity of that factor supplied exceeds the demand for the factor, v_{j} > B_{j¢ }x, but then the corresponding return to factor j must be zero, i.e. w_{j} = 0.
Although in the context of a different model, John von Neumann (1937) adopted the quantity inequalities suggested by Schlesinger and Wald, but also introduced one that they had not thought of: namely, that price-cost_{ }equations should also be transformed into inequalities in order to allow for "non-produced" goods. Translated into the Walras-Cassel context, this implies that we ought to have in equilibrium:
p £ Bw
so that, in equilibrium, for a particular good i either its price is equal to its cost of production (i.e. p_{i} = B_{i}w) or its price is less than cost of production, p_{i} < B_{i}w, but then the corresponding amount of good i produced is zero, i.e. x_{i} = 0.
As we see, the Schlesinger-Wald-von Neumann inequalities allow that, in equilibrium, some factors may not have value and some goods may not be produced at all. In this way we will be guaranteed that there will never be negative prices nor negative quantities in equilibrium. In place of these, we will have "free factors" and "non-produced goods". Now, Walras himself had been aware of this problem, but feared replacing equalities with inequalities because then his number of equations would fall below the number of unknowns in the system. As the mathematics of linear and non-linear programming were undeveloped during his time, Walras effectively felt that he had to impose equalities in order to obtain a determinate solution.
It was up to Abraham Wald (1935, 1936) to show that this was not necessary and that existence was possible when these inequalities are introduced. The basic insight is that we can decompose our equilibrium conditions into a linear programming problem with a primal and a dual. The primal is:
max p¢ x
s.t.
v ³ B¢ x
x ³ 0
where we are maximizing output revenue, p¢ x, subject to the constraint that factor demand (B¢ x) not exceed factor supply (v). Thus, from the primal problem, given p, v and B¢ , we obtain the solution x*. As any standard linear programming text shows (e.g. Takayama, 1974), the primal can be rewritten as a Lagrangian problem L = p¢ x + l[v - B¢ x] where l is an m ´ 1 vector of Lagrangian multipliers. The "solution" to this is then a pair (x*, l*) where
(i) p¢ x* ³ p¢ x for all other x which fulfill the constraints (i.e. v ³ B¢ x, x ³ 0)
(ii) l*[v - B¢ x*] = 0 where l* ³ 0.
Let us now turn to the dual problem. This is:
min w¢ v
s.t.
p £ Bw
w ³ 0
where we are minimizing returns paid to factors w¢ v, subject to the constraint that cost of production (Bw) not fall below output price, p. Thus, in the dual problem, given p, v and B, we obtain the solution w*. We can also express the dual problem as a Lagrangian L = w¢ v + m[Bw - p] where m is a n´ 1 vector of Lagrangian multipliers. The solution to this problem is a pair (w*, m*) where:
(i) w*¢ v £ w¢ v for all other w which fulfill the constraints (i.e. p £ Bw, w ³ 0)
(ii) m*[Bw - p] = 0 where m* ³ 0.
The duality theorem of linear programming (e.g. Gale, 1960: p.78; Lancaster, 1968: p.29; Takayama, 1974: p.156) claims then the following three results:
(i) there exists a solution to the primal x* if and only if there exists a solution to the dual, w*.
(ii) the maximized values of the objectives of the primal and dual are the same. In our case, this means that p¢ x* = w*¢ v, which is another way of imposing pure competition (i.e. zero profits) at equilibrium.
(iii) The Lagrangian multipliers which satisfy the primal problem is the solution vector to the dual (i.e. l* = w*) and the multipliers which solve the second problem is the solution vector in the primal (i.e. m* = x*). Thus, this implies that in equilibrium, we have:
w*[v - B¢ x*] = 0
x*[ Bw - p] = 0
which are known as the complementary slackness conditions.
This last condition (iii) is extraordinarily important - as these complementary slackness conditions replace the market-clearing and price-cost equality conditions for equilibrium and allow situations of free goods and non-produced outputs outlined earlier. The first complementary slackness condition, w*[v - B¢ x*] = 0, means that for every factor j = 1...m, either markets clear (so v_{j} = B_{j¢ }x* = 0) or some factors are in excess supply (v_{j} > Bj¢ x*) - but if the latter is true, then the complementary slackness condition requires than that w_{j} = 0 (the factor in excess supply is free). Thus, w_{j} > 0 only if v_{j} = B_{j¢ }x*, i.e. a factor earns positive returns only if the market for that factor clears. If v_{j} > B_{j¢ }x*, then that factor must be free.
Conversely, the second complementary slackness condition, x*[Bw - p] = 0, means that for every produced good i = 1, ..., n, either price equals cost of production (so p_{i} = B_{i}w*) or price is less than cost of production (p_{i} < B_{i}w*), but if the later is true, then the complementary slackness condition requires that x_{i} = 0 (the corresponding level of output of that good is zero). Thus, x_{i} > 0 only if p_{i} = B_{i}w*, i.e. a good is produced at a positive amount only if its price equals its cost of production. If p_{i} < B_{i}w* for a particular good i, then that good is not produced.
The shadow values are even more interesting than they seem. In principle, the shadow values of any programming problem - i.e. the Lagrangian multipliers -- refer to the gain the optimized value of the objective from a marginal release in the relevant constraint. For the primal problem max p¢ x s.t. v ³ B¢ x, the Lagrangian multiplier l _{j}* denotes the gain in p¢ x* when the jth factor supply constraint is slightly released, thus:
l _{j}* = ¶ p¢ x*/¶ v_{j}
This expression can be interpreted as the marginal value product of factor j: i.e. the increase in the maximal revenue (prices times maximal outputs) from a marginal increase in the supply of factor j. Using the duality theorem, we know that l _{j}* = w_{j}*, thus the Wald program yields the solution that ¶ p¢ x*/¶ v_{j }= w_{j}, i.e. the jth factor's return (w_{j}) is equal to its marginal value product!
Thus, in effect, the duality theorems have resurrected the marginal productivity theory of distribution! The complementary slackness conditions reinforce this intuition: as w_{j}*[v_{j} - B_{j}¢_{ }x*] = 0, then the jth factor will not be paid anything if it is in excess supply in equilibrium. In other words, if a factor is not scarce, its marginal value product will be zero (i.e. if the factor constraint is not binding, then adding more of the factor will not increase marginal value product) and, consequently, it will not be paid anything.
This result has been historically somewhat surprising. For a long time, many economists supposed that marginal products were not defined unless the production function was differentiable with continuous substitution among techniques, etc., and that consequently fixed proportions technology would not yield the marginal productivity theory of distribution. Even Friedrich von Wieser (1889), who had initially announced that factor prices were determinate with fixed coefficient production technology, doubted that the marginal productivity theory would be met. An interesting historical aside relates to the Soviet economist and pioneer of linear programming, Leonid Kantorovich. Kantorovich obtained many of these results in the 1950s, yet feared that they might compromise his status in Soviet Russia. Consequently, when he published his results (e.g. Kantorovich, 1959), he buried them in a mountain of heavy Marxian jargon -- e.g. referring to the Lagrangian multipliers as "objectively determined evaluations" -- lest he be accused of lending credence to the Neoclassical marginal productivity theory.
The analogous exercise for the dual problem, min w¢ x s.t. p £ Bw, is no less interesting. Here, the Lagrangian multiplier m _{i}* refers to the reduction in the optimal w¢ *v when the ith output price constraint is released, i.e.
m _{i}* = ¶ w¢ *v/¶ p_{i}
which can be interepreted as the increase in minimum costs given a rise in the price of good i. Using the duality theorem, m _{i}* = x_{i}*, thus we obtain the solution that ¶ w¢ *v/¶ p_{i} = x_{i}*. This can be thought of, in effect, as a version of the famous Hotelling's Lemma.
(3) A Simple Graphical Illustration
The linear production conditions of the Wald system can be illustrated using the same two-sector model we employed earlier in the Walras-Cassel model. We have two outputs, x_{1} and x_{2}, and two factors, v_{1} and v_{2}. Output prices are p_{1}, p_{2} and factor prices are w_{1}, w_{2}. Technology is given by the unit input coefficients b_{11}, b_{12}, b_{21}, b_{22}. The primal problem, can now be written as:
max p_{1}x_{1} + p_{2}x_{2}
s.t.
v_{1} ³ b_{11}x_{1} + b_{12}x_{2}
v_{2} ³ b_{21}x_{1} + b_{22}x_{2}
x_{1}, x_{2} ³ 0.
This is illustrated for this case Figure 2 below. The factor market inequality constraints permit us form a feasible region, shown in Figure 2 as the shaded area below the curves V_{1} and V_{2} (where, recall, V_{1} is where the locus of output levels, x_{1}, x_{2}, where the first constraint is met with equality and V_{2}, where the second constraint is met with equality). The condition x_{1}, x_{2} ³ 0 ensures that the feasible region is above the axes.
The objective in the primal problem is the maximization of total revenue, p¢ x, where p is given exogenously. Thus, letting R be a given amount of total revenue, we can define a linear locus R = p_{1}x_{1} + p_{2}x_{2} depicted in Figure 2 by the straight line denoted R. This line represents the combinations of x_{1} and x_{2} which, for a given set of prices p_{1} and p_{2}, yields total revenue R. The locus R can be rewritten as:
x_{2} = R/p_{2} - (p_{1}/p_{2})x_{1}
thus it has negative slope -(p_{1}/p_{2}) and vertical intercept R/p_{2} (and horizontal intercept R/p_{1}). In Figure 2, the higher locus R* denotes the combinations of x_{1}, x_{2} points which, for the same prices p_{1}/p_{2}, yields the higher revenue R*.
Figure 2 - The Primal: Output Level Determination
With prices p_{1}, p_{2} given, then the primal problem will yield the solution x* = (x_{1}*, x_{2}*), shown in Figure 2 by point e. This represents the highest locus R* which meets the constraints imposed by factor market clearing. Notice that in Figure 2 this happens to be at the corner point e in Figure 2 which lies on both the V_{1} and V_{2} locii, thus both constraints are binding, v_{1} = b_{11}x_{1}* + b_{12}x_{2}* and v_{2} = b_{21}x_{1}* + b_{22}x_{2}*.
However, it is not necessary that the solution to the primal problem yields market clearing. For instance, suppose output prices were such that the revenue locii were much steeper (e.g. R¢ in Figure 2). In this case, the solution would be a corner point such as f in Figure 2. Notice that at f, the V_{1} locus is binding (thus v_{1} = b_{11}x_{1}* + b_{12}x_{2}*), but we are clearly below the V_{2} locus (thus v_{2} > b_{21}x_{1}* + b_{22}x_{2}*). Thus, at these new output prices, the revenue-maximizing problem yields a solution where we have market-clearing for factor v_{1} and excess supply of factor v_{2}. Notice also that at f, we produce no output of good x_{2}.
We can trace a relationship between the output price ratio p_{1}/p_{2} and the solution x* to the revenue-maximization problem by recalling that the slopes of the V_{1} and V_{2} constraints in Figure 2 are -(b_{11}/b_{12}) and -(b_{21}/b_{22}) respectively, where (b_{11}/b_{12}) > (b_{21}/b_{22}), which translates into saying that ouput x_{1} is relatively v_{1}-intensive in output x_{2} is relatively v_{2}-intensive. Consequently, when p_{1}/p_{2} is very high (i.e. the revenue curves are very steep), so that p_{1}/p_{2} > (b_{11}/b_{12}) > (b_{21}/b_{22}) then the solution will always be the corner solution f in Figure 2, where a positive amount of good x_{1} and no amount of good x_{2} is produced and where factor v_{2} is in excess supply, but v_{1} is market-clearing. Notice that if it happens that p_{1}/p_{2} = (b_{11}/b_{12}) so that the revenue curve is of exactly the same slope as factor constraint V_{1}, then any point on V_{1} between e and f will be considered optimal. Thus, in this case, the output levels are indeterminate (x_{2} ranging somehwere between 0 and x_{2}* in Figure 2, x_{1} ranging from x_{1}* to f), but we are certain that, except for point e itself, the factor v_{2} will always be in excess supply and factor v_{1} will be at market-clearing. Conversely, if output prices are such that the revenue curve is flatter than the V_{2} constraint, then the solution will be g in Figure 2, where V_{2} is binding and V_{1} not binding.
The following table gives a breakdown of the different types of solutions we would obtain in Figure 2 depending on the slope of the revenue curve (i.e. output prices p_{1}/p_{2}). We go from steepest to flattest revenue cure:
Slope of Revenue Line |
Position in Figure 2 |
Output |
Output |
Factor |
Factor |
p_{1}/p_{2} > b_{11}/b_{12} |
f |
f |
0 |
Clears |
Ex. Supply |
p_{1}/p_{2} = b_{11}/b_{12} |
between e and f |
between x_{1}* and f |
between 0 and x_{2}* |
Clears |
Ex. Supply |
b_{11}/b_{12} > p_{1}/p_{2} > b_{21}/b_{22} |
e |
x_{1}* |
x_{2}* |
Clears |
Clears |
p_{1}/p_{2} = b_{21}/b_{22} |
between e and g |
between 0 and x_{1}* |
between x_{2}* and g |
Ex. Supply (except at e) |
Clears |
b_{21}/b_{22} > p_{1}/p_{2} |
g |
0 |
g |
Ex. Supply |
Clears |
Table 1 - Solutions to Primal Problem
Notice that in Table 1, we only obtain indeterminacy in output levels when output prices are exactly such that the slope of the revenue curve is identical to V_{1} or V_{2}, otherwise output levels are clearly defined.
Let us now turn to the dual problem. The dual for this two-factor, two-output case is:
min w_{1}v_{1} + w_{2}v_{2}
s.t.
p_{1} £ b_{11}w_{1} + b_{21}w_{2}
p_{2} £ b_{12}w_{1} + b_{22}w_{2}
w_{1}, w_{2} ³ 0.
which is illustrated in Figure 3. The price-cost inequalities for the two production processes create a feasible region, represented in Figure 3 by the shaded area above the curves P_{1} and P_{2}. The lines P_{1} is the locus of factor prices, w_{1}, w_{2}, where the first constraint is met with equality and P_{2} is the corresponding locus for the second constraint. The condition w_{1}, w_{2} ³ 0 ensures that the feasible region does not stretch below the axes.
The objective in the dual problem is the minimization of total factor payments, wv, where factor supplies v = (v_{1}, v_{2}) are given exogenously. Thus, letting C be a given amount of total factor payments we can define the locus C = w_{1}v_{1} + w_{2}v_{2} depicted in Figure 3 by the straight line C, which represents the combinations of factor prices w_{1} and w_{2} which, for a given set of factor supplies (v_{1}, v_{2}), yields the same costs C. The locus C can be rewritten as:
w_{2} = C/v_{2} - (v_{1}/v_{2})w_{1}
and so has negative slope -(v_{1}/v_{2}) and vertical intercept C/v_{2}. In Figure 3, the lower locus C* represents the combinations of factor prices which, for the same factor supplies, yields lower total cost C*.
Figure 3 - The Dual: Factor Price Determination
The analysis proceeds in an analagous manner. With factor supplies v_{1}, v_{2} given, the dual problem will yield the solution factor prices w* = (w_{1}*, w_{2}*) in Figure 3 at point e¢ , the lowest cost locus C* which fulfills price-cost constraints. As e¢ it is on the corner of P_{1} and P_{2}, then both constraints are binding, p_{1} = b_{11}w_{1}* + b_{21}w_{2}* and p_{2} = b_{12}w_{1}* + b_{22}w_{2}*.
However, once again, this need not be the case. If factor supplies are such that the C locus is quite steep (C¢ in Figure 3), then the solution will be a corner such as f¢ in Figure 3. Notice that at this corner, P_{1} is binding but P_{2} is not, so p_{1} = b_{11}w_{1} + b_{21}w_{2} and p_{2} < b_{12}w_{1} + b_{22}w_{2}, i.e. price equals cost of production for output x_{1} (thus production of x_{1} breaks even, or is "viable") but output process x_{2} is running at a loss. Notice, correspondingly, that at f¢ , factor prices w_{1} = 0 and w_{2} = g¢ > 0. If factor supplies were such that the cost locii are very flat, the corresponding solutions will be at g¢ , where P_{1} is non-binding and thus x_{1} is running at a loss, but P_{2} constrains and thus x_{2} is viable.
Recall that the slope of P_{1 }is -(b_{11}/b_{21}) and of P_{2} is -(b_{12}/b_{22}). The fact that P_{1} is steeper than P_{2} indicates that b_{11}/b_{21} > b_{12}/b_{22} which, it can be noticed at once, is consistent with the inequalities we assumed in the primal problem earlier depicted in Figure 2. We can trace the relationship between factor supplies (v_{1}/v_{2}) and the solutions to the cost-minimizing problem as depicted in Figure 3. These is laid out in Table 2 below. We go from the case where the cost locus is the steepest to when it is the flattest:
Slope of Cost Line |
Position in Figure 3 |
Factor Price |
Factor Price |
Output |
Output |
v_{1}/v_{2} > b_{11}/b_{21} |
g¢ |
0 |
g¢ |
Viable |
Loss |
v_{1}/v_{2} = b_{11}/b_{21} |
between g¢ and e¢ |
between 0 and w_{1}* |
between g¢ and w_{2}* |
Viable |
Loss |
b_{11}/b_{21} > v_{1}/v_{2} > b_{12}/b_{22} |
e¢ |
w_{1}* |
w_{2}* |
Viable |
Viable |
p_{1}/p_{2} = b_{12}/b_{22} |
between e¢ and f¢ |
between w_{1}* and f¢ |
between w_{2}* and 0 |
Loss |
viable |
b_{12}/b_{22} > v_{1}/v_{2} |
f¢ |
f¢ |
0 |
Loss |
viable |
Table 2 - Solutions to Dual Problem
Once again, note that we only obtain indeterminacy in factor prices when factor supplies are such that the slope of the cost curve happens to be identical either to the slope of P_{1} or P_{2}. Notice also that we have the intuition that when a particular factor price is low, the output process which is relatively intensive in that factor will be viable, whereas when a factor price is high, the output process relatively intensive in that factor will be running at a loss (compare factor price w_{1} and the viability of the v_{1}-intensive output process x_{1} or factor price w_{2} and the viability of the v_{2}-intensive output x_{2}).
(4) Existence and Uniqueness of Equilibrium
We are not finished, however. The linear programming problems set out above yield us (modified) market clearing conditions (the complementary slackness equations), pure competition and solutions for output (x*) and factor returns (w*) - given a particular technology (B), a price vector (p) and a factor supply vector (v). To assume technology given is not too difficult (although issues of choice of technique are relevant, they would take us too far afield at this point), but where do the given p and v come from?
As in the Walras-Cassel model, we need consumer output demand functions and factor supply functions to close the system. Recall that from utility-maximization we obtained market commodity demand functions D(p, w) and market factor supply functions, F(p, w). For simplicity, we shall ignore factor supply functions F(p, w) and assume factors are supplied inelastically (i.e. v is fixed) for the rest of this section.
Now we turn to the existence question posed by Abraham Wald (1935, 1936). Consider the equlibrium price vector p*. If we plug this into our primal and dual problems, we obtain solutions x* and w*. Taking these derived factor prices w* and our original prices, p*, we can plug them into our demand function to obtain the quantity demanded, D(p*, w*). If we are indeed in equilibrium, then it would better be that the output levels x* we found in the primal problem be equal to this demand, i.e. D(p*, w*) = x*. If not, then p* could not have been equilibrium to begin with. The existence question is then this: is there indeed a p* such that the w* and x* we obtain as solutions to the linear programming problems will lead us to an equilibrium?
Wald proved that indeed such an equilibrium exists although he did not have price-cost inequalities and did not possess the linear programming duality theorems as a weapon. Consequently, shall not replicate his rather complicated proof, but instead take the avenue set out in Lancaster (1968: Ch. 9) (for an alternative proof, see Dorfman, Samuelson and Solow (1958: Ch. 13)).
As v is given, then we insert any set of prices p, we obtain, from the primal, a solution x and solution w from the dual. Henceforth, we shall refer to the solution to the primal problem as x(p) and the solution the dual problem as w(p). We shall reserve the terms p*, x* and w* for equilibrium prices and quantities.
Let us first confine prices to a price simplex. We know that if p is given, then w(p) is the associated solution in the dual. Consequently, if kp is given, then it can be easily shown that kw(p) is the solution in the dual. Demand is homogeneous of degree zero in output and factor prices, then D(p, w) = D(kp, kw) and the solution to the primal problem is also homogeous of degree zero in output prices, x(p) = x(kp) (i.e. notice in Figure 2 that doubling output prices will leave the slope of the objective functions and thus the solution x(p) unchanged). As a consequence, we can normalize p so that E/font> _{i=1}^{n} p_{i} =1. Thus, let us defined the price simplex P = {p | p ³ 0, E/font> _{i}p_{i} = 1}.
We now turn to the following preliminary lemma:
Lemma: If p* is the equilibrium price vector, then the equilibrium outputs and equilibrium factor prices (call them x* and w*) will be solutions to the linear programming problems.
Proof: As x* and w* are the equilibrium outputs and factor prices, then by definition of equilibrium we must have it that p*D(p*, w*) = w*v by the aggregation of the consumer budget constraints. By output-market clearing, D(p*, w*) = x*, thus this can be rewritten as p*x* = w*v. Now, if we insert equilibrium ouptut prices p* into our linear programming problem, we obtain solutions (x(p*), w(p*)) where, by the duality theorem, w(p*)[v - B¢ x(p*)] = 0 and x(p*)[Bw(p*) - p*] = 0 which translates to: p*x(p*) = w(p*)v. It is noticeable from before that equilibrium outputs x* and factor prices w* fulfill this condition, thus they are solutions to the linear programming problems.§
The purpose of the Lemma was to demonstrate that once we know the equilibrium prices p*, we know by extension that the equilibrium outputs x* and factor prices w* will be solutions to the linear programming problem with p* given. As a further note, it worth remembering the following from the duality theorem p*x(p*) = w(p*)v. If the optimal solution w(p*) is indeed an equilibrium factor price vector w*, then from the budget constraint, we necessarily have it that x(p*) = D(p*, w*), i.e. output market clears, so the solution x(p*) is the equilibrium output. Conversely, if optimal solution x(p*) is equilibrium output x*, then, again from the budget constraint, necessarily the solution w(p*) will be the equilibrium factor price, w*.
Now, suppose any p is given. Then, we can define the resulting vectors w(p) and x(p) as the solutions to the linear programming problem. In principle, solutions to the linear programming problem are not necessarily unique, thus let us define W(p) and X(p) as the associated sets of optimal solutions - which, note, will be convex. Thus, for any w Î W(p) and any x Î X(p), we have it by the duality theorem that px = wv or, more succinctly, pX(p) = W(p)v.
Now, demand vector is defined as D(p, w). As w = w(p) by solution, then we can reduce this to D(p, w(p)) = D(p) as the market demand vector when prices are p. Naturally, many demand vectors may be chosen at a particular price, thus let us define D(p) as the set of demand vectors D(p). Thus, we can now define the set of excess demand vectors at price p as:
Z(p) = D(p) - X(p).
Pre-multiplying every vector in this set by p, we have:
pZ(p) = pD(p) - pX(p)
Now, since the budget constraint is fulfilled for any D(p) Î D(p), we thus have it that pD(p) = W(p)·v and since pX(p) = W(p)v by the linear programming problem, then this can be written:
pZ(p) = W(p)·v - W(p)·v = 0
which is simply Walras's Law.
Now, recall that X(p) and W(p) are mappings from prices to optimal sets in the linear programming problems. Now, as we know from standard programming theory, these will be upper semicontinuous correspondences. In constrast, as we know that D(p) is a continuous function of p, then D(p) is upper semicontinuous. The sum of upper-semicontinuous correspondences will be upper semicontinuous, thus Z(p) must be upper semi-continuous. We would also like to show it is convex. This is easily done with the assistance of Walras's Law.
Theorem: (Existence): In the Walras-Cassel-Wald system outlined above, a competitive equilibrium exists, i.e. there is a set of equilibrium prices (p*, w*) and equilibrium quantities (x*) such that:
(1) D_{i}(p*, w*) £ x_{i}* for all i = 1, .., n |
(output market clearing) |
(2) If D_{i}(p*, w*) < x_{i}*, then p_{i}* = 0 |
(rule of free goods) |
(3) v_{j} ³ B_{j¢ }x* for all j = 1, .., m |
(factor market clearing) |
(4) If v_{j} > B_{j¢ }x*, then w_{j} = 0 |
(rule of free factors) |
(5) p_{i}* £ B_{i}w* |
(competition) |
(6) If p_{i}* < B_{i}w*, then x_{i} = 0 |
(rule of viability) |
Furthermore, if B > 0 strictly, then (1) and (2) are replaced with the following:
(1¢ ) D_{i}(p*, w*) = x_{i}* for all i = 1,.., n (output markets clear, no free goods).
Proof: Defining Z(p) = D(p) - X(p), we know that Z(p) is a convex, upper semi-continuous correspondence from the price simplex P. By the Gale-Nikaido Lemma, there is consequently a price-vector p* such that Z(p*) Ç (- W ) is not empty. This last implies that there is a z* Î Z(p*) where z* £ 0.
We now wish to prove that conditions (1)-(6) are met at z*. Condition (1) is automatic from the Gale-Nikaido lemma: namely, z* £ 0 implies z_{i}* = D_{i}(p*, w*) - x_{i}* £ 0. Condition (2) follows from Walras's Law pZ(p) = 0. As z* Î Z(p*), then p*z* = 0,which can be rewritten for the ith market as p_{i}*[D_{i}(p*, w*) - x_{i}*] = 0, so if D_{i}(p*, w*) < x_{i}*, then necessarily p_{i}* = 0.
Condition (3) follows from the definition of X(p). At equilibrium prices, p*, we defined X(p*) as the set of solutions to the primal problem, so by definition as x* Î X(p*), then v ³ B¢ x*, so for the jth factor, (3) follows. Condition (4) follows from the Lemma: as p* and x* are equilibrium prices and outputs, then as we know from the duality theorem, the solution to the dual problem w* will be equilibrium factor prices. Consequently, optimality conditions establish that w*[v - B¢ x*] = 0, thus for the jth factor, w_{j}*[v_{j} - B¢ _{j}x*] = 0, so if v_{j} > B¢ _{j}x*, then necessarily w_{j}* = 0.
Condition (5) follows by the same logic. At equlibrium prices p*, we defined W(p*) as the solution to the dual problem, so by definition w* Î W(p*), so p* £ Bw*, so for the ith output, (5) follows. Finally, condition (6) is obtained also from the Lemma: as p* and w* are equilibrium output prices and factor prices, then by the duality theorem, the solution to the primal problem x* will be equilibrium outputs. Consequently, optimality conditions establish that x*[Bw - p] = 0, so for the ith output, x_{i}*[B_{i}w* - p_{i}*] = 0, so if p_{i}* < B_{i}w*, then necessarily x_{i}* = 0.
Finally, let us turn to (1¢ ). If we assume B > 0 strictly (all elements positive), then B_{i} > 0 strictly, then necessarily B_{i}w* > 0, so if p_{i}* = 0, then necessarily, B_{i}w* - p_{i}* > 0. But then, by condition (6), we must have it that x_{i}* = 0. This can be interpreted as follows: if an output is free (p_{i}* = 0) , then that output will not be produced (x_{i}* = 0). But in equilibrium (1), we have it that z_{i}(p*) = D_{i}(p*, w*) - x_{i}* £ 0. As D_{i}(p*, w*) ³ 0, then it must be that if x_{i}* = 0, then D_{i}(p*, w*) = 0. Thus, in sum, if p_{i}* = 0, then necessarily z_{i}(p*) = 0, i.e. there are no free outputs in equilibrium. This is what (1¢ ) states.§
Let us now turn to one of Wald's major concerns: uniqueness of equilibrium. Wald (1936) introduced the following famous assumption:
Weak Axiom of Revealed Preference (WARP): Let p, p¢ be two prices and D, D¢ be their associated market demand vectors, if pD ³ pD¢ , then p¢ D¢ < p¢ D.
This can be interpreted as follows: if D is chosen at price p when D¢ was affordable, then if D¢ is chosen at p¢ , then it must be because D¢ was not affordable at those prices. There are two forms of WARP introduced by Wald: one for the individual demand functions (what he called Condition 6w) and one for the market demand functions (what he called Condition 6). However, Wald was careful enough to remind us that these are two different conditions: WARP at the individual level does not imply and is not implied by WARP at the market level (as later confirmed by the Debreu-Sonnenschein-Mantel theorem). However, he writes that "there is a statistical probability that, from the assumption that (6w) holds for every [household], the validity of (6) follows" (Wald, 1936a:p.292), but he did not prove this. We should note that Wald (1936b) does go on to prove that WARP will be satisfied if we impose the assumption that all goods are gross substitutes -- a result later much used in stability theory during the 1950s and 1960s.
The proof of uniqueness on the basis of WARP is quite simple. Let p* and p** represent two distinct equilibrium price vectors, thus by definition, z* = D* - x* = 0 and z** = D** - x** = 0. Now, we claimed that D** was feasible when prices were p*, thus x** was feasible in the linear program generated by p*, i.e. in max p*x s.t. v ³ B¢ x. But since x* is the solution to that program, then necessarily p*x* ³ p*x**, where we have it that v ³ B¢ x* and v ³ B¢ x** (both x* and x** are feasible). However, when x** chosen in its own program, max p**x s.t. v ³ B¢ x, notice that the constraints have not changed from the previous program. In other words as v ³ B¢ x*, then x* must be feasible in the second program. Consequently, as x** is the optimal solution, we obtain p**x** ³ p**x*. But this is a clear violation of WARP: x* was chosen when x** was feasible, and x** was chosen when x* was feasible. Thus if WARP holds, then it must be that p* = p** and x* = x**, i.e. the equilibrium is unique.
(5) The Wald System with Capital
We can easily add intermediate goods into the Wald construction, exactly as we did before for the Walras-Cassel model. Let A be a matrix of unit output demands for intermediate goods (i.e. produced factors) and B a matrix of unit output demands for primary factors. Thus, the market-clearing condition for produced goods is now:
x = A¢ x + D(p, w)
i.e. output supplies (x) must equal input demands (Ax) and consumer demands (D(p,w)). In contrast the market-clearing condition for primary factors remains:
v ³ B¢ x
i.e. supply of primary factors is equal to the demand for primary factors. In contrast, the price-cost inequalities now include the following:
p £ Ap + Bw
where Ap is the cost of intermediate goods and Bw is the cost of primary factors. As a result, the primal program of the Wald model can be rewritten as:
max pD(p, w)
s.t.
x = A¢ x + D(p, w)
v ³ B¢ x
x ³ 0
while the dual is:
min wv
s.t.
p £ Ap + Bw
w ³ 0
By redefining prices as net prices, i.e. p° = (I - A)p, our price-cost inequalities become p° £ Bw and by re-expressing our outputs as net outputs, x° = (I - A)x, thus we now have x° = D(p, w) as our output market-clearing condition. As a result, we can rewrite the primal and dual problems as:
max p¢ (I - A¢ )x = p^{0¢ }x
s.t.
v ³ B¢ x
x ³ 0
which is effectively the same as in the old Wald system, only now with adjusted prices. In contrast the dual is:
min wv
s.t.
p^{0} = (I-A)p £ Bw
w ³ 0
which is, again, identical to the old Wald system, except with adjusted prices p^{0}. The solution then proceeds as in the conventional Wald model. For more details, refer to our earlier references on capital and growth in the Walras-Cassel model.
W.J. Baumol and S.M. Goldfeld (1968) Percursors of Mathematical Economics: An anthology. London: London School of Economics.
R. Dorfman, P.A. Samuelson and R.M. Solow (1958) Linear Programming and Economic Analysis. New York: McGraw-Hill.
D. Gale (1960) The Theory of Linear Economic Models. New York: McGraw-Hill.
L. Kantorovich (1959) The Best Use of Economic Resources. 1962 translation, Oxford: Pergamon.
S. Karlin (1959) Mathematical Methods and Theory in Games, Programming and Economics, Vol. I & II. Reprint, New York: Dover.
H.W. Kuhn (1956) "On a Theorem by Wald", in H.W. Kuhn and A.W. Tucker, editors, Linear Inequalities and Related Systems. Princeton: Princeton University Press.
R.E. Kuenne (1963) The Theory of General Economic Equilibrium. Princeton: Princeton University Press.
K.J. Lancaster (1968) Mathematical Economics. 1987 reprint, New York: Dover.
H. Neisser (1932) "Lohnhöle und Beschäftigungsgrad in Marktgleichgewicht", Weltwirtschaftliches Archiv, Vol. 36, p.413-55.
K. Schlesinger (1935) "On the Production Equations of Economic Value Theory", in Menger, editor, Ergebnisse eines mathematischen Kolloquiums, 1933-34. Translated and reprinted in Baumol and Goldfeld, 1968.
H.F. v. Stackelberg (1933) "Zwei kritische Bemerkungen zur Preistheorie Gustav Cassel", Zeitschrift fE Nationalökonomie, Vol. 4, p.456-72.
A. Takayama (1974) Mathematical Economics. 1985 (2nd) edition, Cambridge, UK: Cambridge University Press.
A. Wald (1935) "On the Unique Non-Negative Solvability of the New Production Functions (Part I)", in Menger, editor, Ergebnisse eines mathematischen Kolloquiums, 1934-35. As translated and reprinted in Baumol and Goldfeld,1968.
A. Wald (1936) "On the Production Equations of Economic Value Theory (Part 2)", in Menger, editor, Ergebnisse eines mathematischen Kolloquiums, 1934-35. As translated and reprinted in Baumol and Goldfeld, 1968.
A. Wald. (1936) "On Some Systems of Equations of Mathematical Economics", Zeitschrift fE Nationalökonomie, Vol.7. Translated, 1951, Econometrica, Vol.19 (4), p.368-403.
E. R. Weintraub (1983) "The Existence of Competitive Equilibrium: 1930-1954", Journal of Economic Literature, Vol. , p.1-39.
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