Agent-based economic modeling often requires the determination of an initial equilibrium price vector. Calculating this directly requires algorithms of exponential computational complexity. It is known that a partial equilibrium price can be estimated using a median of trades. This paper explores the possibility of a multivariate generalization of this technique using depth functions.

1. Statistical Estimation of a General Exchange
Equilibrium
John S. Schuler
Department of Economics
George Mason University
March 12, 2019

2. Why is this both Interesting and Difficult?
Economics is usually built around interdependence.
Statistics is usually built around independence or at least
exchangeability.
It is the hope of complexity science that some complexity is
only apparent.
General Equilibrium is an extreme example of interdependence.
What can we sample over?

3. Key Terms
Term Definition
Endowment Initial Allocation
Price Exchange Rate of Goods

4. Modeling the Agent
Each agent ai ∈ A consumes some k goods and has an
endowment k to trade.
We assume each good is infinitely divisible and therefore, the
consumption bundles xi ∈ Rk
+.
The agent has a preference ordering on this commodity space.
It is represented by an equivalence class of utility functions,
Ui : Rk → R identified up to monotonic transformations.
Then, the agent has a demand curve for each of k goods,
Dj
i = f (p) defined as arg maxxi ∈Rk
+
U (xi ) subject to the
contraint that pxi ≤ p, i .

5. The Market Demand Curve
We can add the individual demand curves for a market demand
curve:
Dj
(p) =
n
i
Dj
i=1 (p) .

6. Economic Equilibrium
Q1
P1
D1
S1
Market 1
Q2
P2
D2
S2
Market 2

7. General Equilibrium: j Agents and k Goods
In an economy with k goods, a general equilibrium is a set of
associated vectors:
P =










1
P2
...
Pi
...
Pk










Q =










Q1
Q2
...
Qi
...
Qk










The first entry in the price vector is 1 since we choose a
“numeraire” good and normalize its price. The price vector always
has one fewer dimension that the quantity vector.

8. Indifference Curves
A useful piece of machinery.
For a commodity space x ∈ Rk
+, fixing Ui (xi ) = U implicitly
defines a function xk = fi (x1, . . . , xk−1)
The gradient of this function comes in handy. fi refers to
what economists call the marginal rate of substitution.
This yields an alternative characterization of a general
equilibrium.
Good 1
Good2
•

9. Alternative Characterization
A given 1 × (k − 1) price vector p and a k × n matrix of quantities
[qi,j ] represents a general equilibrium if and only if for all agents i:
p, fi = p, f1 = 1
In other words, we have a general equilibrium when the marginal
rate of substitution of all agents coincides with the price vector.

10. The Edgeworth Box
Good 1
Good2
•
•
η
C

11. A Mathematical Description
There are m agents and n goods,
Each agent i has a utility function Ui : x → R where x ∈ Rn
+
and i ∈ {1, . . . , m}.
Each agent has an endowment i .
Then, the Walrasian general equilibrium η is a function of
these utility functions and endowments.
The contract surface is a function of the Ui alone and consists
of the equilibria for all possible endowments. It has one fewer
dimension than the commodity space.

12. A Mathematical Descriptions
Without loss of generality, we can assume that there is a total
of one unit of each good in the economy.
Then, if J is a ones matrix and X is the allocation matrix with
a column for each good, JX = J.
Thus, everything is normalized. This can be transformed
arbitrarily to match all other exchange equilibrium problems.

13. Characterizing the Contract Curve
Take the perspective of a single agent called agent 1 without
loss of generality.
Fix the utility levels of all other agents 2, . . . , n such that
Ui = Ui . Note that these utility levels are not generally the
same.
This gives us an indifference curve Ii for each agent.
Then, c := arg max Ui (x1) for x ∈ i Ii is a point on the
contract curve.

14. The Statistics
We can sample from the allocation space x ∈ Rk
+.
Then, we evaluate the utilities, Ui of these allocations.
Discretize the utility space m
i=2 Ui (xk).
Within each category, find the maximum utility of agent 1.
This estimates the contract curve!

15. Characterizing the Contract Curve

16. A Change of Perspective

17. An Estimator
Recall that in equilibrium, the marginal rates of substitution of
the agents are coincident with each other and the price vector.
We can calculate the cosine between the price vector and the
MRS of each agent. This gives us a vector C of length m.
We can calculate the variance of this vector. The closer the
point is to equilibrium, the lower the variance.
Thus, we can use 1
σ2
C
to weight an average. Points closer to
the equilibrium will contribute more.

18. Results

19. Can We do Better?
At the moment, the statistical properties of this weighted average
are not well-understood.
Insofar as we have estimated the contract curve, we get the
equilibrium price for free.
Looking closely at the Edgeworth box will give us a hint.

20. What Edgeworth Hides
Good 1
Good2
•
•
η
C

21. Income Invariant Exchanges
Let E be the endowment matrix and X be an arbitrary equilibrium
allocation matrix.
An exchange is income preserving if:
[E − X] p = Tp = 0
For ones matrix J, the only constraint on X and E is that
JX = JE = J. T is not of full rank and so is not invertible.

22. The Moore-Penrose Pseudo-Inverse
Derived from the singular value decomposition.
The pseudo-inverse of T is denoted T+.
It characterizes the solution set p.
p = [I − T+T] w for arbitrary w ∈ Rk.
The difficulty: the economics is more constrained than the
mathematics. p must be in the positive orthant.
Without loss of generality, we can restrict it to the set of
vectors of norm 1 in this orthant.
To do: construct a basis in w-space that maps to a probably
smaller basis in p-space. This will allow us to generate valid
price vectors.

23. Income-Invariant Allocations
This technique allows us to define the subsets of the
allocation space that are income preserving with respect to a
fixed price vector.
These form affine subspaces of the allocation space.
Thus, we can randomly generate w’s to get the corresponding
p’s.
Now, we can track the failure rate of each income preserving
allocation space; that is, what percentage of the sample falls
outside the core.
The allocation space with the minimal failure rate estimates
the major axis of the core.
Thus, the allocations that are both on the contract curve and
in this axis are good estimates of the general equilibrium.

24. Results

25. Conclusions
Using order statistics introduces bias but is necessary given
that the support of the distributions is unknown.
More work remains to be done but progress comes from
figuring out what set to sample over.
In particular, here, the useful structure is in the utility space
and the income families.
This is a highly non-linear transformation of the uniform
allocation space.
In future, ideally we can add production.