4.1. Assumptions and Equilibria
Borrowing for speculative purposes was modeled in [
2] by modifying the debt dynamics Equation (
4) to
where the additional term
F corresponded to the flow of new credit to be used solely to purchase existing financial assets. It is implicit in the formulation of [
2] that this is a model for speculation by firms rather than households, as the flow
F is added directly to the net borrowing of firms from banks. This additional borrowing is used in turn to buy financial assets held by the Firms sector itself. In other words, they correspond to assets issued by some firms and held by others. As such, they do not appear either in the consolidated balance sheet for the Firms sector in
Table 1, nor in the transactions and flow of funds matrices, because their purchases correspond to intra-sector transactions.
This way to model speculation is consistent with the passive role attributed to households in [
2], with the accommodating consumption profile expressed in (
23). Far from being an artificial feature, however, trade of financial securities between firms, for example in the form of shares and derivatives, was an important aspect of the financialization of the global economy observed in the lead up to the financial crises. In terms of modeling, our specification of firms buying securities from other firms corresponds, for example, to one of the “edge cases” discussed in [
9].
The dynamics of
F was modeled in [
2] as
where
is an increasing function of the observed growth rate
of the economy.
In the analysis presented in [
4], this was changed to
in order to ensure positivity of
F. It was then shown that the extended system for the variables
, where
, admitted
as a good equilibrium, with
,
,
defined as in (
17)–(
19), but with local stability requiring that
in addition to the previous condition (
20). Moreover, [
4] also provide the conditions for local stability for the bad equilibria corresponding to
and
, and showed that these were wider than the corresponding conditions in the basic Keen model. In other words, the addition of a speculative flow of the form (
48) makes it harder to achieve stability for the good equilibrium and easier for the bad equilibrium.
In this paper, we revert back to the original formulation in [
2], because it allows for more flexible modeling of the flow of speculative credit, which as we will see can be either positive or negative at equilibrium. In addition, in accordance with the previous section, we continue to assume a wage-price dynamics of the form (
24) and (
25) and modify (
47) to
where is now an increasing function of the growth rate of nominal output.
There is a problem, however, with the specification of speculative flow proposed in [
2]. As firms borrow from banks to buy financial assets held by other firms, the firms selling these assets receive a flow of payments in addition to their profits, which should either reduce their own need to borrow from banks or lead to an addition in their bank deposits. In other words, at the level of the firm sector, there cannot be an additional increase in net borrowing
B, but rather only an increase in gross debt
L. We address this problem by proposing the following modifications to the model in [
2].
First we allow the interest rate
charged on loans to be distinct from the interest rate
paid on deposits. This means that, in accordance with
Table 1, the expression for profits presented in (
3) needs to be modified to the more general form
which naturally reduces to (
3) when
.
Next we consider separate dynamics for loans and deposits for the firm sector. Notice that the only accounting constraint imposed by
Table 1 is that
as in (
4). Apart from this constraint, the exact way in which financial balances are allocated between the two asset classes
L and
depend on portfolio preferences of the firm sector. One possibility is
where
is a constant repayment rate and
F is the speculative flow defined in (
50). Observe that, according to this definition, the speculative flow affects loans and deposits simultaneously and therefore has no impact on the overall net borrowing by firms. While it is possible to have different allocations of financial balances between
L and
, the broad qualitative results obtained in this paper are likely to remain valid. In particular, for simplicity, we take
so that (
53) and (
54) reduce to
and consider the state variables
and
, where
. It then follows that the model corresponds to the four-dimensional system
with
. Observe that, when
, we have that
and the first three equations above decouple from the last, so that the model reduces to that of
Section 3 with an added speculative flow
f that has no effect on the other economic variables
. In other words, we introduce speculation by firms as a smooth perturbation of the previous model thorough the spread parameter
. Observe that, after solving the system (
57), it is possible to retrieve the the trajectories for
b and
(and consequently both net and gross debt levels) by solving the auxiliary system
Similarly to the model (
26), new equilibria emerge along with familiar ones. With the addition of the speculative dimension
f, we see that the point
obtained by defining
as in (
16), so that
, and setting
is a good equilibrium for (
57). Finding this point requires solving (
58), (
60) and (
61) simultaneously using the definition of
. Considering the change of variable
, this is equivalent to solve the following equation for
X:
Since the polynomial part of (
62) is of order three, it crosses the non-decreasing term
at least once, implying the existence of at least one solution to (
62). The good equilibrium satisfies
if and only if the corresponding solution to (
62) verifies
. The stability of this equilibrium is analyzed in
Section 4.2.1.
Similarly to [
4], the change of variables
and
allows us to study the bad equilibria given by
and
where
There are thus two possible crisis states for the speculative flow. One corresponding to a finite ratio
which corresponds to a financial flow
(since
whenever
). The other corresponds to the explosion of
f, but at lower speed compared to
c. We refer to [
4] for a full interpretation. The local stability of these two types of equilibria are studied in
Section 4.2.3 and
Section 4.2.2, respectively.
Next we consider the equilibrium
where
is given as in (
33), whereas
and
solve the system
where
.
As in
Section 3, this equilibrium must be interpreted as a bad equilibrium despite the finite values taken by state variables. The interpretation extends to
f, which leads to a financial flow
. The stability of this equilibrium points is analyzed in
Section 4.2.1.
The final possibilities corresponds to the equilibrium points
and
, where
is given as in (
33) and
The stability of these equilibria is studied in
Section 4.2.2 and
Section 4.2.3. Overall, the system exhibits one good equilibrium point and seven different bad equilibria, confirming Tolstoy’s dictum on the multiplicity of states of unhappiness.