Accounting Identities are Useful

Nick Rowe directs us today to an eyebrow-raiser. Economist Ankit Mital writes

In the current system of cross-border bookkeeping, unilateral transfers are recorded in the current account (income from trade and foreign business interests) and not the capital account (financial transactions that end up balancing the current account deficit or surplus) side of the transactions.

This is really interesting to those of us who mind our accounting identities. In its simplest form, the balance of payments identity states that the current (CA) and capital account (KA) balances sum to zero. That is, any transaction which increases the CA balance must also decrease the KA balance. If you see the CA balance changing with no corresponding change in the KA balance then you are failing to correctly record the transaction.

So what is Mital missing? First, the KA balance is defined to be the change in foreign ownership of domestic assets less the change in domestic ownership of foreign assets. So if you— residing in the UK— send me £1, then the UK sees foreign ownership of domestic assets rise. That is, the KA rises by £1– exactly balancing the £1 fall in the CA.

It does not matter exactly how the transfer takes place. If you send me \$1.40 instead, then the UK sees domestic ownership of foreign assets fall by \$1.40, raising the KA by \$1.40 and exactly balancing the \$1.40 fall in the CA. If in my name you hand a £1 coin to a London bank to open a deposit account, then again the UK sees foreign ownership of domestic assets rise. (The relevant asset being the deposit and not the coin which moves only between UK actors.)

And there lies the value in minding accounting identities. They force one to think carefully about what else changes.

Depreciation and Income Shares

I would like now to wade briefly into a debate over the gap between growth in productivity and wages by introducing a bit of modeling fun. It seems clear that— in recent decades— although wage income has grown more slowly than GDP there has been little difference between the growth rate of NDP (GDP net of capital depreciation) and the growth rate of total labor compensation.

On the other hand, inequality of compensation has increased quite a bit— driving a large wedge between pay at the top and pay of the ordinary worker. None of this is news. Somewhat less clear is whether net product is more or less appropriate as a comparison. At first blush, workers still have to produce the whole of output no matter how much investment goes to replacing depreciating capital. It might make sense for labor compensation to rise in step with gross production. But...

Differential confusion and a note on “Standard Neoclassical pedagogy”

In my Comment to Standish and Keen, I asserted that it must be that if $P$ is defined as a function $P\!\left(Q\right)$ then it must be that ${\partial P}/{\partial q_i}$ must be zero because $P$ is not a function of $q_i$, but one of $Q$ alone. Standish and Keen wish to argue that if we hold $Q=\sum_i{q_i}$, then ${\partial P}/{\partial q_i}={dP}/{dQ}$, which is assumed to be negative. It is my contention that this may appear correct at first blush, but this compact notation masks hidden assumptions about the underlying economic model.

I tried to explain this in Section 4.2, but it appears my message was lost. Here, I am going to clarify the notation a bit. When specifying the inputs to a function, I am going to use square brackets. Evaluation of a function will employ parentheses. That is, $f\!\left[x\right]$ should be read “$f$— a function of $x$” while $f\!\left(y\right)$ should be read “$f$— a function of one variable evaluated at $y$.”

For example, though Wilfred Kaplan states in Advanced Calculus (3^rd ed.) that

If $z=f\!\left(x,y\right)$ and $x=g\!\left(u,v\right)$, $y=g\!\left(u,v\right)$, then $$ \frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u} $$

Kaplan also clarifies on the following page that $z\!\left[u,v\right]=f\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$ “is the function whose derivative with respect to $u$ is denoted by ${\partial z}/{\partial u}$.”

That is, when $z$ is evaluated at $\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$ the result is an entirely new function. Kaplan’s chain rule could be written more clearly (if pedantically)

If $z\!\left[x,y\right]=f\!\left(x,y\right)$ and $x\!\left[u,v\right]=g\!\left(u,v\right)$, $y\!\left[u,v\right]=h\!\left(u,v\right)$, then $$ \frac{\partial z^*\!\left[u,v\right]}{\partial u}=\frac{\partial z\!\left[x,y\right]}{\partial x}\frac{\partial x\!\left[u,v\right]}{\partial u}+\frac{\partial z\!\left[x,y\right]}{\partial y}\frac{\partial y\!\left[u,v\right]}{\partial u} $$ where $z^*\!\left[u,v\right]=z\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)=f\!\left(g\!\left(u,v\right),h\!\left(u,v\right)\right)$

“Rationality” in the Theory of the Firm... Conclusion

Previously: Introduction; Part 1; Part 2; Part 3; Part 4; Part 5; Part 6; Part 7; Part 8

As we have seen, the response of Keen and Standish is sorely lacking. They continue to misunderstand or misrepresent textbook models, fail to recognize how competitive profit maximization lowers profits, offer no coherent theory to support their contrary arguments, and continue to insist– despite all evidence– that their simulations support their claims.

Defending this failure, the authors write

Admittedly, the agents specified in our paper are not sophisticated enough to do this, but this was a deliberate choice, since we wanted to show that agents following a simple iterative and non-collusive algorithm would choose output levels that clustered around the true profit-maximizing level of output, and not the Cournot level.

The words “not sophisticated” and “simple” are perhaps telling. Keen and Standish seem to argue that their firms are rational profit-maximizers to but insufficiently intelligent to figure out that someone else might have a better strategy. The most frustrating thing about their work is that the claim they fail to support– the existence of a competitive equilibrium at the collusive level within the infinitely-repeated Cournot-Nash game– is itself textbook. Far from debunking textbook economics, they simply do a terrible job of catching up to it.

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Read my original Comment (including Technical Appendix) at World Economic Review.

“Rationality” in the Theory of the Firm... Part 8

Previously: Introduction; Part 1; Part 2; Part 3; Part 4; Part 5; Part 6; Part 7

Keen and Standish write

This focuses on the simple agent model we use to explore this issue. Rosnick compares the model to agents dancing in a wagon careering downhill. Whilst poetic, it is not a useful metaphor, chiefly because with the wagon, the system’s dynamics are largely determined by the trajectory of the wagon, not the individual agents, whereas with our model, the dynamic behaviour is completely determined by the actions of the agents.

Here, they misunderstand the story. Each firm is– in their simulations– out of control. The rise or fall in its profits is almost entirely determined by the actions of all its competitors and not at all its own actions. Thus, it is not rational for the firm to make decisions based on the assumption that it is in control. Much of the time a firm thinks it made a decision which raised its profits, the firm actually made the wrong choice. Much of the time a firm thinks it made a decision which lowered its profits, the firm actually made the correct choice. It is a funny kind of rationality, that.

They continue,

In §6.3, Rosnick introduces a variant of our agents where the decision is made to reverse the usual decision with small probability that increases the closer to equilibrium the system is. Undoubtedly, this could be done so as to benefit the agent concerned, to the detriment of the other agents in the system, however this action cannot be considered rational. Rational agents always choose the optimum action — the only time rational agents are permitted to act stochastically is when multiple equally valued courses of action are available.

First of all, in the strategy I laid out in this Section 6.3 the decision to reverse is based not on closeness to equilibrium, but the change in profits from the previous period. That aside, this critique cuts against Keen and Standish. Far from always choosing the “optimum action” their firms make inferior choices quite often. By their own argument, then, their agents are irrational. The whole point of a firm reconsidering is to avoid letting the rest of the industry fool it into making inferior choices. There is no reason to rule out stochastic strategies, but even if we disallowed those, chance is not required to accomplish the goal. Keen and Standish simply ignore Section 6.4 of my Comment and the entire Technical Appendix. This only points the way, of course. As discussed in Section 6.1 of my Comment and in Part 4 Keen and Standish still leave undefined the infinite-horizon utility for the firm; without knowing how the firm aggregates profits accumulated over the entire course of the game, there is no way to prove that it has employed an optimal strategy to maximize such profits.

This passage does underline the notion that Keen and Standish do not understand the theory they criticize. Agents may be allowed to choose randomly among choices to which it is indifferent. However, agents may not be indifferent between deterministic and stochastic strategies. It may certainly be that a stochastic strategy pays better than a deterministic one. It may also be that a second deterministic strategy pays even better than the stochastic strategy, but that is actually irrelevant; if any strategy pays better than the one suggested by Keen and Standish, then the equilibrium falls apart.

Most importantly, when Keen and Standish concede “this could be done so as to benefit the agent concerned, to the detriment of the other agents in the system” they concede the whole critique. If– given the strategies employed by the other firms– a superior strategy may be found, then a competitive profit-maximizing firm must use the superior strategy. How can it be rational to leave profits on the table for another firm to collect– except by agreeing with other firms that nobody will ruin their good thing by thinking of their own self-interest?

Contrary to the authors’ claim, their firms do not pursue clearly a strategy that is both rational and non-collusive. As best as may be determined in the absence of a clear objective for the full game, either firms fail to recognize that there is money on the table, or they agree with the others to leave it there.

Conclusion

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Read my original Comment (including Technical Appendix) at World Economic Review.

“Rationality” in the Theory of the Firm... Part 7

Previously: Introduction; Part 1; Part 2; Part 3; Part 4; Part 5; Part 6

In their response, Keen and Standish propose the following dynamics:

[A]round a Keen outcome ${\bf q}^K$, we get $$ \frac{d{\bf q}}{dt}=\sum_j{\frac{\partial\pi_i}{\partial q_j}\left(q_j-q^K_j\right)} $$

Right off this math is confusing. Apparently, what they mean is $$ \frac{dq_i}{dt}=\sum_j{\frac{\partial\pi_i}{\partial q_j}\left(q_j-q^K_j\right)} $$ This suggests that they have in mind $$ {\bf F}\!\left({\bf q}\right)\equiv\boldsymbol{\pi}\!\left({\bf q}\right)-\boldsymbol{\pi}\!\left({\bf q}^K\right) $$ which, by construction yields ${d{\bf q}}/{dt}=0$ when ${\bf q}={\bf q}^K$. This much is true no matter what ${\bf q}^K$ is chosen. It is also a very strange strategy for firms to employ. Firm $i$ expands only if it has profits in excess of its benchmark profits $\pi_i\!\left({\bf q}^K\right)$, and contracts only if it has profits below its benchmark. Clearly, the symmetric collusive oligopoly is not stable according to this dynamical system. If all firms start by underproduce by some tiny amount, then they will all reduce their outputs to zero.

If marginal costs are constant, then the eigenvalues are determined by the zeroes of $$ \left|\begin{array}{cccc}\left(\lambda-P+\mathrm{MC}\right)-q_1P'&-q_1P'&\cdots&-q_1P'\\-q_2P'&\left(\lambda-P+\mathrm{MC}\right)-q_2P'&\cdots&-q_2P'\\\vdots&\vdots&\ddots&\vdots\\-q_nP'&-q_nP'&\cdots&\left(\lambda-P+\mathrm{MC}\right)-q_nP'\end{array}\right| $$ so $\lambda\in\left\{P-\mathrm{MC}+QP',P-\mathrm{MC}\right\}$. That is, profitability ($P>\mathrm{MC}$) implies instability.

Even if marginal costs are not constant, if any two firms have identical cost structures, there is no ${\bf q}$ which is both stable and symmetric. Stability is in general achieved when smaller firms are driven out of the industry and the market is monopolized by a single firm.

In Figure 1, we see 25 simulations of a duopoly ($n=2$) with costs and demand parameterized as in their earlier example. The initial output for each firm is randomly selected to be within 2 percent of the proposed equilibrium. Each simulation is seen as a line, with a marker at the end of each run– the ${\bf q}$ at time $t=0.1$ from simulation start.

Figure 1: Simulations of Keen-Standish Dynamics

Keen and Standish write

The condition describes not an equilibrium point, but rather an equilibrium manifold of constant total market production, which is stabilised by the agents ensuring that if any agent were to cause the system to stray from this manifold, then all agents will follow suit, causing that agent to not enjoy its advantage for long. Any rational agent will then return to the fold.

which is a really funny way of saying that they believe firms with higher initial production ruthlessly drive out the competition until the industry turns into a monopoly. Note that this is nothing like their earlier agent simulations in which equilibrium market shares are predictable without dependence on the initial firm choices. Keen and Standish may invoke (pdf) the movie War Games (“A strange game. The only winning move is not to play.”) but their response brings to mind Highlander. “There can be only one!”

However, the math in their response makes no sense to begin with, so the dynamical system Keen and Standish have in mind may be differ from the one analyzed above. Still, a Cournot-like strategy where firms move toward their best response to the current price such as $$ {\bf F}\!\left(q_1,q_2\right)\equiv\left[\begin{array}{c}a-c\\a-c\end{array}\right]-\left[\begin{array}{cc}2b+d&b\\b&2b+d\end{array}\right]\left[\begin{array}{c}q_1\\q_2\end{array}\right] $$ shows globally stable dynamics. Figure 2 shows this strategy simulated with each firm’s output initialized randomly to within 50 percent of the Cournot-Nash result.

Figure 2: Simulation of a Cournot-like Dynamics

This alone does not imply that either dynamic system describes optimal firm behavior; such would require still an appropriately defined firm objective such as $$ \Pi_i\!\left[{\bf q}\!\left(t\right)\right]=\int^\infty_0{\rho^t\pi_i\!\left[{\bf q}\!\left(t\right)\right]dt} $$ Rather, the examples demonstrate that Keen and Standish’s response respecting this thread is in fact math salad.

In Part 8, we will consider their response to my criticism of their computer simulations.

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Read my original Comment (including Technical Appendix) at World Economic Review.

“Rationality” in the Theory of the Firm... Part 6

Previously: Introduction; Part 1; Part 2; Part 3; Part 4; Part 5

In Part 5, we saw that for $$ \frac{d{\bf q}}{dt}={\bf F}\!\left({\bf q}\right) $$ the choice of $$ {\bf F}\!\left(q_1,q_2\right)\equiv\left[\begin{array}{c}a-2q_1-q_2\\a-q_1-2q_2\end{array}\right] $$ led to a globally stable $$ \frac{d{\bf q}\!\left({a}/{3},{a}/{3}\right)}{dt}=0. $$ We could just as easily have chosen $$ {\bf F}\!\left(q_1,q_2\right)\equiv\left[\begin{array}{c}0\\a-q_1-2q_2\end{array}\right] $$ with eigenvalues determined by $$ 0=\left|\begin{array}{cc}\lambda&0\\1&\lambda+2\end{array}\right|=\lambda\left(\lambda+2\right) $$ so that $\lambda\in\!\left\{-2,0\right\}$. The zero eigenvalue here means Jacobian alone does not guarantee a stable ${\bf q}$ in the sense that a small change in initial $q_1$ assures convergence to a different ${\bf q}$. In fact there are a family of stable output pairs at ${\bf q}=\left(q_1,{\left(a-q_1\right)}/{2}\right)$. If this leads– as outlined in the previous Part– to a price $P=a-q_1-{\left(a-q_1\right)}/{2}={\left(a-q_1\right)}/{2}$ and profits $$ \boldsymbol{\pi}=\frac{a-q_1}{2}\left(q_1,\frac{a-q_1}{2}\right) $$ then firm 1’s ideal staring point is $q_1={a}/{2}$. That yields a price $P={a}/{4}$ and profits of $\pi_1={a^2}/{8}$ where the old strategy yielded a price $P={a}/{3}$ and profits of only ${a^2}/{9}$.

The point is, even though both sets of strategies lead ${\bf q}$ to converge to a fixed point, the first pair of strategies are not stable. If either firm follows the initial strategy, the other has incentive to respond by fixing production at $q={a}/{2}$. The strategies do not form a competitive equilibrium. Firm 2 may now profit from a change of strategy, but that is well and truly beside the point that the initial pair of strategies are not competitive.

On the other hand, if both firms employ the strategy of fixing production at $q={a}/{3}$, then neither has incentive to pursue an alternative strategy. If firm 1 fixes production at $q_1={a}/{3}$ there is nothing firm 2 can do in any period to generate greater profits than it would receive by fixing its own production at $q_2={a}/{3}$. No matter how much weight is given to any period in calculating the firm’s objective function, it loses. Therefore, given the strategy of firm 1, firm 2 must reciprocate. Such a (symmetric!) pair of strategies is both competitive and stable– dynamically and strategically.

In this way, a firm’s strategy for profit-maximization depends critically on the strategies employed by its competition. If no firm has incentive to change strategy, only then can the set of strategies form a competitive equilibrium. Importantly, as my Comment and Technical Appendix demonstrate, the strategy laid out by Keen and Standish cannot form the basis for a competitive equilibrium. There must be firms with incentive to pursue higher profits and so there are only two possibilities. Either

• all firms collude by agreeing not to pursue any individually-superior strategy, or
• the proposed equilibrium must fall apart.

Keen and Standish do, however, present a new set of strategies in their response. We turn to that new arrangement in Part 7.

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Read my original Comment (including Technical Appendix) at World Economic Review.