A new model of fluctuations and growth: Case with underutilized factor of production

We now return to the two-sector model of Section 4.1, and develop it more fully.

As we mentioned there, fluctuations of aggregate economic activities or business cycles have long attracted attention of economists.

Resources are stochastically allocated to sectors in response to excess de­mands or supplied of the sectors. We show that the total outputs of such an economy fluctuate, and that the average level of aggregate production (or GDP) depends on the patterns of demand.^[VIII] Because we assume zero adjustment cost for the sizes of sectors, our model is a model of an economy with underutilized factors of production, such as hours of work of employees.

In the literature, economic fluctuations are usually explained as a direct outcome of the individual agents behavior. The focus is thus on individual agents. Often, elaborate microeconomic models of optimization or rational expectations are the starting points. The more strongly one wishes to interpret aggregate fluctuations as something “rational” or “optimal,” the more one is led to this essentially microeconomic approach.

The model of this section proposes a different approach to explain economic fluctuations. The focus is not on individual agents, nor on elaborate microeco­nomic optimization modeling. Rather, the focus is on the manner in which a large number of agents interact.

Although studies of macroeconomies with many possibly heterogeneous agents are not new, the dynamic behavior of economies in disequilibrium is not satisfactorily analyzed. Clower (1965) and Leijonhufvud (1968) pointed out that quantity adjustment might be actually more important than price adjustment in economic fluctuations. Although this insight spawned a vast literature of the so-called “non-Walrasian” or “disequilibrium” analysis, this approach suffers from the basically static or deterministic nature of the analysis.

8.6.1 The model

Our model is a simple quantity adjustment model composed of a large number of sectors or agents. Resources are stochastically allocated to sectors in response to excess demands or supplies of the sectors. We show that the total output of such an economy fluctuates, and that the average level of aggregate production (or GDP) depends on the patterns of demand. Because we assume zero adjust­ment cost for the sizes of sectors, our model is a model of an economy with underutilized factors of production, such as hours of work of employees. For empirical studies of such economies, see Davis et al. (1996).

We assume that sector i has productivity coefficient c_i, which is exogenously given and fixed. Assume, for convenience, that the sectors are arranged in decreasing order of productivity. Sector i employs N_i units of the factor of production. It is a nonnegative integer-valued random variable. We call its value the size of the sector. When N_i (t) = n_i, i = 1, 2,..., K,the output of sector i is c_in_i, and the total output (GDP) of this economy is

Demand for the output of sector i is denoted by s_i Y(t), where s_i > OL the share of sector i, and ∑_i s_i = 1. The shares are also assumed to be exogenously given and fixed.

We denote the excess demand for goods of sector i by

i = 1, 2,..., K.We keep the cs and demand shares fixed exogenously. Denote the set of sectors with positive excess demands by

and similarly for the set of sectors with negative demands by

for the sum over the sectors with negative excess demands.

Sectors with nonzero excess demands attempt to reduce the sizes of excess demands by adjusting their sizes, up or down, depending on the signs of the excess demands. Section 8.6.3 makes this precise.

8.6.2 Transition-rate specifications

The transition probabilities are such that

for i ∈ I₊, and

for i ∈ I_-, where the transition rates, γ and ρ, of the jump Markov process are specified later.

We assume that the γ's and ρ's depend on the total number of sizes and the current size of the sector that adjusts:

and

This is an example of applying W E. Johnson's sufficientness postulate. We have discussed specifications of entry and exit probabilities in Aoki (2000b). See also Costantini and Garibaldi (1979, 1989), who give clear discussions on reasons for these specifications. As explained fully by Zabell (1992), there is a long history of statisticians who have discussed this type of problems. There are good reasons for γ_i to depend only on n_i and n, and similarly for ρ_i. See Zabell for further references on the statistical reasons for this specification.

We specify the entry rate, that is, the rate of size increase, by

and that of the exit rate, namely, the rate of size decrease, by

So long as θ is kept constant, the above expression implies that the choices of K and α do not matter, provided α is much smaller than K.

We set α = 0 to discuss economies with fixed numbers of sectors, and set it to a positive number to allow for new sectors to emerge. In the latter case, a new sector emerges with probability θ/(θ + n), while the size of sector i increases by one when the sector has positive excess demand with probability (α + n_i)∕(θ + n). See Ewens (1972).

8.6.3 Holding times

We assume that the time it takes for sector i to adjust its size by one unit (up or down), T_i, is exponentially distributed:

where b_i is either γ_i or ρ_i, depending on the sign of the excess demand. This time is called the sojourn time or holding time in the probability literature. We

⁴ There is an obvious interpretation of this approximate expression in terms of the Ewens sampling formula (Ewens 1972). assume that the random variables T_i of the sectors with nonzero excess demand are independent.

The sector that adjusts first is the sector with the shortest holding time. Let T' be the minimum of all the holding times of the sectors with nonzero excess demands. Lawler calculates that for a ∈ I₊

8.6.4 Aggregate outputs and demands

After a change in the size of a sector, the total output of the economy changes to

where a is the sector that jumped first by the time t + h.⁵

After the jump, this sector's excess demand changes to

Other, nonjumping sectors have the excess demand changed to

for

These two equations show the effects of an increase of size in one sector.

(8.3) shows a source of externality for this model that affects the model behavior significantly The index sets I₊ and I- also change in general.

⁵ For the sake of simplicity, we may think of the skeleton Markov chain, in which the directions of jump are chosen appropriately but the holding times themselves are replaced by a fixed unit time interval. The limiting behavior of the original and that of the skeletal version are known to be the same under certain technical conditions, which hold for this example. See Cinlar (1975).

We later give examples to show that simulation results support the analytical calculations remarkably well. Simulations show that, after the initial transient

periods, the model moves around the equilibrium or near-equilibrium values of the sector sizes and outputs, in other words, we have equilibrium cycles when α = 0, and growth with cycles with positive α.

8.6.6 Behavior out of equilibrium: Two-sector model

The expected level of total output, which is the equilibrium level in a deter­ministic model of the kind given in the previous subsection, is indeterminate. Here, we explore the behavior of the economy out of equilibrium in a stochastic model.

Table 8.1.

* marks non-applicable or theoretically not possible combina­tions irrelevant situation.

After a change in n₁ by ±1

above L₁, and so on.

The signs of the excess demands and how the sign changes by a change in size in sector 1 and 2 are summarized in Table 8.1.

The symbol * marks entries that do not apply. Note that signs of f₁ are reversed for regions 4 to 6.

The probability of a size increase in sector 2 is larger than that of a size decrease in sector 2 when

With α = 0, this inequality holds when n₁ < n₂.

From a state (n₁, n₂) hi R₁, consecutive jumps in sector 1 will bring the state to the boundary L₁ by increasing n₁; then the model state enters R₂, and the nature of the dynamics changes. This is so because f₁ continues to be positive after jumps in sector 1. Similarly, consecutive jumps in sector 2 from a state in R1 also eventually bring the state to the same boundary by descreasing n₂. In general, we can calculate the various combinations of jumps in sector 1 and sector 2 to bring the state to the boundary, L₁. We thus see that the state leaves R₁ with probability 1. From a state in R₂ consecutive decreases in n₂ are possible until the state enters R₃. From a state on L₂ a jump in sector 2 brings the states to L₃.

The sector that jumps first is determined by the sector with the shortest holding time. Given that sector i changes its size, if f_i(t) is positive, then we assume that n_i will increase by one. If the excess-demand expression is negative, we assume that n_i will decrease by one. Since no adjustment cost is included in the model, the sizes n_i may be interpreted as some measure of the capacity utilization factor in situations where the capacity constraint is not binding. With fixed numbers of employees in each sector, hours worked per period are an example of units of production factor entering and leaving production processes.

We show by simulation that cycles are possible in this model, and that the average level of output responds to demand patterns, that is, larger demand shares for more productive sector outputs tend to produce higher average output of the economy as a whole than smaller demand shares.

8.6.7 Stationary probability distribution: the two-sector model

Here, we derive the stationary probability distribution for the sizes of the two- sector model.

A general discussion of dynamics is conducted via the master (Chapman- Kolmogorov) equation. Here, we report on the derivation of the stationary prob­ability distribution near the equilibrium states represented by L₃.

By the detailed-balance conditions between states e and b, and those between b and c, we derive the relations for the stationary probabilities:

whereand

By repeating the process of expressing the ratios of probabilities, we obtain

with

with the obvious upper limits of summation, which we drop for simplicity. Note

that

Substituting (1 + β)b∕2 for l, we obtain, after some algebra,

Proposition. The expected value of Y will increase as the demandfor sector 1 is increased in the range of β > 1.

Writing c₂β as c₁z,with z = (1 - s)∣s, and rewriting β as κz where κ = c₁∣c₂, we see that the second derivative of E(Y) with respect to β is negative in the range of z where

For example, with c₁ = c₂, f (z) > 0 for z > z* with z* somewhere between 0.6 and 0.7. This means that for s ≤ 0.5 so that β ≥ 1, the sign of this second derivative is negative. We can combine tlιis result wth that of the first derivative and conclude the following fact.

Fact. E (Y) is Uconvexincreasingfunctionof s intherange 0 < s < c₁∣(c₁ + c₂z*). When z* ≤ 1, β ≥ 1 in this range.

Analogous proposition may be established for the range β < 1 in similar manner.

8.6.8 Emergence of new sectors

Next, suppose that new sectors appear at a rate proportional to θ∣(θ + n₊). This transition rate may be justified as a limiting case in which the parameter α goes to zero, while Kα approaches a positive value θ. More in detail, we assume that either one of the sectors with positive excess demand increases size by one with probability (α' + n.j)∣(K₊α + n₊), where K₊ denotes the number of sectors with positive excess demand, and n₊ is the total size of such sectors, or a new sector emerges with rate proportional to (K₊ - 1)α∣(K₊α + n₊). In the limit of letting α go to zero, and assuming that K₊α approaches a common positive value for the sake of simplicity, we have a model in which either one of the existing sectors with positive excess demand increases size by one, or a new sector emerges.⁶ That is, (8.1) is now modified to read that the conditional change in Y (t + h) given Y (t) consists of two terms, the first conditional on the event of the new sector appearing, which occurs with probability θ ∣(θ + n₊), and the second conditional on the event that no new sector appears.

We assume that a new sector, when it emerges, inherits the characteristics - that is, c and s - of one of the existing sectors with equal probability. That is, if there are L sectors, then with probability 11L, the value of c and s of randomly selected sector is inherited. The s's are then renormalized so that they sum to one, including the newborn sector. This is merely for convenience. Other schemes may also be tried.

⁶ We could assume that K+α converges to θ+, which may change each epoch. This would lead to a slight modification of the Ewens sampling formula.

8.6.9 Simulation runs for multi-sector model

This section summarizes our findings of the model's behavior by simulation.

What is most striking is the fact that the production levels, that is, the sizes of the different sectors of the model, are such that high-productivity sectors are constrained by demands. By starting the simulation with the initial condition of equal sizes for all sectors, we see that inflows of production factors into high-productivity sectors are clearly constrained, and the sizes of more produc­tive sectors actually shrink in simulation. This is consistent with the views of Yoshikawa expressed in some of his writings (Yoshikawa 1995,2000). We keep the total number of sectors at K = 10. We have done simulations with K = 15 and 20, but do not report them, since we have not observed any substantive dif­ferences. As our discussion above indicates, for small value of θ, which ranges from 0.2 to 0.6 in our experiments, there is not much loss of generality in keep­ing the value of K fixed. We also keep fixed the order of the productivities from c₁ = 1 to c_κ = 1/K at equal intervals. We start the simulation runs with the initial condition n_i = 10 for all sectors, i = 1, 2,..., 10. In the graphs below, we skip the first 150 or 200 periods to avoid transient responses.

We vary the demand patterns for the outputs of the sectors as follows. We try five patterns, P_i, i = 1,..., 5:

Pattern P₁ has s = (5, 5, 4, 4, 3, 1, 1, 1, 1, 1)/26;

Pattern P₂ has s = (5, 3, 2, 1, 1, 1, 1, 1, 1, 1)/17;

Pattern P₃ has s = (2, 2, 2, 2, 2, 1, 1, 1, 1, 1)/15;

Pattern P₄ has s = (1, 1, 1, 3, 3, 3, 3, 1, 1, 1)/18;

Pattern P₅ has s = (2, 2, 2, 1, 1, 1, 1, 1, 1, 1)/13.

The sum of the shares of the top five sectors are 0.8, 0.7, 0.66, 0.5, and 0.61 respectively. Our analysis of the two-sector model may be adapted to these five patterns by lumping the top five sectors and the bottom five sectors separately to produce a two-sector model. The simulation confirms what the anaysis predicts, that is, the output is the largest for P₁, followed by those of P₂, P₃, and so on.

All patterns were run 200 times for 500 periods with θ = 0.6 except as we note below. Pattern P₂ has also been run with θ = 0.2. Pattern P₅ was also run for 1000 periods 400 times.

Runs of 200 are small for Monte Carlo experiments. Our interest here, how­ever, is not in accurate estimates of any statistical properties of output variations, but rather in exhibiting possibilities of cyclical behavior in this simple model, and showing that output levels respond to demand pattern shifts.

One of the clear effects of different patterns is the dependence of average output levels on the patterns. Mean outputs are approximately in the order of P₁ to P₅. By putting larger demand shares in higher-productivity sectors, the averge output shifts up. Because the standard deviations of outputs are still large due to the small numbers of runs, effects of different demand patterns on the statistical features of cycles are not so clear cut. Peak-to-peak swings are about 2 percent of the mean levels of outputs.

Figure 8.1 shows outputs, averaged over 200 Monte Carlo runs, with demand shares P₁ through P₅, each for the case of θ = 0.6. Figure 8.2 is the plot of P₂ outputs averaged over 200 runs with θ = 0.2. Figure 8.3 shows 1000 time periods of outputs averaged over 400 runs, with θ = 0.6. Figure 8.4 shows per-unit output Y/n averaged over 200 runs. The equilibrium value is y^e∣n^e = 0.4196. This value is independent of θ. Figure 8.5 shows the outputs for θ = 0.1 and θ = 1 with P₃ demand share pattern. These two figures are included to give some feel for the effects of the magnitude of θ on the outputs. As is pointed out by Feller (1968, Chap. 3), the random walks generated by fair coin tosses show much counterintuitive behavior. The numbers of periods and runs are not large enough to draw any precise conclusions. These simulation experiments serve to show the existence of equilibrium cycles even in this extremely simple quantity adjustment model.

The four panels of Fig. 8.6 show a sample of how the number of sectors increases, together with the total number of sizes, total output, and output per unit size, for the demand pattern P₃ with θ = 0.3. As the value of θ is increased, the number of new sectors increases more quickly. For small values such as θ = 0.01, new sectors come in much more slowly.

8.6.10 Discussion

Instead of assuming that resources are instantaneously reallocated to equalize productivities in all sectors, the model of this section assumes that (re)allocation of resources takes time. The model calculates the holding times of all sectors, which determine the probability of the sector that actually increases output in response to positive excess demands for goods of sectors of the economy. As parts of this calculation, the probability of a new sector emerging is also determined. The model solves the conceptual problem, in the usual agent-based simulation models, of which agent moves first and by how much.

Without building microeconomic structures into models, this model shows that cyclical fluctuations and growth with fluctuations are possible. What is most striking is the fact that production levels, that is, the sizes of the different sectors of the model, are such that high-productivity sectors are constrained by demands. By starting the simulation with the initial condition of equal sizes for all sectors, we see that inflows of production factors into high-productivity sectors are clearly constrained and sizes of more productive sectors actually

Fig. 8.1. Total outputs of five demand patterns, P₁ through P₅, all for 500 time periods, average of 200 Monte Carlo runs, and θ = 0.6.

Fig. 8.2. Total output with P₂ pattern with 500 times periods, average of 200 runs, θ = 0.2.

Fig. 8.3. Total output with P₅ pattern with 1000 times periods, average of 400 runs, θ = 0.6.

Fig. 8.4. Per unit output, Y/n, with pattern P₃. Upper panel for θ = 0.1 Lowerpanelwith θ = 1.

shrink in simulation. This is consistent with the views in Yoshikawa (2000, 1995). Wemayalso call the reader's attention to Davisetal. (1996, p. 83), which seems to lend support to the kind of modeling described in this section. They complain about downplay between cycles and the restructuring of industries and jobs in the traditional economics literature, and call for going beyond the stress placed on the role of aggregate shocks in business cycles.

Fig. 8.5. Total output with pattern P₃. Upper panel θ = 0.1. Lower panel θ = 1.

We have taken the entry and exit probabilities to depend on the sizes of the sectors. An alternative specification will specify them to depend on the excess demands themselves. This possibility is definitely worth pursuing.

Also, we note that changing the outputs from linear ones in (8.1) to concave ones c_inY, with 0 β(n₁ + 1) is replaced with (n₂)^γ > β(n₁ + 1)^γ, that is, with n₂ > β^i/Y (n₁ + 1).

The regions A₁ through R₆ are analogously defined by lines L₁ through L₅ with slope β¹'^γ. Arguments to derive the stationary distribution go through with β replaced by β^l'^γ. Since the Proposition in Section 8.6.7 holds for all values of β, it also holds for economies with c_in^γ_i, i = 1, 2,..., K.

8.7