A highly peaked and symmetric pattern of the frequency distributions of
and
suggest that there is a central tendency in the distribution and structured deviations from it. The observed pattern of technical progress departs from the single-state equilibrium, such as the Walrasian equilibrium, in that the deviation from the central tendency (that is, the mode of the frequency distribution) is persistent and is not properly explained by the normal distribution as is in the case of many stochastic Walrasian models. (The actual fit of the observed data of
is better explained by the Laplace distribution
, where
and
b are the location and the scale parameter. There are a few notable economic studies deriving and applying the Laplace distribution from the statistical equilibrium perspective. For example, see [
17,
18] for their studies on the firm profit rates and growth rates.) A more proper concept of equilibrium compatible with the pattern of the observed data is one that predicts the equilibrium as a distribution of different states.
While the notion of equilibrium as a probability distribution is relatively unfamiliar to economists, it has been widely accepted in physics and information theory under the name of
statistical equilibrium [
1,
2]. Statistical equilibrium represents the most likely state of the system in the form of a probability distribution
, which can be derived by maximizing the entropy of the system,
(For the discussions on the derivation of different statistical equilibrium from the maximum entropy, see [
19,
20].) The Walrasian single-state equilibrium is an unattainable special case of this equilibrium when only one state is assigned a positive probability. The statistical equilibrium of most systems in reality does not collapse to a degenerate distribution but assigns a positive probability to multiple states. Therefore, introducing statistical equilibrium to economics suggests that the goal of economic models is to find a non-degenerate probability distribution of the target system that explains the central tendency along with inherent fluctuations around it.
Based on the statistical equilibrium approach, we will introduce one class of model called the QRSE model [
15]. Suppose a finite set of outcomes
and a finite discrete set of actions
. In our example of the firm’s choice of technique, the discrete action variable is a binary set
consisting of two complementary actions with
a and
indicating the adoption and non-adoption of a new technology, while the set of outcomes consists of the rate of cost reduction expected from the new technology. The key dynamics of this model is that the outcome and action variables interact with each other. First, the quantal response part of the model predicts that the firm decision on the adoption of a new technology is determined in response to the outcome variable,
, the degree of potential rate of cost reduction. The impact of the potential cost reduction on the probability of adoption is expressed through the conditional probability of
A on
,
. Second, the model predicts that the rate of cost reduction itself is also affected by the firm’s act of adopting a new technology. The impact of adoption of technology on outcome variables is expressed as
.
Different economic theories of technical change can lead to different specifications of these two-way interactions expressed by and . The following subsections discuss the QSRE model of ITC to specify these interactions and derive the statistical equilibrium distribution of the rate of cost reduction.
3.1. The Impact of the Cost Reduction on Adoption of a New Technology
We employ the induced technical change (ITC) model [
21,
22,
23,
24,
25] as the baseline behavioral model for a choice of technique. In this model, a typical firm maximizes
by adopting a new technology constrained by the
innovation possibilities frontier (IPF), which defines a trade-off between increases in labor and capital productivity. To maximize
, the firm adopts a technology that affects
and
in response to changes in
and
. For example, if there is an increase in unit labor cost
, the firm will be better off introducing a labor-saving technology, which is expressed by increasing
and decreasing
.
Following the logic of the ITC model, our model assumes that the probability of the firm adopting a new technique depends on how much cost reduction
it can achieve. Therefore, the quantal response function is expressed by the conditional distribution of the adoption decision on the potential rate of cost reduction,
. The quantal response function has an associated payoff function
representing the payoff to the typical firm of adopting a new technique. The model assumes that the typical firm maximizes the expected payoff with a mixed strategy of
. This boils down to a simple maximization problem as follows:
With no further constraint, the solution to this problem is the Dirac Delta function, choosing to either adopt or not adopt:
where
is the choice of adoption or non-adoption that maximizes the payoff. Therefore, the resulting frequency distribution of
puts unit weight on the payoff-maximizing action and zero weight on the other.
From a statistical equilibrium point of view, however, this result is extremely unlikely to happen because it requires the entropy of the system to be zero. In economics, the zero entropy case can be understood as a
perfect rationality model, in which the typical firm has a full capacity to process all relevant market signals, resulting in a complete certainty about her decision. In the context of technical change, the perfect rationality model implies that any changes in the input costs will induce an optimal response of technical change so that the potential rate of cost reduction is fully exhausted. Since the zero entropy is not attainable in the real world and remains only as an unrealistic theoretical entity, we need to generalize the model by introducing a positive minimum entropy
. This simple modification is one way to model the
bounded rationality of the economic agent. Consequently, the model yields a different maximization problem with an entropy constraint as follows:
A Lagrangian function of this maximization problem is:
The resulting frequencies of
at a behavior temperature or “shadow price”
T is:
where the partition function
. The result suggests that the optimal technical change is not a single rate of cost reduction, but a probability distribution of all the possible rates of cost reduction. The behavior temperature
T, which was originally called
entropy prices in Foley’s seminal work on the statistical equilibrium approach to economics [
4], plays an important economic role because it determines the overall intensity of the payoff of economic actions in the market. As we will see in more detail in the subsequent discussions, higher behavior temperature
T implies lower intensity of the payoff and thus more uncertainty over the possible economic actions. This form of Quantal Response (QE) model has been used to model bounded rationality in economics. A general survey of QE models can be found in [
26,
27,
28]. One major difference of the QRSE model from the previous models in the QE literature is that our QE model is obtained as an implication of the maximum entropy principle, not a behavioral assumption. (For a seminal work on the QE models from the game-theoretic perspectives, see [
29,
30,
31]. A discussion on the entropy theoretic approach to the bounded rationality can be found in [
32]. I appreciate the anonymous referees of this journal for this point.)
Further deriving
for a binary variable
A, we have:
The recovered conditional distribution gives the probability of a particular action given observed economic variable . Except for the case when the behavior temperature T is zero, the link function is not degenerate and assigns positive probabilities to heterogeneous responses.
The payoff difference
(that is, the difference in payoff between adoption and non-adoption) can be modeled to satisfy the following conditions. First, a higher
increases the payoff of adoption while lowering the payoff of non-adoption, and therefore increases the payoff difference. Second, the payoff difference becomes zero so that the firm is indifferent to the adoption of a new technology when
is equal to a shift parameter
that determines the indifference point or the hurdle point of the rate of cost reduction where the probability of adopting a new technology is 50%. This shift parameter
can be understood as a “premium” on the rate of cost reduction required for the firm to adopt the technology. If
is high, this implies that the firm needs a higher premium on the rate of cost reduction to adopt the technology with a higher than 50% probability. The simplest linear function that reflects these two constraints is:
Therefore, the impact of the rate of cost reduction on adoption of a new technology is modeled as
Figure 4 shows the quantal response function with the behavior temperature
T:
This figure shows that the higher the behavior temperature, the more uncertain the decision. Except for the unattainable case when
, the function predicts a gradual increase in the frequency of action in response to a higher rate of cost reduction. When
T is sufficiently large, the quantal response becomes uniform across different actions (the red line). When the behavior temperature
T is close to zero, the function becomes a step function (the brown line) [
14].
3.2. The Impact of the Adoption of a New Technology on the Rate of Cost Reduction
The key assumption of the ITC model is that the firm faces a limit in technical progress, the IPF, through which available technologies are constrained by the trade-off between the rates of increase of labor and capital productivity. Under the IPF constraint, an increase in productivity of one input is made possible at the cost of a decrease in productivity of the other input. For example, higher labor productivity growth is coupled with lower capital productivity growth on the innovation possibilities frontier. The trade-off between
and
can be represented by the concave function:
The key property of the IPF is that the tangent of the function is the negative factor-price ratio,
[
22]. Depending on a particular
, the firm chooses the corresponding cost-minimizing techniques on the frontier.
The second primary assumption of the QRSE model of technical change is that the adoption of new technology has an impact on the rate of cost reduction through changes in the factor-price ratio. In the model, the typical firm adopts a new technology with a higher than average rate of cost reduction, and therefore, the expected rate of cost reduction conditional on adoption is greater than the expected rate of cost reduction conditional on non-adoption. Now suppose that the act of adopting a new technology does not make any impact on the factor market. In this case, the difference in the conditional expectations of the rate of cost reduction will not be corrected, so that the firm with a new technology will continue to have a higher rate of cost reduction. However, if the new technology changes the factor prices in the market and makes the input initially saved become more expensive, the initial gain in the increased rate of cost reduction will be lost unless the firm finds another new optimal technology for the changed factor-price ratio.
To illustrate, suppose that the firm adopts a labor-saving and capital-consuming technology in response to a higher cost of labor input relative to the capital input. As a result, the firm would require less labor and more capital, which, in turn, will twist the factor-price ratio against the firm’s initial choice by making the labor input cheaper relative to the capital input. Following [
15], we can model this negative feedback (or the competitive pressure) in terms of the difference between the two conditional expectation of
weighted by the marginal probability of action variable:
where
represents the degree of the “ineffectiveness” of the competitive pressure. The larger
is, the less effective market response is as to the adoption of new technology. When
, it implies that the factor market is so effective that it completely corrects any impacts from the technical change. In contrast, when
, the cost reduction achieved by a new technology will not be offset by the negative feedback from the adverse changes in the factor-price ratio.
3.3. Maximum Entropy Program of the Quantal Response ITC Model
Up to now we have discussed two model assumptions in terms of the interaction between the action variable
A and the outcome variable
. Additionally, we introduce the mean constraint on
,
, which represents the typical rate of cost reduction deemed by the cost-minimizing firms in a competitive market. (For a detailed discussion on the first moment constraints in economic models, see [
33].) These assumptions can be used as the constraints of our maximum entropy program to find the statistical equilibrium of
A and
. Using the constraints in Equation (
7) and (8), and the mean constraint of the model
, the maximum entropy program can be written as follows:
The solution to this problem is expressed in terms of the joint distribution
which implies two “predicted” marginal distributions,
), and two conditional distributions,
. Since the data on the action variable
A are not observable, we need to express the solution in terms of the marginal distribution of
,
to be able to estimate the unknown parameters of the model. The solution to this program then becomes:
where
and
is the hyperbolic tangent function, written as
. (See [
15] for the proof. The solution is in discrete terms and requires the coarse-grained bins of
.) There are four unknown parameters,
and
.
and
T are the location and the shape parameter of the quantal response function,
is the Lagrangian multiplier of the
constraint and implies the impact of adoption
A on the cost reduction
, and
is the Lagrangian multiplier of the mean constraint
, which determines the skewness of the distribution. A graphical characterization of distribution (10) is provided in
Figure 6.