An evolutionary model of economic growth must be able to explain the patterns of aggregate outputs, inputs, and factor prices that neo classical theory “explains .” In the exercise here, the standard of ref erence is provided by Robert Solow’ s classic article “Technical Change and the Aggregate Production Function” (Solow, 1957) . The data addressed in that article comprise gross national product (GNP), capital i nput, l abor input, and factor prices, over a forty-year period. Data beneath these macro aggregates is ignored. Our simulation model must be capable of generating those macro aggregates, but through the route of “building them up” from microeconomic data. And our model must eschew neoclassical analytic components based on well- specified production functions and profit-maximizing behavior and employ in their place the evolutionary theory compo nents of decision rules, search, and selection.

The model involves a number of firms, all producing the same homogeneous product (GNP), by employing two factors: labor and physical capital. In a particular time period, a firm is characterized by the production technique it is uSing-described by a pair of input coefficients (a_{l} , a_{k}) -and its capital stock, *K. *As in the model pre sented in Chapter 6, to enable us to exploit the mathematics of finite Markov chains, capital stock is assumed to come in discrete packets. A firm’s production decision rule is simply to use all of its capacity to produce output, using its current technique -no slow-down or shut-down decision is allowed for. Thus, at any time, the “state” of a firm can be characterized by a tripIe (a_{l} , a_{k} , K) indexed by time and the identification number for a particular firm . The industry state at time t is the (finite) list of firm states at time t. Given the basic behav ioral assumption, aggregate output and labor demand are directly determined by the industry state. The wage rate is endogenous, and is determined in each time period by reference to a labor supply curve. The gross returns to capital are simply output (at price equal to one) minus labor payments. Thus, the model can generate or explain the macroeconomic data that Solow addressed.

Changes in the industry state are generated by applying probabi listic transition rules, independently, to the individual firm states. These transition rules result from our specification of search pro cesses and investment rules. In turn, the way we characterize partic ular transition mechanisms reflects our desire to capture, in stylized form, some of the salient aspects of technical advance and Schumpe terian competition as they have been identified by micro economic studies. We discuss, first, the transition rules for firms “in business” – that is, with a positive capital stock. Assumptions gov erning entry will be mentioned later. In the following discussion, a parenthetical delta (0) will identify parameters that have been varied in the experimental runs. The assumptions below, which determine the form of the general model, reveal the kinship of this model with that analyzed in Chapter 6. Yet they differ in important ways.

### 1. Technical Change

Use of the term “search” to denote a firm’s activities aimed at im proving on its current technology invokes the idea of a preexisting set of technological possibilities, with the firm engaged in exploring this set. This connotation seems natural when one is considering R&D aimed to find, say, a seed variety with certain properties or a chemical compound with certain characteristics. It seems less natural when one is considering R&D aimed to develop a new aircraft, or, more generally, R&D activities where the terms “invention” or “de sign” seem appropriate. Instead of exploring a set of pre existing pos sibilities, R&D is more naturally viewed in these contexts as creating something that did not exist before. And surely modern research on hybrid seeds and pharmaceuticals involves creating as much as dis covering.

But for the purposes of our evolutionary modeling, the distinction here is one of semantics not substance. The R&D activities of our firms will be mod eled in terms of a probability distribution for coming up with different new techniques. We will discuss this in terms of sampling from a distribution of existing techniques. But alternatively we could discuss it in terms of a distribution of things that a firm might “create.” In either case, that distribution might be a function of time (opportunities might evolve over time), a firm’s R&D policy (some firms might spend more or perform different kinds of R&D than others), the firm’s existing technique (search may be largely local), and other variables.

In the particular model explored in this chapter, time *per* *se *is not an element; there is a given set of techniques to be found; a fi rm’s R&D “policy” is modeled as involving “satis ficing.” And what a firm comes up with as a result of its R&D is much influenced by its pre vailing technique and the prevailing techniques -of other firms.

*Satisficing. *To highlight the similari ty of the model employed here to the equilibrium-seeking model of Chapter 6, we assume that if firms are sufficiently profitable they do no “searching” at all. They simply attempt to preserve their existing routines, and are driven to consider alternatives only under the pressure of adversity. Their R&D activity should thus be conceived as representing an *ad hoc *organizational response rather than a continuing policy commit ment. This satisficing assumption is a simple and extreme represen tation of the incentives affecting technical change at the firm level. We dispense with this assumption in the dynamic competition models in Part V, in which the differential profitability of alternative levels of commitment to R&D expenditure is a major focus of con cern, but we believe it is adequate for our present purposes. In fact, it seems useful to demonstrate that in an evolutionary model with such conservative firms, there can be contin uing innovation in the economy as a whole.

In the simulation runs here, only those firms that make a gross re turn on their capital less than the target level of 16 percent engage in search. Given that a firm is searching, it either seeks incremental improvements to its present methods or looks to what other firms are doing, but not both at the same time.

*Local Search. *There is a given constant set of technological possi bilities, and each technique is characterized by coefficients *a _{l}*

*and a*

_{k}

*.*Technical progress occurs as this set gradually is explored and dis-covered. For any firm engaging in such exploration, search is ulocal” in the sense that the probability distribution of what is found is con centrated on techniques close to the current one. The formula used for the distance between techniques h and

*h’*is

That is, distance is a weighted average of the absolute differences in the logs of input coefficients. This gives rise to diamond-shaped equal-distance contours in the space of logs of input coefficients. Em ployment of different values of WTL (δ) permits us to treat search with differing degrees of “bias” toward discovering labor- or capital saving technologies. Probabilities for transitions from a given tech nique to others are then determined as a decreasing linear function of distance, subject to obvious nonnegativity conditions, an appro priate normalization, and introduction of a probability that no alter native technique will be found. The slope of this linear function is IN(δ), where IN stand mnemonically for “ease of INnovation.” The larger (less negative) the value of IN, the more likely it is that the search process will uncover technologies with input coefficients sig nificantly different from the initial ones.

*Imitation.* A searching firm may look to what other firms are doing.

If it does, the probability that it will find a particular technique is proportional to the fraction of total industry output produced by that technique in the period in question. Alternatively we might have as sumed that imitation is focused on Jlbest practice,” and we do so in models presented later. The assumption here is more consonant with models of diffusion, where what is best practice is not obvious to a firm *ex ante *but where widely used techniques attract attention .

The actual probabilities of “finding” different techniques for a firm that is searching are, then, a weighted average of the probabili ties defined by “local search” and the probabilities defined by “imi tation .” The rel ative weights on local search and imitation are char acterized by the parameter IM (δ), where IM is a mnemonic for liemphasis on IMitation .” A high value of IM denotes a regime where search is more likely to be over what other firms are doing and less likely to be of the lilocal search” type than it would be in regimes where the value of IM is low.

An alternative rule turned up by the search process is adopted by the firm only if it promises to yield a higher return, per unit capital, than the firmfs current rule. (Since the firm’s capital stock is inde pendently determined, the retum-per-unit-capital criterion gives the same result as a test based on anticipated total profit .) The wage rate employed in this comparison is the one associated with the current industry state. There is an element of random error in the com parison: the capital and labor input coefficients employed in the test are not the true values for the alternative technique, but the products of the true values and realizations of i ndependent normal deviates. A firm in business misjudges the input coefficient of an alternative technique by an amount that exceeds 20 percent about a third of the time.

### 2. Investment

Our characterization of the determinants of changes in the sizes of firms can be described much more compactly. The capital stock of a firm with positive capital in the current state is first reduced by a random depreciation mechanism; each unit of capital is, indepen dently, subj ect to a failure probability of D = 0. 04 each period . The capital stock, thus reduced, is then increased by the firm’s gross in vestment in the period. Gross investment is determined by gross profit, where gross profit πK is revenue Q minus wage bill WL minus required dividends RK. (More precisely, gross investment is gross profit rounded to the nearest integer, the rounding being necessary because capital stock is integer-valued and gross profit is not.) This rule is applied even when gross profit is negative, subj ect only to the condition that the resulting capital stock not be negative. The higher the value of R (δ), the smaller the investment the firm is able to finance .

### 3. Entry

As indicated above, we make special assumptions about entry. A firm with zero capital in the current state is a potential entrant and IIcontemplates” the use of a production decision rule. If i ts decision rule implies a gross rate of return to capital in excess of 16 percent cal culated at current prices, it becomes an actual entrant with proba b il ity 0. 25. If it does enter, its capital stock is determined by a draw on a distribution that is uniform over the integers from five to ten. (Entry is relatively infrequent, and the contribution it makes to gross

investment is minor when averaged over several periods. ) Other firms (those contemplating rules that do not meet the rate-of-return test) remain at capital stock zero with probability one. The assump tions about search by potential entrants differ slightly from the as sumptions about search by firms already in the industry; these will be mentioned when needed.

### 4. The Labor Market

The price of labor is endogenous to the model, b eing determined by the exogenous supply and endogenous demand for labor. The pre vailing wage rate influences the profitability of each firm, given the technique it is using, and, in turn, the behavior of the industry as a whole is a powerful, but not unique, influence on the wage rate. The simulation program admits all wage determination equations of the

where t is the time period, L_{t} is the aggregate labor use in the period, and a, b, c, and g are constants. When g = 0, labor supply conditions are constant over time, and the model as a whole is a Markov process with constant transition probabilities. A nonzero g corresponds to changing labor supply conditions; the model as a whole remains a Markov process, but with time-dependent transition probabilities. The Markov process defined by the above relations may be sum-marized as follows . At any moment the capital stocks of extant firms, together with their techniques, determine their required labor inputs and their outputs. Industry output and total labor employ ment then are determined. Total labor employment determines the industry wage rate. Given the wage rate, the gross profitability of each firm is determined.

Firms that make a gross rate of return of less than the target level engage in search. Of those firms that are searching, some attempt to innovate and others to imitate the techniques used by more profit able firms. Firms screen the techniques that they have uncovered by search, and if they deem them more profitable they are adopted and the old ones discarded. Firms that had been earning more than the target level, or that do not come up with techniques they deem better than the ones they had, keep their old techniques.

Extant firms invest in the purchases of new capital the earnings they have left after paying wages and required dividends. Their net investment equals gross investment minus depreciation. New firms may enter the industry at positive capital stock if the profitability of the technique they were contemplating exceeds the target level.

Thus, the next-period techniques of all firms are determined (probabilistically), and so are the next-period capital stocks. The “industry state” for the next period then has been established.

### 5. Calibration

The model will generate a time path of firm and industry inputs and output, and a time path of the industry wage rate and firm and in dustry rates of the return on capital, the labor share, and the capital share. One central question we are exploring is whether a model of the sort described above is capable of generating time paths of the macroeconomic variables that are similar to the actual observed time paths of these variables (in particular to those displayed in the data analyzed by Solow).

The initial conditions of the model were set so that they roughly corresponded to the conditions revealed in Solow’s data for 1909.2 Thus, we initially endowed our firms with techniques that, on average, had roughly the input coefficients displayed by the Solow data for 1909 . We assigned an initial amount of capital to each firm and positioned the labor supply curve so that, given the implied labor requirements fo r the initial period, the wage rate equaled the 1909 wage rate and the initial capital-labor ratio roughly matched the 1909 data. (For reasons of convenience we chose an initial total capi tal stock of three hundred units. ) Given that wage rate and the choice of input coefficients, the initial average rate of return on capital of our firm must be roughly equal to that in the Solow data for 1909 . And labor’s and capital’s shares of income under initial conditions of the model also will be consonant with the actual Solow data for 1909. The data analyzed by Solow also determined the set of possible techniques (input coefficient pairs) built into the model. The tech niques were determined by random choice from the uniform distri bution over a square region in the space of logarithms of input coef ficients.3 The region includes, with room to spare, all of the historical coefficients implied by Solow’ s data. We judged that distinguishing one hundred possible techniques in this region would permit ade quate representation of cross-sectional diversity and historical change. This scatter is displayed in Figure 9. 1, along with the actual time paths of input coefficients from the Solow data. An important question being explored is whether the (average) input coefficients of our simulation model can be induced to display a time path that is similar to the actual one.

The time path of the input coefficients, and of related variables like the capital-labor ratio, obviously will depend on how labor and capital grow over time in the model. Given the broad specification of the model’s logic, this will depend on the particular parameter set tings of some of the key variables. Thus, in the runs reported here we have assumed that the labor supply curve shifted to the right over the period of time at a rate of 1. 25 percent per year. This is roughly con sistent with the observed historical rate.

9.1 Input coefficient pairs for unit output, with Solow’s historical input coefficient values.

### 6. Varying Some Key Parameter Values

One question we are asking about the model is whether, under plau sible parameter settings of the sort described above, it can generate time paths of the m acroeconomic variables comparable to those actu ally observed. Another range of issues being explored involves the connections between vari ables defined at the microeconomic level and the macroeconomic time path.

The “localness of innovative search” assumption built into the model implies that, at a microeconomic level, most innovations are relatively minor. It is possible, however, to vary the localness of search -to make it easier (more l ikely) or harder (less likely) for a firm to discover a technique significantly d ifferent (far away) from the one it has. (Specifically, the relevant parameter here is the slope of the relationship between the distance of an alternative technique from the current one and the probability that the alternative will be discovered.) If search is less local, if m ajor innovation is easier, a firm is more likely to come up with innovation that is markedly inferior. But given that its profitability checks are reasonably reliable, it will not adopt such innovations. On the other hand, the innovations that it does adopt are likely on average to be b igger (involving a larger de-cline in the input coefficients). To what extent would the ease of major innovation, in the above microeconomic sense, show up in, say, a faster rate of growth of labor productivity or of total factor pro ductivity? One’s faith in the model’s ability to represent micro-macro links would be severely strained unless there were some such associ ation. By choosing different settings of the *II *ease of m ajor innova tion” parameter, it is possible to explore this question.

It also is possible to vary the parameter that determines what frac tion of a firm’s “searching” will be directed to what other firms are doing, rather than toward possible innovations . What differences would this make? The logic of the system at the micro level would suggest that if more search is directed toward imita ting and less toward innovating, the production techniques of firms will tend to be bound together more closely. The competitive race would be “closer.” And one implication of this might be that firms tend to re main together in size, as well as in technology. By calculating some measure of industry concentration at the beginning and end of the simulation runs, one can explore the effect of different degrees of emphasis on imitation on the extent to which concentration evolves over time in the model economy, and perhaps on some other vari ables. If interesting and plausible connections show up in the simu lation results, these might form hypotheses to be tested against real-world data.

One can also vary the required dividend rate. If the dividend payout is low, the rate of growth of the capital stock ought to be higher than it would be if the payout were higher. This higher capital stock might be expected to lead to higher labor demand, thus to higher wages and to a tendency to adopt less labor- intensive tech niques, when the cost of capital is low than it would when the cost is high. Another influence on the evolution of the capital-labor ratio might come from the extent to which search is easi er in a capital saving direction or in a labor-saving direction. One can vary this within the model by choosing different weights on the distance mea sure regarding innovative search .

In our simulation models we employed two different settings for e ach of the variables discussed above: the ease of major innovation, the emphasis on imitation, the cost of capital, and the labor-saving bias of search. That is, we undertook runs (of fifty periods each) with sixteen different sets of parameter settings. The sixteen runs com prise all possible combinations of levels of the four experimental factors, with two levels for each factor.

All of the experimental runs were initiated with the same assign ments of techniques to thirty-five firms. In the eight runs with a high dividend payout rate, the fifteen firms in business each had twenty units of capital. In the runs with low required dividends, firms in business each had twenty- two units of capital. These initial capital values were chosen to put the system in approximate “equiIib ri um”- that is, with roughly zero expected net investment in the initial period. To have started all runs at the same indus try state, ignoring the implications of the different parameter values, would have been a straightforward but nai’ve approach to the problem of achieving “identical” initial conditions for the different runs. Drastic differences in the aggregate outcomes in the early periods would then have been implied by the differences in parameter values; no such strong effects are visible in the results as they stand .

Source: Nelson Richard R., Winter Sidney G. (1985), *An Evolutionary Theory of Economic Change*, Belknap Press: An Imprint of Harvard University Press.