This study integrates Cournot competition into the Solow-Uzawa neoclassical growth model with Zhang’s concept of disposable income and utility function. For the sake of simplicity, we deal with a duopoly here. The model can be generalized in a straightforward manner in case of any number of firms in Cournot competition. Most aspects of the model are basically the same as those in the Solow-Uzawa two-sector growth model, except for the modelling of household behavior and duopoly behavior. The economy supplies final goods and products of the duopoly. The final goods sector is perfectly competitive and produces capital goods as in the Solow model. Capital goods are invested and consumed. The final goods sector is the same as the one in the Solow model. We follow the Uzawa two-sector model regarding the modelling of the economic structure. In the Uzawa model, it is assumed that the consumer goods sector is composed of two firms and characterized by Cournot competition. In our model, all input factors are supplied competitively. The duopoly’s product is solely consumed by consumers. The final goods sector and duopolists use capital and labor as inputs to produce final goods and duopoly’s products. In perfect markets (homogenous), firms have zero profit, while a duopoly might have positive profits. This study assumes that the homogeneous population shares profits equally. There is no free entry in a duopoly market. In addition, the final good is expressed in numeraire, which serves as a medium of exchange. It is assumed that capital depreciates at a fixed depreciation rate of δ_k

Let F_i (t), K_i (t), and N_i (t) stand for output, capital input, and labor inputs of the final goods sector, respectively. The production function of the final goods sector is specified as follows (1):

(1)

Where A_i, a_i and β_i are parameters. We use w(t) and r(t) to denote the wage and interest rates, respectively. The input prices are equal for all the producers due to perfect competition in input factors markets. The profit of the final goods sector is

π_i (t) = F_i (t) - (r(t) + δ_k ) K_i (t) - w(t) N_i (t)

The marginal conditions imply (2):

(2)

Where

r_δ(t) ≡ r(t) + δ_k

This study applies the approach to modeling household behavior proposed by Zhang (1993, 2005). We use k ̄ (t) to denote wealth per household. In addition, we have k ̄ (t) = K(t) / N ̄ , where K(t) is the total capital. It is assumed that profits, due to the duopoly, are equally shared among the population. Note that profits, in the literature of industrial economics, are often assumed to be invested in innovation. Conceptually, it is not difficult to make profit distribution more realistic by including endogenous technologies. Here, we use to represent human capital and π_j (t) to denote duopolist j's profit. The current income of the representative household is obtained as follows (3):

(3)

The household’s disposable income, i.e., y ̂ (t), equals the current disposable income and the value of wealth as follows (4):

(4)

where

R ̃(t) ≡ R(t) k ̄(t) + h w(t), R(t) ≡1 + r(t)

The representative household distributes the total available budget among the consumption of the duopoly’s product c_s(t), the consumption of final goods c_i(t), and savings s(t). The budget constraint is (5):

(5)

where p(t) is the price of the duopoly’s product It is assumed that the U(t) utility level is related to c_s(t), c_i(t), and s(t) as follows:

U(t)=c^η0 _s (t) c^ξ0 _i (t) s^λ0 (t), η₀, ξ₀, λ₀ > 0

where λ₀ denotes the propensity to save. We solve the optimal problem (6):

(6)

Where

There is a proportional relationship between the disposable income and the total value of the variable. This simple relationship is due to the presumed utility functional form. We conclude that, once we determine p(t) and y ̂ (t), we can determine the behavior of the household.

According to the definition of s(t), the change in the representative household’s wealth is calculated as follows (7):

(7)

According to this equation, the change in wealth equals saving minus dissaving.

We denote the output of duopolist j using F_j (t) The equilibrium condition for the duopoly’s product is (8):

(8)

Duopolies are characterized by Cournot competition. Companies in the consumer goods industry compete over the amount of output they will supply. Each firm decides independently of other firms at the same time. They do not cooperate and compete in quantities. They choose quantities simultaneously. There is no product differentiation between the firms. Firms have market power as each firm’s output decision influences the price of the duopoly’s goods. They behave economically rationally, and each firm designs its strategies with the aim of maximizing profit given the other firm’s decisions on quantities.

From (8) and (6), the demand function for the duopoly’s product is given by (9):

(9)

Where F_d(t) ≡ F₁(t) + F₂(t). We use K_j(t) and N_j(t) to represent duopolist j’s capital input and labor input, respectively. The production functions of the duopoly are taken on the following form (10):

(10)

Where A_j, a_j , and β_j are parameters. The profit of duopolist j is given by (11):

(11)

Inserting (9) in (11), we get (12):

(12)

Adding the two equations in (12) yields (13):

(13)

where

K_d (t) ≡ K₁ (t) + K₂ (t), N_d (t) ≡ N₁ (t) N₂ (t)

Inserting (13) in (12), we obtain (14):

(14)

Each duopolist maximizes its profit given the other duopolist’s output (and input factors). The marginal conditions are the following (15):

(15)

According to (15), each duopolist decides on the labor and capital inputs as functions of the wage rate, interest rate, wealth, and the other duopolist’s output and input factors. Each duopolist’s output and profit are determined by (10) and (14), respectively. The price of the duopoly’s product is given by (9).

The change in capital stock equals the output of the final goods sector minus the depreciation of the capital stock and total consumption. The physical capital accumulation equation is given as follows (16):

(16)

where

C_i (t) = c_i (t) N ̄

In an equilibrium of the labor market, we have (17):

(17)

For capital markets, we have (18):

(18)

We have thus constructed the model proposed here, which is founded on the Solow-Uzawa model (Solow, 1956; Uzawa, 1961), Cournot-Nash equilibrium model (Nikaido, 1975), and Zhang’s concept of disposable income and utility function (Zhang, 2020). The rest of this paper examines the properties of this model.

We first examine how the national economy changes its path of development when one duopolist’s total factor productivity is enhanced as follows: Δ ̄A1 = 3.3. The effects of this modification on the variables are plotted in Figure 2. The output of duopolist 1 is increased. Duopolist 1 employs less labor and capital inputs. Duopolist 2 produces less and employs less labor and capital inputs. Duopolist 1 gets more profit, while duopolist 2 gets less of it. The final goods sector produces more and employs more labor and capital inputs. The growth rate is enhanced. The national output is augmented. The interest rate falls. The wage rate rises. The price of the duopoly’s product is reduced. The representative household has more wealth and disposable income. The utility is enhanced. The household consumes more final goods and duopoly’s product.

Figure 2. Duopolist 1’s total factor productivity is enhanced Figure 2. Duopolist 1’s total factor productivity is enhanced Figura 2. La productividad total de los factores del duopolista 1 mejora Source: Created by the author.

We now study how the national economy is affected when the propensity to consume the duopoly’s product is augmented as follows: Δ ̄n0 = 10. The effects on the variables are plotted in Figure 3. The output levels of the two duopolists are increased. Each duopolist employs more labor and capital inputs. Their profits are enhanced. The final goods sector produces less and employs less labor and capital inputs. The growth rate is reduced. The national output is augmented. The interest rate rises, and the wage rate falls. The price of the duopoly’s product is enhanced. The utility is enhanced initially but reduced in the long term. The household has less wealth and disposable income. The household consumes fewer final goods but more duopoly’s product.

Figure 3. The propensity to consume the duopoly’s product is augmented Figure 3. The propensity to consume the duopoly’s product is augmented Figura 3. Se aumenta la propensión a consumir el producto del duopolio Source: Created by the author.

We now study how the movement of the national economy changes when the propensity to save is enhanced as follows: Δ ̄ λ ₀ = 1.15. The effects on the variables are plotted in Figure 4. The national physical capital is increased. The national output falls initially but rises in the long term. The output levels of the two duopolists are reduced initially but increased in the long term. Each duopolist employs less capital input initially but more of it in the long term. Each duopolist employs less labor input. Their profits are reduced initially but enhanced in the long term. The final goods sector produces more and employs more labor and capital inputs. The growth rate is enhanced. The interest rate falls, and the wage rate rises. The price of the duopoly’s product is reduced. The utility is enhanced. The household has more wealth and disposable income. The household consumes fewer final goods and duopoly’s product initially but more final goods and duopoly’s product in the long term.

Figure 4. The propensity to save is augmented Figure 4. The propensity to save is augmented Figura 4. La propensión a ahorrar aumenta Source: Created by the author.

We now study how the national economy is affected when the propensity to consume the duopoly’s product is augmented as follows: Δ ̄ A_i = 5. The effects on the variables are plotted in Figure 5. The final goods sector produces more and employs more capital input. The sector employs more labor force initially but does not change labor input in the long term. The output levels of the two duopolists are reduced initially but increased in the long term. Each duopolist employs less capital input initially but more of it in the long term. It employs less labor input initially but it does not change in the long term. Their profits are enhanced. The growth rate is augmented. The national output is augmented. The interest rate rises initially but does not change in the long term. The wage rate rises. The price of the duopoly’s product is enhanced. The household has more wealth and disposable income. The utility is enhanced. The household consumes more final goods. The household consumes less duopoly’s product initially but more of it in the long term.

Figure 5. The final goods sector’s total factor productivity is enhanced Figure 5. The final goods sector’s total factor productivity is enhanced Figura 5. La productividad total de los factores del sector de bienes finales mejora Source: Created by the author.

We now analyze how the national economy is affected when duopolist 1’s output elasticity of capital input is augmented as follows: Δ ̄ a_i = 10. The effects on the variables are plotted in Figure 6. The national capital stock and national output rise initially and do not change in the long term. The output level of duopolist 1 is enhanced. Duopolist 1 produces and employs more capital input and less labor input. Duopolist 2 produces more initially but produces the same amount in the long term. It employs more capital input initially and less of it in the long term. It uses more labor input. Duopolist 1 has more profit, while duopolist 2 has less profit. The final goods sector produces more initially and almost the same amount in the long term. Said sector uses more labor input. It employs more capital input initially and less of it in the long term. The interest rate rises, and the wage rate falls. The price of the duopoly’s product is reduced. The utility is enhanced initially but changed slightly in the long term. The household has more wealth and disposable income initially but almost the same amount in the long term. The household consumes more final goods initially but almost the same number in the long term. The household consumes more duopoly’s product.

Figure 6. Duopolist 1’s output elasticity of capital input is increased Figure 6. Duopolist 1’s output elasticity of capital input is increased Figura 6. La elasticidad de salida de la entrada de capital del duopolista 1 se incrementa Source: Created by the author.

This section compares the dynamics of the model proposed with Cournot competition and the two-sector model with perfect competition. When the system is perfectly competitive, firms take the price as given, and the equilibrium condition of demand and supply determines the price. We here describe the growth model when the consumer goods market is perfectly competitive. One of the main differences is that the profit is zero in perfect competition, i.e., π_j (t)=0. The profits and marginal conditions for the two firms are the following, respectively:

where Equations (11)’ and (15)’ correspond to (11) and (15), respectively. The rest of the equations in Section 2, except those related to the duopoly’s profits and marginal conditions, hold as well for the perfectly competitive case. In Appendix A-2, we present a computational program to plot the movement of the competitive model. We plot the movement of the two systems under the same parameter values in (20). The result is plotted in Figure 7. In the latter, we do not plot profits as no firm achieves a positive profit in the perfectly competitive economy. In our model, firm 2 produces nothing in the case of the perfectly competitive economy. Hence, firm 1’s behavior represents the sector’s behavior. From the figure, we conclude that, in the Cournot competition, the national output and national capital (and thus household wealth) are higher than in the perfectly competitive economy. In the Cournot competition, the final goods sector produces more and employs more labor and capital inputs, while the consumer goods sector produces less and employs less labor and capital inputs. The interest rate is lower, the wage rate is higher, and the price of consumer goods is higher. The household has more disposable income, consumes more final goods, consumes fewer consumer goods, and has a higher level of utility. We see that, if the profits of the duopoly are equally distributed among the households, the welfare of the latter is higher welfare when the consumer goods market is characterized by Cournot competition rather than perfect competition.

Figure 7. Comparing the movement of the two economic systems

Note: Solid lines represent the Cournot competition, and dashed lines denote perfect competition