Abstract

The paper uses a new technique to find a unique solution to a delay differential cobweb model (formulated from a joint supply-demand function of price) with real model parameters via the Lambert W-function without considering any complex branches. The dynamics of the model are demonstrated with simulations and found to complement previous studies using literature values. However, the condition for instability in the previous studies was defied by our model due to the time delay associated with the supply function. The practical application and advantage of this model over the existing models are that the stability of this model is not limited to only the ratio of price elasticity of demand and supply but also the time-delay parameter (i.e., a missing link in the previous models). Our model, on the other hand, loses its stability when the time delay associated with the supply function is fixed at . Since most of the physical systems, including economical systems, are time-delay inherent and such stability conditionalities should not limit their performance, it is recommended that such systems be modelled using delay differential functions. The novelty of this research is that there has not been a definite general solution to the cobweb model with a time delay whose price dynamics mimic the behaviour of the existing cobweb models in the literature. An illustrative example in a delayed fractional-order differential equation also buttressed the importance of the time delay in the model, aside from the impact of the ratio of the price elasticity of supply and demand.

1. Introduction

The cobweb model is an economic model, which explains the reasons for fluctuations in the prices of commodities in certain markets. The model usually characterizes the equilibrium price in a single market with a time lag between supply and demand functions. It is applied in agricultural markets where there is a time lag between planting and harvesting. Generally, the quantity of produce is determined by the suppliers’ future price expectations (i.e., naive price expectations) due to the earlier price trend [1, 2] (see also Ezekiel [3]). Traditionally, the cobweb model is derived from the supply function with a time lag and demand functions of price to determine the current price based on the previous prevailing market price, using the equilibrium market principle in a single market with a single good [2, 4–6]. Let us consider the following piecewise functions of price for supply and demand:where the demand curve gives the total amount of farm produce demanded at each price at a particular time t by the consumers in the market, and the supply curve tells us the total amount farmers are willing to supply at each price. A production time delay (i.e., farm produce requires a period to grow to maturity and get harvested for the market) is introduced into the supply function of price. Concerning the formation of price expectations, we use the nave price expectation, where the expected price at time is the market price realized at a time with . We assume that the constants and (i.e., the price elasticity of demand and supply, respectively), in this case, make the demand curve slope downward and the supply curve slope upward. We also need the constant to ensure that there is a demand for the farm produce if the price is low enough, and we also assume that . On the assumption that the relative change in market price at any time is the balance between demand and supply [2, 5], then we obtainwhere is the adjustment coefficient.

As already stated, the inclusion of a time delay in the mathematical formulation is necessary to account for the unknown details underlying the processes from planting to harvest [7]. Thus, we consider a continuous-time price model developed from delay differential equations, which characterizes equilibrium price dynamics in a single commodity market (i.e., a restructured form of demand and supply price functions in [2, 5, 6]). From equation (2), we have the following initial value problem:where , , , , with assumption that and delay . The general solution of a mathematical formulation involving supply and demand functions of price has taken different forms depending on the formulation of either the supply function or the demand function or the nature of time around which the system is modelled (i.e., discrete or continuous). If (1) is redefined asthen at market clearing equilibrium , we obtain the following general solution for an initial value of :where the initial price and the equilibrium price . This is a discrete-time cobweb model, see [5, 8]. The stability conditionality of the model (5) is described as follows:(i)That the model (1) exhibits converging oscillations towards the equilibrium price if and only if (ii)That the model (1) exhibits alternating oscillations around the equilibrium price if and only if and(iii)That the model (1) exhibits explosive oscillations around the equilibrium price if and only if [3, 8].

If we incorporate in (1), a conformable fraction derivative with in the demand function and also let in (1), then we have a piece-wise function of the form.where a positive now replaces a negative in the demand function in equation (1), and the general solution (6) is expressed bywhere . This price function formulation is known as the conformable fractional cobweb model [5]. The stability conditionality of the model (7) is defined by Theorem 3 in [5] as for convergence. By implication, we can say for divergence, as outlined in [3, 8]. If the price expectation in (1) is formed instantaneously or the supply is instantaneous , then we have an ordinary differential cobweb model [9] whose unique solution is given bywhere and . Since , the equilibrium price has always been locally stable. Thus, the market price will converge to the equilibrium price as provided the initial price is close enough to the equilibrium [9]. However, the general solution for (3) cannot be derived with any particular fixed rule as applied to the various models given in literature. Hence the motivation for this study so that we use a new technique to compute the general solution, which complements the previous research. In this study, we apply the Lambert function with a different technique to find only one solution, contrary to the infinite solutions from the Lambert W function, and use only the real branches since our model parameters are all real values. Thus, the key contribution is that there are general solutions to the cobweb model, whether they are discrete-time solutions defined or continuous-time solutions like equations (5), (7) and (8). All these equations converge to the equilibrium price, irrespective of the initial price value set for the model. Also, irrespective of the solution method one uses, the general solution and price dynamics are always alike (unique). However, there is no single defined general solution to the time-delayed cobweb model. The method with the closest general solution has been the Lambert W function, but it has infinite solutions, hence the motivation of the research is to come out with only one general solution defined in the purview of the model parameters (i.e., if they are real or complex values).

The rest of the paper is structured as follows: Section 2 discusses the existence, stability, and bifurcation of equation (3). The section also considers conditions for the positivity of the solution and the general solution of the equation (3). In Section 3, detailed numerical solutions and stability analysis are done by comparing model (3) to models given in the literature. Section 4 outlines the findings and Section 5 provides conclusions derived from the analyses of the models.

2. Materials and Methods

2.1. Preliminary Analysis

It is widely known that for any delay, there is a differential equation of the formand via the Lambert W function, the solution can be written aswhere is determined from the history function, . Every function , such that , is known as a Lambert function, and it is complex-valued, with a complex argument , which has an infinite number of branches , with , including the real branch [10].

On the contrary, this study presents an approach that brings out the real value argument so that is also be considered real and has only a real branch for its solution. This technique is a deduction from [11], and we are using it because the parameters in both the demand and supply functions of price are all real values. The technique provides us with advanced knowledge of the type of W branch to use to save time, taking into account the infinite number of branches.

2.2. Stability Analysis of Equation (3)

To find the stability of the differential equation in (3), we linearize it in the neighbourhood of the equilibrium point and obtain the following expression:where and . We can observe from equation (12) that is the only equilibrium point. If we assume an exponential solution , where , then substituting it into equation (12) yields a characteristic equation (13), from which the stability of equation (12) is determined through the locations of its eigenvalues .

Now, we seek for a nontrivial solution from equation (12). If we let , then for the steady state is asymptotically stable. On the other hand, when we assume that then there exists with such that the real and the imaginary of equation (13) are derived as given by

Moving the constants and the circular functions to either side of the equations, and then adding the squares of the resulting equations provides the following expression:

The necessary condition for stability change should be . From equation (15), if , then it contradicts the fact that , making the delay parameter harmless. We adopt the theorem from [9].

Theorem 1. If , then steady state of equation (12) is asymptotically stable for any positive value of .

On the contrary, if , then we can define from equation (15) as follows:

If the results in equation (16) are substituted into the first equation of the piecewise equation (14), then we obtain the threshold value of the delay parameter as

2.2.1. Stability Switches and Hopf Bifurcation of (3)

In determining the direction of the stability switch, we assume that the roots of equation (13) are a continuous function of the time-delay parameter. Therefore, we take the derivative of equation (13) with respect to the time delay and solve for to arrive at the following results:

Then,

Therefore,

The results in equation (20) indicate that at on the imaginary axis, all roots of the characteristic equation (13), near the critical value , will cross from left to right as the delay parameter varies continuously from a number less than to that greater than .

Furthermore, we partially differentiate equation (17) with respect to the values and to determine the critical value of the delay as follows:with

Therefore, as and then it implies that by increasing the value of and decreasing the value of , the stability switching curves are shifted upwards. Thus, the variation of the parameters in the directions indicated has significant effects on stability. Therefore, the condition for transversality is met, and that for Hopf bifurcation is also satisfied at . Theorem 2 summaries this assertion as follows:

Theorem 2. If and , then there exists such that the steady-state of equation (3) is asymptotically stable for , loses stability at , and becomes unstable or bifurcates to a limit cycle if .

See full proof in [9, 12].

2.2.2. Existence and Uniqueness Solution of (3)

For simplicity, we make and in equation (12), all positive values, so that it is transformed toand then we can apply the method of steps to find a nonnegative solution equivalent to equation (12) from equation (23). From the interval , we now generate a nonnegative solution expressed by

Since , the solution exists and it is unique and nonnegative in the neighbourhood considered. Again in the interval , if we let be the solution of equation (23), then for it implies that

Thus, it is observed that for every nonnegative initial function , the solution of equation (23) is defined for in [13].

2.2.3. Existence and Positivity Solution of (3)

We now study the conditions, which guarantees nonnegative solutions of equation (23) for every positive initial function . Following the results in Theorem 1.2 [14], the solution of the equation (23) can have negative values for a positive initial condition. This study is very important due to economic reasons. We now transform equation (23) into the following form:where , with but the bars have been dropped for convenience’s sake. We can further transform equation (27) into two composite functions, as expressed bythen we consider the following auxiliary problem from the given composite function:

From the proof of Theorem 1.1 in [13], for every , equation (30) has a nonnegative solution. Therefore, we will turn our attention towhere and the has the following properties:(i),(ii),(iii),(iv) is unimodal, i.e., there exists with , for every , see [14].

Using the method of steps, then for the interval , the solution is defined by

Now, we define the auxiliary function such thatand let be the solution to . Let us consider the following lemma:

Lemma 1. If and then the corresponding solution to the function (32) is positive.

Proof. For each , if we have , then . This implies that the solution is increasing and approaches 1 as . Hence, the solution is positive. Also, if there exists such that , , with and , for , then , with .
We can observe that for , the solution is decreasing; however, it approaches 1, which implies a nonnegative solution. Let be the first point, such that and , for , then on the assumption that there is a sequence , such that , there exists , where has a maximum value. We can see that as , for . Consequently, andSimilarly, if there exists within which has a minimum value, then just as in the previous case of maximal value, we have andSince , for , the function decreases from 0 to on this interval. We now examine the auxiliary function . We can see that and . Therefore, it is apparent that and there is a unique positive zero solution of the function . Let denote this point, then for every , the solution to equation (32) is positive, because for the next maximal and minimal points, the inequalities (36) and (37) are satisfied [14].

2.2.4. General Solution of Equation (3)

If we divide through the differential equation in (3) by , then by letting and , we obtain the following differential equation:where is the equilibrium price. We can further denote and as deviations from equilibrium price, so that equation (38) is transformed to the form given bywhere .

This gives us a transcendental equation of the following form if we assume an exponential solution and then multiply through the resultant equation by .

Let and , then we havewhere the range of values of determines the type of solution of the system [11]. To find m from equation (41), we also let so that then we derive the real and imaginary parts of the systems, respectively, as given by

From equation (43), , ; this gives us

Therefore, as , . Then, from equation (42), we obtain , which means that (41) has exactly one real zero solution [11]. The deduction here implies that we have a Lambert W function with real parameters satisfied by the following function if we multiply through equation (40) by where and the W satisfies the property , with . Therefore, we findwhich provides a solution to equation (39) and in turn results in the particular solution of equation (38) in the form given bywhere , , and . Unlike its counterpart (8) with no time delay and the other models reviewed in the study, the market price dynamics of this model are determined not only by the parameters and but also by the time-delay associated with the supply function. The Lambert W branch satisfying the equation (48) is , the principal branch. Hence, one does not need to use other W branches for a solution. This is the knowledge gap this paper sought to establish.

3. Results and Discussion

3.1. Numerical Solution: Price Dynamics of Model (3) and Model (2)

In this section, we adopt the same parameter values used in [5] and compare the stability analysis of our model (3) to the stability analyses of the existing price adjustment models ((4), (6), and (8) reviewed in this study). In all the models, except (8), the convergence of the market price was dependent on the condition that , which conforms to the condition of convergence in the classical cobweb model Ezekiel in [3]. We now use the following parameters , , , , and , with . All the numerical simulations will be done using MATLAB software.

It is observed in Figure 1 that, for the same parameter values except for the time-delay parameter of , the market price converges to the equilibrium price set for the system. The dynamics of model (49) complement the assertions of the papers in [3, 5, 8, 15, 16], including models (4) and (8).

Figure 1 

Price dynamics of model (3) for different values of .

3.2. Numerical Solution: Price Dynamics of Model (3) for

In the case of the condition , the two models reviewed, models (4) and (6), will have the market price diverged from the equilibrium price fixed for the price adjustment model, with the exception of models (3) and (8). These models (3) and (8) do not have the same defined conditions for stability as the others. Even though the price of an agricultural commodity is affected by the ratio of the price elasticity of supply and demand, the time to supply the commodity cannot be factored out but that will be reserved for future research. The time delay is a paramount factor missing in the other models. We now attest the condition for .

In Figure 2, we interchanged the values for and and had the condition for convergence now . However, instead of divergence of the system, it is converging. We observe that if , then the time delay associated with the supply is multiplied by the inverse of that proportion and we will achieve stability. We can loosely infer that the supply of commodities is also affected by time delay aside from the ratio of price elasticity of supply and demand parameters.

Figure 2 

Price dynamics of model (3) for different values of and .

3.3. Numerical Solution: Limitation of Model (3)

In this section, we study conditions under which model (3) loses its stability concerning parameters used in the numerical simulation. We have already seen that does not have much impact on the stability of the model, but the time-delay parameter does, so we should try values greater than one.

In Figure 3, we maintained the literature values and varied the time-delay parameters to assess their impact on the stability of the model (3). We increased the delay value to , and for the two initial prices, we had all prices converged to the equilibrium price.

Figure 3 

Price dynamics of model (3) for different values of and .

For , the display in Figure 4 conforms to the behaviour of model (23) in [5] with incorporated in the supply function of price with assigned a negative value. However, the highest time-delay parameter permissible for the supply function is to be . Above this limit value, model (3) becomes unstable; the market price diverges from the equilibrium price set for the system.

Figure 4 

Price dynamics of model (3) for and highest value.

3.4. Illustrative Application of the Technique

In this section, we provide an illustrative example of the technique proposed in the study applied to a common ordinary differential model, transform it into a delay differential equation with integer-order and fractional-order, and then discuss the dynamics of the solutions using numerical simulations. Thus, we will consider the general solution technique of the following differential equation (49) and then apply the technique proposed in the study to both the delay differential equation and the fractional-order differential equation.

The variable is farm produce supplied at a unit time . The general solution of equation (49) is given bywhere stability is guaranteed if . When a delay parameter is incorporated in (49), such that we have , then with the proposed technique, we can express the general solution aswhose stability condition is such that for all real values of , , and .

Similarly, we can change both the left and the right-hand sides of (49) to the form known as the delayed fractional order differential equation (DFODE) and apply the Laplace transform to derive its characteristic equation and in turn obtain the following general expression via the Lambert function:where is a fractional-order (noninteger), with , , and as real values.