The Cobweb Model of the Price Adjustment
Introduction
The Cobweb model of supply and demand is one immediate extension from a static supply and demand model. Suppose that the producer-seller uses the price of a unit of product at t-1 when (s)he supplies the product on the market at time t. In this case, the supply equation becomes a function of the price of the product one time period earlier, and the supply and demand model is composed of one linear demand equation and one difference equation for supply. The solution to this model is no longer a pair of two numbers but instead becomes a function. The exact behavior of the solution curve of this function depends on the parameter. This note presents the basic cobweb model of supply and demand and shows how we can find and visualize the solution curve.
The model structure
The quantity demanded remains a negative function of a price of the product, whereas the quantity supplied is a positive function of the price of the product one time earlier.
Q[t]=d0-d1P[t] Q[t]=s0+s1P[t-1]
where all parameters d0,d1, s0 and s1 are positive constants. The parameter d1 is the price sensitivity of the quantity demanded, whereas s1 is the price sensitivity of the quantity supplied.
At equilibrium the two quantities equal with one another, which gives rises to the following linear first-order difference equation.
Clear[Q, P, d0, d1, s0, s1, t]; Solve[d0 - d1 P[t] == s0 + s1 P[t - 1], P[t]]
If the initial value of P[t], P[0] is given by P0, the analytical solution to this difference equation becomes:
RSolve[{P[t] == (d0 - s0 - s1 P[-1 + t])/d1, P[0] == P0}, P[t], t] // Simplify
A numerical example 1
Suppose we have the following parameters and the initial value of P(t):
d0 = 4; d1 = 1.7; s0 = 0.5; s1 = 1.1; P0 = 1;
The numerical solution becomes:
RSolve[{P[t] == (d0 - s0 - s1 P[-1 + t])/d1, P[0] == P0}, P[t], t]
Using a built-in DiscretePlot [ ] function, we can draw a following graph that shows the behavior of the P[t]:
?DiscreetPlot
DiscretePlot[-0.25(-5. + 1.(-0.6470588235294118)^t), {t, 0, 10}]
We can find and visualize the same solution curve using different Mathematica functions, RecurrenceTable [ ] and ListLinePlot[ ] and as follows:
d0 = 4; d1 = 1.7; s0 = 0.5; s1 = 1.1; P0 = 1;
?RecurrenceTable
sol1 = RecurrenceTable[{P[t] == (d0 - s0 - s1 P[-1 + t])/d1, P[0] == P0}, P, {t, 0, 10}]
?ListLinePlot
ListLinePlot[sol1]
A numerical example 2
Suppose this time that the producer's supply response is changed, s1 = 1.9. Then the solution becomes:
d0 = 4; d1 = 1.7; s0 = 0.5; s1 = 1.9; P0 = 1; sol2 = RecurrenceTable[{P[t] == (d0 - s0 - s1 P[-1 + t])/d1, P[0] == P0}, P, {t, 0, 10}]; ListLinePlot[sol2]
Instead of converging, the equilibrium price continue to diverge away from its notional steady state value.
Conclusion
The cobweb model of the price adjustment is one insightful extension of a simple supply and demand model. By introducing a time delay in the quantity supplied on the product market, the model shows how the equilibrium price changes over time (either converging to or diverge away from the steady state value), depending on the parameters.
One can easily extend this model by introducing an additional time delays. For example, we can think of a possibility that the producer and supplier adjusts the price of the product based upon the market prices of the product observed in the past two time periods, instead of the price at time t-1. In this case, the supply equation and the resulting equilibrium condition yields a linear second-order difference equation.
We can also introduce series of inventory-workforce adjustment variables. If we assume that the producer's sales is equal to the demand for this product and the producer wants to hold some positive inventory stock in order to meet the unexpected surge in the demand, we can successively derive various inventory variables such as the desired inventory stock, inventory coverage period, time to fill the inventory gap by engaging in active production activities, etc., and tie them to the dynamic supply and demand adjustment process. Since the supply and the inventory management requires this producer-seller to hire a desired number of workers (given the average capital-labor ratio), we can also introduce the desired workforce, actual workforce, and time delay for hiring new workers, etc., to construct a more comprehensive dynamic model. In this way, we can easily expand this simple cobweb model to account for the dynamic adjustment in both inventory stock and workforce.
Reference
Michael Hoy et al. Mathematics for Economics, The MIT Press: 2022, pp.752-754 and pp.798-802. Ronald Shone, An Introduction to Economic Dynamics. Cambridge University Press: 2001, pp.27-35. Srinivas Raghavendra and Petri T. Piiroinen, An Introduction to Economic Dynamics - Modelling, Analysis and Simulation, Routledge: 2023, pp.15-32.
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