Question

One way to generate disequilibrium prices in a simple model of supply and demand is to incorporate a lag into producer's supply response. To examine this possibility, assume that quantity demanded in period \(t\) depends on price in that period \(\left(Q_{i}^{p}=a-b P_{t}\right)\) but that quantity supplied depends on the previous period's price-perhaps because farmers refer to that price in planting a crop \(\left(Q_{t}^{s}=c+d P_{t-1}\right)\) a What is the equilibrium price in this model \(\left(P^{*}=\right.\) \(P_{t}=P_{t-1}\), for all periods, \(t\) b. If \(P_{0}\) represents an initial price for this good to which suppliers respond, what will the value of \(P_{1}\), be? c. By repeated substitution, develop a formula for any arbitrary \(P_{t}\) as a function of \(P_{0}\) and \(t\) d. Use your results from part (a) to restate the value of \(P_{r}\) as a function of \(P_{0}, P^{\prime \prime},\) and \(t\) e. Under what conditions will \(P_{t}\) converge to \(P^{*}\) as \(t \rightarrow \infty ?\) f. Graph your results for the case \(a=4, b=2, c=1\) \(d=1,\) and \(P_{0}=0 .\) Use your graph to discuss the origin of the term cobweb model.

Short answer

Answer: The price will converge to the equilibrium price as the number of periods goes to infinity if and only if the absolute value of the coefficient of the lag price in the supply function is smaller than the absolute value of the coefficient of the price in the demand function (|d| < |b|).

Step-by-step solution

Equilibrium price

To find the equilibrium price \(P^*\), we need to equate the quantity demanded and quantity supplied under the condition that the price does not change between periods i.e., \(P_t = P_{t-1}\). Set \(Q^p_t=Q^s_t\), i.e., \(a−bP_t=c+dP_{t−1}\). Since \(P_t = P_{t-1}\), we can rewrite the equation as \(a−bP^* = c+dP^*\). Now, we can solve for \(P^*\): \(P^* = \frac{a - c}{b + d}\)

Value of \(P_1\)

Given an initial price, \(P_0\), suppliers react according to \(Q^s_t = c + d P_{t-1}\). We want to find the value of \(P_1\) where the quantity demanded equals the quantity supplied: \(a - b P_1 = c + d P_0\). Solving for \(P_1\), we get: \(P_1=\frac{a-c+dP_0}b\).

Formula for any arbitrary \(P_t\)

By repeated substitution, we can express any \(P_t\) as a function of \(P_0\) and \(t\). First, notice that the equation for \(P_t\) with the lag can be expressed as: \(P_t = \frac{a-c+dP_{t-1}}b\). Using this recursive formula, we can calculate \(P_t\) iteratively as: \(P_t = \frac{a-c}{b}+\frac{d^n}{b}P_0\) , where \(n\) denotes the number of periods.

Restating value of \(P_t\)

Using the result from step 1, we can restate the value of \(P_t\) as a function of \(P_0\), \(P^{*}\), and \(t\). Using the formula from step 3, we get: \(P_t = \frac{a-c}{b}+\frac{d^n}{b} (P_0 - P^*)\).

Conditions for convergence

The price will converge to \(P^*\) when \(t\) goes to infinity if and only if the term \(\frac{d^n}{b}\) goes to zero: \(\lim_{t\to\infty} \frac{d^n}{b} = 0\). This will occur if \(|d|<|b|\), which means that the coefficient of the lag price in the supply function is smaller in absolute value than the coefficient of the price in the demand function.

Graphical representation and cobweb model

Graph the results for the case \(a=4, b=2, c=1, d=1,\) and \(P_0=0\). The convergence properties can be best visualized with a cobweb model. The cobweb model is a graphical representation of the sequence of prices in this lagged supply-demand setting. A cobweb diagram is drawn as follows: First, a supply curve, \(a - bP = c + dP_{t-1}\), and the demand curve, \(Q_t^p = a - bP_t\), are plotted, along with the path of prices over time using the recursive formula derived above. Step by step, prices are represented as a movement along the demand curve and supply curve with the lag. The name "cobweb model" comes from the graphical representation of the sequence of prices, forming a web-like pattern as the price converges or diverges over time. In this particular case, the price will not converge to equilibrium because the values of coefficients \(b\) and \(d\) are equal: \(|d|=|b|\). Thus, the price will oscillate between \(0\) and \(\frac{(a-c)}{b}\) indefinitely, forming a web pattern on the graph.