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 \(\left(Q_{t}^{D}=a-b P_{t}\right)\) 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 \(\left(P^{*}=P_{t}=P_{t-1}\right)\) 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 \(P_{t}\) as a function of \(P_{0}, P^{*},\) 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

The equilibrium price (\(P^*\)) in this model is \(\frac{a-c}{b+d}\). b. Calculate the value of \(P_{1}\) given an initial price \(P_{0}\). The value of \(P_1\) given an initial price \(P_0\) is \(\frac{a - c - d P_{0}}{b}\). c. Develop a formula for any arbitrary \(P_{t}\) as a function of \(P_{0}\) and \(t\). The formula for any arbitrary \(P_t\) as a function of \(P_0\) and \(t\) is \(P_t = \frac{(a - c)(b + d)^{t-1} - d^t P_0}{b^t}\). d. Restate the value of \(P_{t}\) as a function of \(P_{0}, P^{*},\) and \(t\). The value of \(P_t\) as a function of \(P_0, P^*\), and \(t\) is \(P_t = P^* + \frac{d^t (P_0 - P^*)}{b^t}\). e. Determine the conditions under which \(P_{t}\) will converge to \(P^{*}\) as \(t \rightarrow \infty\). The conditions under which \(P_t\) will converge to \(P^*\) as \(t \rightarrow \infty\) is when \(0 < \frac{d}{b} < 1\). f. Graphing the model and analyzing it will show a cobweb-like pattern, which demonstrates the lag in producer response in the supply and demand model.

Step-by-step solution

Set quantity demanded equal to quantity supplied

At equilibrium, the quantity demanded and the quantity supplied are equal, so we have: \(a - b P^* = c + d P^*\).

Solve for \(P^*\)

Now, we will solve the equation for the equilibrium price \(P^*\): \(a - b P^* - c = d P^*\) \((b + d) P^* = a - c\) \(P^* = \frac{a-c}{b+d}\). #b. Value of \(P_1\)

Substitute \(P_0\) into the supply equation

Using the supply equation, substitute \(P_0\) for \(P_{t-1}\): \(Q_1^{S} = c + d P_0\).

Set quantity demanded equal to quantity supplied at \(t=1\)

At \(t=1\), we have: \(Q_1^{D} = a - b P_1 = Q_1^{S}\). Thus, \(a - b P_1 = c + d P_0\).

Solve for \(P_1\)

Now, we will solve for \(P_1\): \(P_1 = \frac{a - c - d P_0}{b}\). #c. Formula for \(P_t\)

Express \(P_t\) in terms of \(P_{t-1}\)

By equating the quantity demanded and supplied, we get the following relationship: \(P_t = \frac{a - c - d P_{t-1}}{b}\). By repeated substitution, we can obtain a formula for \(P_t\).

Express \(P_t\) recursively

As an example of how this is done, we will compute \(P_3\). 1. We have \(P_2 = \frac{a - c - d P_1}{b}\). 2. Similarly, \(P_3 = \frac{a - c - d P_2}{b}\). Substitute expression of \(P_2\) into expression for \(P_3\): \(P_3 = \frac{a - c - d \left(\frac{a - c - d P_1}{b}\right)}{b} = \frac{(a - c)(b + d) - d^2P_1}{b^2}\). We can see a pattern forming with each value of t as powers of d and fractions of b in the denominator. Using this pattern, we can write the general formula for \(P_t\) as: \(P_t = \frac{(a - c)(b + d)^{t-1} - d^t P_0}{b^t}\). #d. Restate \(P_t\) as a function of \(P_0, P^*\), and \(t\)

Substitute the equilibrium price \(P^*\) into the formula for \(P_t\)

Using the equilibrium price \(P^*\), we can rewrite the formula for \(P_t\): \(P_t = P^* + \frac{d^t (P_0 - P^*)}{b^t}\). #e. Convergence of \(P_t\) to \(P^*\) The sequence \(P_t\) will converge to \(P^*\) if \(P_t\) approaches \(P^*\) as \(t\rightarrow\infty\). From the expression derived in part d, we can see that for convergence to happen, the term \(\frac{d^t (P_0 - P^*)}{b^t}\) should approach zero as \(t\rightarrow\infty\). This will happen if \(0 < \frac{d}{b} < 1\). In this case, the sequence \(P_t\) will converge to \(P^*\) as \(t\rightarrow\infty\). #f. Graphing and Analysis Using the given values, we have: 1. \(P^* = \frac{a-c}{b+d} = \frac{4-1}{2+1} = 1\). 2. \(P_0 = 0\). Now we calculate a few iterations of \(P_t\): 1. \(P_1 = \frac{3}{2} = 1.5\). 2. \(P_2 = \frac{1}{2}\). 3. \(P_3 = 1\). Now, plotting these values and their iterations, we can observe a cobweb-like pattern in the graph. This is the origin of the term "cobweb model" in reference to this type of supply and demand model with lags in producers' responses.

Key concepts

Disequilibrium Prices

Disequilibrium prices occur when the market price is not at a level where supply equals demand. This situation is typical in models where there is a lag in the response of producers to market conditions. In our exercise, the quantity supplied is based on the price from a previous period, while the quantity demanded responds to the current price. This mismatch can lead to fluctuations in the price as the market seeks an equilibrium point which satisfies both consumers and producers.

A clear understanding of disequilibrium prices is crucial for students of economics, as it provides insight into how real markets can deviate from the idealized models of perfect competition. The cobweb model is a useful illustration of how disequilibrium can persist over time, especially in markets where production decisions are made in advance, such as agriculture or manufacturing with long lead times.

Supply and Demand Lag

Supply and demand lag is a core concept that explains the delay in the adjustment of supply and demand in response to price changes. In our textbook exercise, the lag is represented by the producers' supply being dependent on the previous period's price. This can often be seen in agricultural markets where the decisions about what and how much to plant are based on past prices, not on real-time or future market information.

Understanding Lag in Markets

• Lag can cause oscillating prices as producers try to 'catch up' with the market.
• It provides context as to why prices don't always immediately adjust to changes in demand or supply conditions.
• Learning about the lag helps students to understand market dynamics in industries where production cannot be rapidly altered in the short term.

Equilibrium Price Calculation

Calculating the equilibrium price is a fundamental skill in economics, referring to the price at which the quantity demanded by consumers equals the quantity supplied by producers. In the provided exercise, the equilibrium price calculation is obtained by setting the quantity demanded equal to the quantity supplied and solving for the price. The equilibrium price, denoted by \(P^*\), serves as a reference point for understanding how prices will adjust from different starting conditions over time.

Understanding how to calculate the equilibrium price helps students predict the long-term outcome in a market. It also forms the basis for further analysis, such as the impact of government interventions or the effects of external shocks on market prices.

Convergence Condition

The convergence condition in the cobweb model refers to the specific circumstances under which the prices in the model will stabilize towards the equilibrium price over time. In our problem, the condition for convergence is that the ratio of the producers' response coefficient to the demand slope coefficient, \(\frac{d}{b}\), must fall between 0 and 1. When this criterion is met, the oscillations of the price after each period become progressively smaller, eventually reaching the equilibrium price.

Key Points for Convergence

• Convergence is not guaranteed; certain market conditions must be met.
• Understanding convergence helps in analyzing the stability of various markets.
• Students learn to appreciate the dynamic nature of economic models and the importance of the underlying parameters that determine market outcomes.

Through examining the convergence condition, students can appreciate that in real-world markets, achieving equilibrium can be a complex and gradual process influenced heavily by the behavior of participants and the time lag in their responses to price changes.