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Chapter 12: Problem 12

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

Expert verified
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

01

Set quantity demanded equal to quantity supplied

At equilibrium, the quantity demanded and the quantity supplied are equal, so we have: \(a - b P^* = c + d P^*\).
02

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 \(P_1\)
03

Substitute \(P_0\) into the supply equation

Using the supply equation, substitute \(P_0\) for \(P_{t-1}\): \(Q_1^{S} = c + d P_0\).
04

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, \(a - b P_1 = c + d P_0\).
05

Solve for \(P_1\)

Now, we will solve for \(P_1\): \(P_1 = \frac{a - c - d P_0}{b}\). #c. Formula for \(P_t\)
06

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 \(P_t\).
07

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 \(t\)
08

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

The perfectly competitive videotape-copying industry is composed of many firms that can copy five tapes per day at an average cost of \(\$ 10\) per tape. Each firm must also pay a royalty to film studios, and the per-film royalty rate \((r)\) is an increasing function of total industry output (Q): \\[ r=0.002 Q \\] Demand is given by \\[ Q=1,050-50 P \\] a. Assuming the industry is in long-run equilibrium, what will be the equilibrium price and quantity of copied tapes? How many tape firms will there be? What will the per-film royalty rate be? b. Suppose that demand for copied tapes increases to \\[ Q=1,600-50 P \\] In this case, what is the long-run equilibrium price and quantity for copied tapes? How many tape firms are there? What is the per-film royalty rate? c. Graph these long-run equilibria in the tape market, and calculate the increase in producer surplus between the situations described in parts (a) and (b). d. Show that the increase in producer surplus is precisely equal to the increase in royalties paid as \(Q\) expands incrementally from its level in part (b) to its level in part (c). e. Suppose that the government institutes a \(\$ 5.50\) per-film tax on the film-copying industry. Assuming that the demand for copied films is that given in part (a), how will this tax affect the market equilibrium? f. How will the burden of this tax be allocated between consumers and producers? What will be the loss of consumer and producer surplus? g. Show that the loss of producer surplus as a result of this tax is borne completely by the film studios. Explain your result intuitively.

The handmade snuffbox industry is composed of 100 identical firms, each having short-run total costs given by \\[ S T C=0.5 q^{2}+10 q+5 \\] and short-run marginal costs given by \\[ S M C=q+10 \\] where \(q\) is the output of snuffboxes per day. a. What is the short-run supply curve for each snuffbox maker? What is the short-run supply curve for the market as a whole? b. Suppose the demand for total snuffbox production is given by \\[ Q=1,100-50 P \\] What will be the equilibrium in this marketplace? What will each firm's total short-run profits be? c. Graph the market equilibrium and compute total short-run producer surplus in this case. d. Show that the total producer surplus you calculated in part (c) is equal to total industry profits plus industry short-run fixed costs. e. Suppose the government imposed a \(\$ 3\) tax on snuffboxes. How would this tax change the market equilibrium? f. How would the burden of this tax be shared between snuffbox buyers and sellers? g. Calculate the total loss of producer surplus as a result of the taxation of snuffboxes. Show that this loss equals the change in total short-run profits in the snuffbox industry. Why do fixed costs not enter into this computation of the change in short-run producer surplus?

Suppose that the market demand for a product is given by \(Q_{D}=A-B P .\) Suppose also that the typical firm's cost function is given by \(C(q)=k+a q+b q^{2}\) a. Compute the long-run equilibrium output and price for the typical firm in this market. b. Calculate the equilibrium number of firms in this market as a function of all the parameters in this problem. c. Describe how changes in the demand parameters \(A\) and \(B\) affect the equilibrium number of firms in this market. Explain your results intuitively. d. Describe how the parameters of the typical firm's cost function affect the long-run equilibrium number of firms in this example. Explain your results intuitively.

The domestic demand for portable radios is given by $$Q=5,000-100 \mathrm{P}$$ where price \(\mathrm{(P)}\) is measured in dollars and quantity \(\mathrm{(Q)}\) is measured in thousands of radios per year. The domestic supply curve for radios is given by $$Q=150 \mathrm{P}$$ a. What is the domestic equilibrium in the portable radio market? b. Suppose portable radios can be imported at a world price of $$\$ 10$$ per radio. If trade were unencumbered, what would the new market equilibrium be? How many portable radios would be imported? c. If domestic portable radio producers succeeded in having a $$\$ 5$$ tariff implemented, how would this change the market equilibrium? How much would be collected in tariff revenues? How much consumer surplus would be transferred to domestic producers? What would the deadweight loss from the tariff be? d. How would your results from part (c) be changed if the government reached an agreement with foreign suppliers to "voluntarily" limit the portable radios they export to \(1,250,000\) per year? Explain how this differs from the case of a tariff.

Throughout this chapter's analysis of taxes we have used per-unit taxes-that is, a tax of a fixed amount for each unit traded in the market. A similar analysis would hold for ad valorem taxes-that is, taxes on the value of the transaction (or, what amounts to the same thing, proportional taxes on price). Given an ad valorem tax rate of \(t(t=0.05 \text { for a } 5\) percent tax), the gap between the price demanders pay and what suppliers receive is given by \(P_{D}=(1+t) P_{S}\) a. Show that for an ad valorem tax \\[ \frac{d \ln P_{D}}{d t}=\frac{e_{S}}{e_{S}-e_{D}} \quad \text { and } \quad \frac{d \ln P_{S}}{d t}=\frac{e_{D}}{e_{S}-e_{D}} \\] b. Show that the excess burden of a small tax is \\[ D W=-0.5 \frac{e_{D} e_{S}}{e_{S}-e_{D}} t^{2} P_{0} Q_{0} \\] c. Compare these results with those derived in this chapter for a unit tax.
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