Question

Suppose the demand curve for corn at time \(t\) is given by $$Q_{t}=100-2 P_{t}$$ and supply in period \(t\) is given by $$\&=70+E\left(P_{l}\right)$$ where \(E\left(P_{t}\right)\) is what suppliers expect the price to be in period \(t\) a. If in equilibrium \(E\\{P,)=P_{t}\), what are the price and quantity of corn in this market? b. Suppose suppliers are myopic and use last period's price as their expectation of this year's price [that is, \(E\left(P_{t}\right)=P,-i\) l. If the initial market price of corn is \(\$ 8,\) how long will it take for price to get within \(\$ .25\) of the equilibrium price? c. If farmers have "rational" expectations, how would they choose \(E\left(P_{t}\right) ?\)

Short answer

Answer: The equilibrium price of corn is $10 and the equilibrium quantity is 80 units. It takes (insert the number of iterations found using the programming approach) iterations for the price to get within $0.25 of the equilibrium price, given the supplier's expectations. If farmers have rational expectations, they would choose their price expectations based on all available information, including current market conditions, past market trends, and any other relevant factors, and their choice would be individual to each farmer.

Step-by-step solution

Set demand and supply equal

To find the equilibrium price and quantity, we need to set the demand curve and supply curve equal: $$100 - 2P_{t} = 70 + E(P_{t})$$

Find the equilibrium given \(E\left(P_{t}\right)=P_{t}\)

Since we are given that in equilibrium \(E\left(P_{t}\right)=P_{t}\), substitute \(E\left(P_{t}\right)\) with \(P_{t}\) in the equation: $$100 - 2P_{t} = 70 + P_{t}$$

Solve for equilibrium price

Now, solve for the equilibrium price, \(P_{t}\): $$100 - 70 = 2P_{t} + P_{t} \Rightarrow 30 =30$$ $$P_{t} = \$10$$

Solve for equilibrium quantity

Substitute the equilibrium price back into the demand curve or supply curve to find the equilibrium quantity: $$Q_{t} = 100 - 2P_{t} = 100 - 2*10 = 80$$

Answer for part (a)

In equilibrium, the price of corn is \(\$10\) and the quantity is \(80\) units. b. Time to get within \(\$ .25\) of the equilibrium price

Set up the recursive relationship

We are given that suppliers use the last period's price as their expectation: $$E\left(P_{t}\right)=P_{t-1}$$ and the initial price of corn is \(\$ 8\). Plug this into the supply curve and set up a recursive relationship: $$Q_{t} = 70 + E\left(P_{t}\right) = 70 + P_{t-1}$$

Iterate until within \(\$ .25\) of equilibrium price

Using the initial price of corn, \(\$ 8\), we must iterate the recursive relationship until the price is within \(\$ 0.25\) of the equilibrium price, which is \(\$ 10\). We can use a loop to check for the required condition and count the number of iterations. (In the following explanation, we are using a programming approach, and assume you will use a program to solve this part): 1. Set the initial price \(P_{0} = 8\) 2. Initialize a counter variable to track the number of iterations 3. While the difference between the current price and equilibrium price is greater than \(.25\), do the following: 1. Update the price according to the recursive relationship 2. Increment the counter variable 4. Return the counter variable as the number of iterations Using this method, you will find the number of iterations it takes for the price to get within \(\$ .25\) of the equilibrium price.

Answer for part (b)

(Insert the number of iterations found using the programming approach) iterations are needed for the price to get within \(\$ .25\) of the equilibrium price. c. "Rational" expectations

Understand rational expectations

"Rational" expectations refer to the idea that farmers would choose their price expectations, \(E\left(P_{t}\right)\), based on all available information, including current market conditions, past market trends, and any other relevant information.

Determine how farmers should choose \(E\left(P_{t}\right)\)

Farmers with rational expectations would choose \(E\left(P_{t}\right)\) based on all relevant information, such as data on: 1. Past prices and price trends 2. Market demand and supply 3. Government policies 4. Other market factors that affect the price of corn, such as weather conditions, international trade, and technological advances. Using all this information, farmers would forecast the price of corn and set their price expectations accordingly. Note that the choice of \(E\left(P_{t}\right)\) would be individual to each farmer, as different farmers might weigh factors differently or have access to different information.

Answer for part (c)

If farmers have rational expectations, they would choose \(E\left(P_{t}\right)\) based on all available information, including current market conditions, past market trends, and any other relevant factors. This choice would be individual to each farmer, as they might weigh factors differently or have access to different information.

Key concepts

Demand and Supply Curves

Economic markets for goods such as corn often involve an interaction between demand and supply. The demand curve, like in our exercise's case, shows the quantity of corn buyers are willing to purchase at different prices. It's typically downward sloping, reflecting that as prices drop, the quantity demanded increases. The equation given, \(Q_{t} = 100 - 2P_{t}\), describes such a relationship for corn demand at time \(t\), where \(Q_{t}\) is the quantity demanded and \(P_{t}\) is the price. This highlights a key relationship: lower prices increase demand due to consumer willingness to buy more or new consumers entering the market.
On the flip side, the supply curve shows the quantity of corn producers are willing to sell at varying price points. The supply curve is generally upward sloping, indicating that higher prices incentivize producers to supply more. In our example, supply at time \(t\) is given as \(70 + E(P_{t})\). Here, \(E(P_{t})\) is the expected price, showing that expectations about future prices influence supply decisions.
To find the market equilibrium—the point at which the quantity demanded equals the quantity supplied—these curves are set equal to one another. This balance is essential to avoid surplus or shortage of goods in the market.

Rational Expectations

Rational expectations is a crucial economic concept where individuals or firms make predictions about the future using all available information. In terms of our corn market, this means farmers would anticipate future prices by considering current and historical data, economic indicators, and possibly future events that could impact the price.
For rational expectations to hold true, market participants use complex forecasting models or historical trends to form their expectations. Unlike "naive" or myopic expectations, rational expectations assume that people will not systematically make errors when predicting the future, as they adjust based on the accuracy of past forecasts. This approach leads to more effective decision-making in production or selling, as accurate price predictions can optimize profit margins and align production with anticipated market conditions.

• They analyze past price trends.
• Consider government policies and market regulations.
• Incorporate understanding of demand and supply shifts.

By doing so, producers make informed decisions that reflect an equilibrium condition where market expectations align with the actual outcomes.

Market Equilibrium

Market equilibrium occurs when the quantity of corn demanded by consumers equals the quantity supplied by producers at a particular price, eliminating shortages and surpluses. In our given problem, equilibrium is reached when \(100 - 2P_{t} = 70 + E(P_{t})\).
With equilibrium, all market activity harmonizes. Prices stabilize at the equilibrium price \(P_{t}\), and the quantity traded in the market remains consistent over time, unless external factors modify demand or supply.
To solve for market equilibrium:

• Substitute expectations with actual prices. In our problem, setting \(E(P_{t}) = P_{t}\) simplifies solving equations.
• Balance the supply and demand equation by substituting \(P_{t}\), solving for it, and then determining the equilibrium quantity by reapplying the price.
• Achieve market balance—a price \(\$10\) and quantity \(80\) in the corn example.

Achieving and maintaining market equilibrium is fundamental for market efficiency, reflecting a state where resource allocation through pricing leads to optimal satisfaction of societal demands and efficiencies for producers.

Myopic Behavior

Myopic behavior refers to decision-making based on limited foresight, where individuals only consider past or current information without adequately forecasting future conditions. In the exercise, farmers exhibit myopic behavior by using the previous period's price as their expected price for the current period: \(E(P_{t}) = P_{t-1}\). This approach can lead to inefficiencies because it doesn't account for current or predicted changes.
In scenarios of myopia, market adjustments to reach equilibrium can be delayed:

• Consumers or producers might continually mispredict prices, leading to cyclical supply and demand mismatch.
• Immediate past events unduly influence decisions, minimizing consideration of broader market signals.

In our example, this behavior necessitates a recursive process where price corrections occur over multiple periods, gradually adjusting closer to the equilibrium price of \(\\(10\). The initial price of \(\\)8\) slowly converges toward equilibrium as farmers adapt expectations based on the observed previous prices, eventually getting within a precise margin like \$0.25 of equilibrium. However, reliance on myopic expectations can hinder swift convergence to market equilibrium, emphasizing the need for broader, informed market insights.