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 \(E\left(P_{t}\right) ?\)

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

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: $$100 - 2P_{t} = 70 + P_{t}$$

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

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

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

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: $$Q_{t} = 70 + E\left(P_{t}\right) = 70 + P_{t-1}$$

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.

(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

"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.

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.

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.