2. Suppose that the quantity of corn supplied depends on the price of corn, p; and the amount of rainfall, R: The demand for corn depends on the price of corn, p; and the level of income, Y: The equations that describe the supply and demand relationships are
Qs = 20R+100p Qd = 4000␣100p+10Y
(a) Sketch a graph of supply and demand curves that shows the e§ect of an increase in rainfall on the equilibrium price and quantity of corn. Please label your axes, and all curves, indicating the direction of shift of one or both curves as appropriate.
(b) Sketch a graph of supply and demand curves that shows the e§ect of an increase in income on the equilibrium price and quantity of corn. Please label your axes, and all curves, indicating the direction of shift of one or both curves as appropriate.
(c) Solve for equilibrium price and quantity as functions of R and Y:
(d) Comparative statics: how does the equilibrium price of corn change as R changes? How does the equilibrium price of corn change as Y changes? Interpret these results.
(e) Comparative statics: how does equilibrium quantity change as R changes? How does equilibrium quantity change as Y changes? In- terpret these results.
(a) To sketch the graph of supply and demand curves, we need to plot the quantity of corn (Q) on the vertical axis and the price of corn (p) on the horizontal axis. Let's start with the supply curve.
The supply equation is Qs = 20R + 100p. We can rearrange this equation to solve for p: p = (Qs - 20R) / 100. This equation shows that the price of corn depends on the quantity supplied (Qs) and the amount of rainfall (R).
Now let's plot the supply curve. Choose some arbitrary values for R, such as R = 0, R = 10, and R = 20. Substitute these values into the equation for p to find the corresponding prices. Plot these points on the graph and connect them to form a supply curve.
Next, let's move on to the demand curve. The demand equation is Qd = 4000 - 100p + 10Y. Again, rearrange this equation to solve for p: p = (4000 - Qd + 10Y) / 100. This equation shows that the price of corn depends on the quantity demanded (Qd) and the level of income (Y).
Similarly, choose some arbitrary values for Y, such as Y = 0, Y = 100, and Y = 200. Substitute these values into the equation for p to find the corresponding prices. Plot these points on the graph and connect them to form a demand curve.
Now, find the intersection of the supply and demand curves. This point represents the equilibrium price and quantity of corn. Label this point on your graph.
If the rainfall (R) increases, the supply curve will shift to the right because a higher level of rainfall is expected to increase the quantity supplied at each price. On the graph, this means that the supply curve will shift outward from its original position.
(b) To show the effect of an increase in income (Y) on the equilibrium price and quantity of corn, we need to keep the supply curve constant and only focus on the demand curve.
If income (Y) increases, the demand curve will shift to the right because consumers are willing to buy more corn at each price level. On the graph, this means that the demand curve will shift outward from its original position.
(c) To solve for the equilibrium price and quantity as functions of R and Y, we need to find the point where the supply and demand curves intersect. This point represents the equilibrium.
Equating the supply and demand equations, we get: 20R + 100p = 4000 - 100p + 10Y
Simplifying this equation, we find: 200p = 4000 - 20R + 10Y
To solve for p, we divide by 200: p = (4000 - 20R + 10Y) / 200
To solve for Q, substitute this value of p into either the supply or demand equation.
(d) To analyze comparative statics, we examine how the equilibrium price of corn changes as R changes and as Y changes.
When R increases, the equilibrium price of corn will decrease. This is because an increase in rainfall leads to an increase in the quantity supplied at each price level. As a result, the supply curve shifts outward, causing a decrease in the equilibrium price.
When Y increases, the equilibrium price of corn will increase. This is due to an increase in income, which leads to an increase in the quantity demanded at each price level. Consequently, the demand curve shifts outward, resulting in a higher equilibrium price.
(e) Similarly, we can analyze how the equilibrium quantity changes as R and Y change.
When R increases, the equilibrium quantity of corn will increase. This is because a higher level of rainfall leads to an increase in the quantity supplied. As a result, the supply curve shifts outward, causing an increase in the equilibrium quantity.
When Y increases, the equilibrium quantity of corn will increase. This is due to an increase in income, which leads to an increase in the quantity demanded. Consequently, the demand curve shifts outward, resulting in a higher equilibrium quantity.
In both cases, an increase in either rainfall or income leads to an increase in the equilibrium quantity of corn, indicating a positive relationship between these factors and the quantity produced and demanded.