Question

Suppose that the demand function for a good has the linear form \(Q=D(P, I)=a+b P+c I\) and the supply function is also of the linear form \(Q=S(P)=d+g P\). a. Calculate equilibrium price and quantity for this market as a function of the parameters \(a, b, c, d,\) and \(g\) and of \(I\) (income), the exogenous shift term for the demand function. b, Use your results from part (a) to calculate the comparative statics derivative \(d P^{*} / d I\) c. Now calculate the same derivative using the comparative statics analysis of supply and demand presented in this chapter. You should be able to show that you get the same results in each case. d. Specify some assumed values for the various parameters of this problem and describe why the derivative \(d P^{*} / d I\) takes the form it does here.

Short answer

Answer: In this market, as income increases, the equilibrium price increases as well. Specifically, for each one-unit increase in income, the equilibrium price goes up by 1/6 units. This suggests that consumers are willing to pay more for the good as their income increases.

Step-by-step solution

Equate Demand and Supply Functions to find Equilibrium

First, we need to find the equilibrium price and quantity. At equilibrium, the quantity demanded is equal to the quantity supplied. So, we can equate the demand and supply functions: \(Q = D(P, I) = a + bP + cI\) \(Q = S(P) = d + gP\) Now, equating demand and supply functions: \(a + bP + cI = d + gP\)

Solve for Equilibrium Price and Quantity

Now, we need to solve for the equilibrium price (P) and quantity (Q) in terms of given parameters (a, b, c, d, g) and income (I). To solve for P, rearrange the equation: \(P(b - g) = d - a - cI\) Now divide by \((b - g)\): \(P = \frac{d - a - cI}{b - g}\) Now that we have the equilibrium price, we can find the equilibrium quantity by plugging the equilibrium price into either the demand or supply function. Let's use the demand function: \(Q = a + b\left(\frac{d - a - cI}{b - g}\right) + cI\)

Find the Comparative Statics Derivative dP*/dI

Now, let's find the derivative of the equilibrium price with respect to income, I: \(\frac{dP}{dI} = \frac{-c}{b - g}\)

Verify the Result using Comparative Statics Analysis

To verify that our results are correct, we can use the comparative statics analysis of supply and demand: The demand and supply functions are given by: \(Q_d = D(P, I) = a + bP + cI\) \(Q_s = S(P) = d + gP\) Take the derivative of both the demand and supply functions with respect to P and I: \(\frac{\partial Q_d}{\partial P} = b\) \(\frac{\partial Q_d}{\partial I} = c\) \(\frac{\partial Q_s}{\partial P} = g\) Using the comparative statics formula: \(\frac{dP}{dI} = -\frac{\partial Q_d / \partial I}{\partial Q_d / \partial P - \partial Q_s / \partial P} = -\frac{c}{b - g}\) We can see that both methods yield the same result, which means our analysis is correct.

Assigning Values to Parameters and Explaining the Derivative Form

Let's assume the following parameter values: \(a = 100, b = -1, c = 0.5, d = 50,\) and \(g = 2\) The derivative of the equilibrium price with respect to income is then: \(\frac{dP}{dI} = \frac{-0.5}{(-1) - 2} = \frac{-0.5}{-3} = \frac{1}{6}\) This means that for each one-unit increase in income (I), the equilibrium price (P) will increase by \(\frac{1}{6}\). It implies that as the consumer's income increases, they are willing to pay a higher price for the good, which leads to an increase in the equilibrium price.