Question

Q. Given the demand and supply equations: Q_x^d=s-kP_x-jM, Q_x^s=-h+bP_x + cW where M represents income and W represents the wage rate:

·  Calculate the impact of a change in income on the equilibrium price and quantity.

·  Will this impact be larger or smaller if the value of k is decreased?

Draw diagram(s) indicating all results

Answer 1

The demand equation is : $Q_x^D = s - kP_x - jM$

The supply equation is : $Q_x^S = -h + bP_x + cW$

Here, M represents income and W represents the wage rate.

At equilibrium:

$Q_x^D = Q_x^S s - kP_x - jM = -h + bP_x + cW \\ bP_x + kP_x = s - jM + h - cW \\ P_x(b + k) = s - jM + h - cW \\ P_x* = \frac{(s - jM + h - cW)}{(b + k)} \\ Q* = s - \frac{k(s - jM + h - cW)}{(b + k)} - jM \\ Q* = s - \frac{(ks - kjM + kh - kcW)}{(b + k)} - jM$

Here, $P_x*$ and Q* are the equilibrium price and quantity.

If income (M) changes by dM,

$dP_x* = \frac{-jdM}{(b + k)}$ (since, other parameters are constant)

$\frac{dP_x*}{dM} = \frac{-j}{(b + k)} < 0 \\ dQ* = \frac{kjdM}{(b + k)} - jdM \\ dQ* = dM(\frac{kj}{(b + k)} - j) \\ \frac{dQ*}{dM} = \frac{kj}{(b + k)} - j \\ \frac{dQ*}{dM} = j(\frac{k}{(b + k)} - 1) \\ \frac{dQ*}{dM} = \frac{-bj}{(b + k)} < 0$

Hence, After change in income (M) the equilibrium price and quantity decrease.

If value of k is decreased to k' then the $\frac{-j}{(b + k)} < \frac{-j}{(b + k')}$ and $\frac{-bj}{(b + k)} < -\frac{bj}{(b + k')}$