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

$$Solution$$

$demand \ equation\ is : Q_x\\^D = s - kPx - jM\\ supply\ equation\ is : Q_x{\\^s} = -h + bPx + c W\\$

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

$At\ equilibrium, \ Qx^d = Qx^S$

$s - kPx - jM = -h + bPx + c W$

$bPx + kPx = (s - jM) + (h - c W)$

$Px(b + k) =( s - jM )+ (h - c W)$

$P x^* =\frac {s - jM + h - cW}{b +k}$

$Q^* =\frac{s-k(s-jM+h-cW)}{(b+k)-jM}$

$Q* =\frac {s-(ks-kjM+kh-kcW)}{(b+k)-jM}$

$Here, Px^* and\ Q^* are\ the\ equilibrium \ price\ and\ quantity.$

$If \ income\ (M) \ changes\ by\ dM,$

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

$\frac{dPx^*}{dM} =\frac{ -j}{b + k}< 0$

$dQ^* = \frac{kjdM}{[b + k] - jdM}$

$dQ^* = dM[\frac{(kj)}{[b + k] - j}]$

$\frac{dQ^*}{dM} =\frac{ kj}{(b + k) }- j$

$\frac{dQ^*}{dM} = j[\frac{(k}{(b + k) - 1}]$

$\frac{dQ^*}{dM} = \frac{-bj}{(b + k)} < 0$

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

soln b

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

soln c