Answer to Question #221107 in Macroeconomics for Isaac mensah

Question

If the supply and demand functions are given by and, respectively, find the equilibrium price and quantity, and calculate the consumer’s and producer’s surplus. 

Answer 1

Given,

Supply function:

$p=20e^{ 0.4Q}$

Supply function:

$p=100e^ {−0.2Q}$

Note: the values will be in approx as the exponential function is given.

At equilibrium, quantity demanded is equal to the quantity supplied:

$20e^ {0.4Q} =100e^ {−0.2Q}\\ \frac{e^ {0.4Q}}{ e^ {−0.2Q}} =5 \\e^ {0.4Q+0.2Q} =5\\ e^ {0.6Q} =5\\ 0.6Q=ln5 (e^ {x} =y⇒x=ln(y))\\ 0.6Q=1.6094\\ Q=\frac{1.6094} {0.6}\\$

$Q=2.68 =3$ (nearest integer )

Substitute Q=2.68 in the supply function for price

$p=20×e^ { 0.4×2.68}$

$p=20×2.9212$

$p=58.424 = 58$ (nearest integer)

Note: nearest integer has been taken for easy calculation.

Equilibrium price = 58 

Equilibrium quantity = 3

The formula for consumer surplus (CS) is given below:

$CS=∫^Q_0p(Q)dQ−p×Q (p(Q)=Demand \space function)\\CS=∫^{3}_0(100e^{−0.2Q})dQ−58×3\\CS=100[\frac {e^{−0.2Q}}{−0.2}]^3_0−174\\CS=\frac{100}{−0.2}[e^{−0.2×3}−e^{−0.2×0}]−174\\CS=−500[0.549−1]−174\\CS=−274.5+500−174\\CS=51.5 \space approx$

The value of consumer surplus is 51.5 approx.

The formula for producer surplus (PS) is given below:

$PS=p×Q−∫^Q_0p(Q)dQ (p(Q)=Supply \space function)\\PS=58×3−∫^3_0(20e^{0.4Q})dQ\\PS=174−20[\frac{e^{0.4Q}}{0.4}]^3_0\\PS=174−\frac{20}{0.4}[e^{0.4×3}−e^{0.4×0}]\\PS=174−50[3.320−1]\\PS=174−166+50\\PS=58 The value of pro$

The value of producer surplus is 58 approx.