Answer to Question #220465 in Macroeconomics for Darryl

Question

If the supply and demand functions are given by  p=20e^0.4Q and p=100e^-0.2Q, 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.