Answer to Question #221205 in Macroeconomics for Ben

Question #221205

P=20e^0.4Q and P=100e^-0.2 find equilibrium price and quantity and also consumer and producer surplus


1
Expert's answer
2021-07-29T09:36:01-0400

Given,

Supply function:

p=20e0.4Qp=20e^{ 0.4Q}

Supply function:

p=100e0.2Qp=100e^ {−0.2Q}

Because the exponential function is supplied, the values will be approximate.

The quantity required equals the quantity provided in equilibrium:

20e0.4Q=100e0.2Qe0.4Qe0.2Q=5e0.4Q+0.2Q=5e0.6Q=50.6Q=ln5(ex=yx=ln(y))0.6Q=1.6094Q=1.60940.620e^ {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=3Q=2.68 =3 (nearest integer )

Substitute Q=2.68 in the supply function for price

p=20×e0.4×2.68p=20×e^ { 0.4×2.68}

p=20×2.9212p=20×2.9212

p=58.424=58p=58.424 = 58 (nearest integer)


For ease of computation, the closest integer has been used.

Equilibrium price = 58 

Equilibrium quantity = 3

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

CS=0Qp(Q)dQp×Q(p(Q)=Demand function)CS=03(100e0.2Q)dQ58×3CS=100[e0.2Q0.2]03174CS=1000.2[e0.2×3e0.2×0]174CS=500[0.5491]174CS=274.5+500174CS=51.5 approxCS=∫^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×Q0Qp(Q)dQ(p(Q)=Supply function)PS=58×303(20e0.4Q)dQPS=17420[e0.4Q0.4]03PS=174200.4[e0.4×3e0.4×0]PS=17450[3.3201]PS=174166+50PS=58ThevalueofproPS=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.


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