Given,
Supply function:
p=20e0.4Q
Supply function:
p=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=100e−0.2Qe−0.2Qe0.4Q=5e0.4Q+0.2Q=5e0.6Q=50.6Q=ln5(ex=y⇒x=ln(y))0.6Q=1.6094Q=0.61.6094
Q=2.68=3 (nearest integer )
Substitute Q=2.68 in the supply function for price
p=20×e0.4×2.68
p=20×2.9212
p=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)dQ−p×Q(p(Q)=Demand function)CS=∫03(100e−0.2Q)dQ−58×3CS=100[−0.2e−0.2Q]03−174CS=−0.2100[e−0.2×3−e−0.2×0]−174CS=−500[0.549−1]−174CS=−274.5+500−174CS=51.5 approx
The value of consumer surplus is 51.5 approx.
The formula for producer surplus (PS) is given below:
PS=p×Q−∫0Qp(Q)dQ(p(Q)=Supply function)PS=58×3−∫03(20e0.4Q)dQPS=174−20[0.4e0.4Q]03PS=174−0.420[e0.4×3−e0.4×0]PS=174−50[3.320−1]PS=174−166+50PS=58Thevalueofpro
The value of producer surplus is 58 approx.
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