Find the consumers' surplus and the producers' surplus at the equilibrium price level for the given price-demand and price supply equations.
p = D(x) = 170e^-0.001x
p = S(x) = 35e^0.001x
a) The value of x at the equilibrium is ___
b) The value of p at equilibrium is $___
c) The consumer's surplus at equilibrium is $___
d) The producer's surplus at equilibrium is $___
To find the equilibrium price level, we need to set the price-demand equation (D(x)) equal to the price-supply equation (S(x)) and solve for x.
1) Set D(x) = S(x):
170e^(-0.001x) = 35e^(0.001x)
2) Divide both sides by 35:
(170/35) * e^(-0.001x) = e^(0.001x)
3) Take the natural logarithm (ln) of both sides to eliminate the exponential terms:
ln((170/35) * e^(-0.001x)) = ln(e^(0.001x))
4) Use the properties of logarithms to simplify:
ln(170/35) + ln(e^(-0.001x)) = ln(e^(0.001x))
5) Simplify further:
ln(170/35) - 0.001x = 0.001x
6) Combine like terms:
2 * ln(170/35) = 0.002x
7) Divide both sides by 0.002 to solve for x:
x = (2 * ln(170/35)) / 0.002
Now that we have the value of x at equilibrium, we can calculate the values of p (price), consumer's surplus, and producer's surplus.
a) The value of x at the equilibrium is (2 * ln(170/35)) / 0.002.
To find the value of p at equilibrium, substitute the value of x into either the price-demand equation (D(x)) or the price-supply equation (S(x)).
b) The value of p at equilibrium is p = 170e^(-0.001x) or p = 35e^(0.001x), where x is the value you obtained in step 7.
To find the consumer's surplus at equilibrium, we need to calculate the area under the demand curve above the equilibrium price level. We can use the definite integral to find this area.
c) The consumer's surplus at equilibrium is the integral of D(x) from x = 0 to the equilibrium value of x, subtracted from the equilibrium value of p times the equilibrium value of x:
Consumer's Surplus = ∫[0 to x](170e^(-0.001x)) dx - (p * x)
Substitute the value of x and p obtained in steps a) and b) into the above equation to find the consumer's surplus at equilibrium.
To find the producer's surplus at equilibrium, we need to calculate the area under the supply curve below the equilibrium price level.
d) The producer's surplus at equilibrium is the integral of S(x) from x = 0 to the equilibrium value of x, subtracted from the equilibrium value of p times the equilibrium value of x:
Producer's Surplus = (p * x) - ∫[0 to x](35e^(0.001x)) dx
Substitute the value of x and p obtained in steps a) and b) into the above equation to find the producer's surplus at equilibrium.