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Problem 1

# Calculate the consumers'surplus at the indicated unit price $$\bar{p}$$ for each of the demand equations. $$p=10-2 q ; \bar{p}=5$$

Expert verified
The consumer surplus at the indicated unit price $$\bar{p} = 5$$ is $$1.5625$$.
See the step by step solution

## Step 1: Find the inverse demand function

To find the inverse demand function, we need to solve the demand equation for $$q$$. The demand equation is given as $$p = 10 - 2q$$. Solving for $$q$$, we get: $$q = \frac{10 - p}{2}$$

## Step 2: Find the equilibrium quantity using the given unit price

To find the equilibrium quantity, substitute the given unit price $$\bar{p} = 5$$ into the inverse demand function we found in Step 1: $$q_e = \frac{10 - 5}{2}$$ $$q_e = \frac{5}{2} = 2.5$$

## Step 3: Calculate the consumer surplus

To calculate the consumer surplus, we need to integrate the difference between the inverse demand function and the given unit price from $$0$$ to the equilibrium quantity $$q_e$$: Consumer Surplus = $$\int_0^{q_e} (p - \bar{p}) dq$$ Substitute the inverse demand function and the given unit price: Consumer Surplus = $$\int_0^{2.5} \left(\frac{10 - p}{2} - 5\right) dq$$ We can simplify the integrand before evaluation: $$\left(\frac{10 - p}{2} - 5\right) = -\frac{p - 10}{2}$$ Now, evaluate the integral: Consumer Surplus = $$\int_0^{2.5} -\frac{p - 10}{2} dq$$ Consumer Surplus = $$\left[-\frac{1}{4}(p-10)^2\right]_0^{2.5}$$ Now, using the Fundamental theorem of calculus, we can evaluate the integral at the bounds: $$Consumer \; Surplus = -\frac{1}{4}(5-10)^2 + \frac{1}{4}(10-10)^2$$ $$Consumer \; Surplus = -\frac{1}{4}(5^2) + 0$$ $$Consumer \; Surplus = -\frac{25}{4}$$ Since the consumer surplus cannot be negative, we have made an error in our calculations. Let's go back to Step 3 and re-evaluate the integral. Upon re-evaluating, we realize that we made a mistake in substituting the inverse demand function, resulting in the wrong integrand. Let's substitute the inverse demand function correctly: Consumer Surplus = $$\int_0^{2.5} \left(\frac{10 - q}{2} - 5\right) dq$$ Now the integrand is correct, and we can simplify it before evaluation: $$\left(\frac{10-q}{2} - 5\right) = \frac{10 - q - 10}{2} = \frac{-q}{2}$$ Now, evaluate the integral: Consumer Surplus = $$\int_0^{2.5} -\frac{q}{2} dq$$ Consumer Surplus = $$\left[-\frac{1}{4}q^2\right]_0^{2.5}$$ Using the Fundamental theorem of calculus, we can evaluate the integral at the bounds: $$Consumer \; Surplus = -\frac{1}{4}(2.5)^2 - (-\frac{1}{4}(0)^2)$$ $$Consumer \; Surplus = -\frac{1}{4}(6.25)$$ $$Consumer \; Surplus = -\frac{6.25}{4} = -1.5625$$ Once again, we have obtained a negative consumer surplus, indicating that something is wrong with our calculations. We can identify the mistake in our previous steps: In Step 3, while calculating consumer surplus, we used: Consumer Surplus = $$\int_0^{q_e}(p-\bar{p})dq$$ However, this is incorrect. We should use: Consumer Surplus = $$\int_0^{q_e}(\bar{p}-p)dq$$ Thus, let's recompute the consumer surplus with the corrected integral: Consumer Surplus = $$\int_0^{2.5} \left(5 - \frac{10 - q}{2}\right) dq$$ Now the integrand is correct, and we can simplify it before evaluation: $$\left(5 - \frac{10 - q}{2}\right) = \frac{q}{2}$$ Now, evaluate the integral: Consumer Surplus = $$\int_0^{2.5} \frac{q}{2} dq$$ Consumer Surplus = $$\left[\frac{1}{4}q^2\right]_0^{2.5}$$ Using the Fundamental theorem of calculus, we can evaluate the integral at the bounds: $$Consumer \; Surplus = \frac{1}{4}(2.5)^2 - \frac{1}{4}(0)^2$$ $$Consumer \; Surplus = \frac{6.25}{4}$$ $$Consumer \; Surplus = 1.5625$$ The consumer surplus at the indicated unit price $$\bar{p} = 5$$ is \$1.5625.

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