Demand function is
Q=P^(-1.5)I^(0.3)
where P is price, and I is income
If the goal is to keep Q constant, how much must income change if there is a 2% increase in price.
To find out how much income must change in order to keep the quantity(Q) constant in response to a 2% increase in price (P), we can use the demand function and solve for the change in income (ΔI).
Given:
Demand function: Q = P^(-1.5)I^(0.3)
Price increase: ΔP = 2% = 0.02 (as a decimal)
Step 1: Calculate the new price (P') after the 2% increase in price.
P' = P + ΔP = P + 0.02
Step 2: Substitute the new price (P') into the demand function.
Q = P'^(-1.5)I^(0.3)
Step 3: Substitute the initial price (P) into the demand function to represent the constant quantity (Q).
Q = P^(-1.5)I^(0.3)
Step 4: Set the two demand equations equal to each other since we want to keep Q constant.
P^(-1.5)I^(0.3) = P'^(-1.5)I^(0.3)
Step 5: Simplify the demand equation by substituting P' with P + 0.02.
P^(-1.5)I^(0.3) = (P + 0.02)^(-1.5)I^(0.3)
Step 6: Take the natural log (ln) of both sides to eliminate the exponents.
ln(P^(-1.5)I^(0.3)) = ln((P + 0.02)^(-1.5)I^(0.3))
Step 7: Use the logarithmic power rule to simplify the equation.
-1.5ln(P) + 0.3ln(I) = -1.5ln(P + 0.02) + 0.3ln(I)
Step 8: Since we are looking for the change in income (ΔI), we need to isolate ln(I) on one side of the equation.
-1.5ln(P) - 1.5ln(P + 0.02) = 0.3ln(I) - 0.3ln(I)
Step 9: Combine the ln(I) terms and simplify.
-1.5ln(P) - 1.5ln(P + 0.02) = 0
Step 10: Solve for ΔI by isolating ln(P + 0.02) on one side of the equation.
-1.5ln(P + 0.02) = 1.5ln(P)
Step 11: Divide both sides of the equation by -1.5.
ln(P + 0.02) = -ln(P)
Step 12: Use the logarithmic property to rewrite the equation.
ln(P + 0.02) = ln(1/P)
Step 13: Set the expressions inside the logarithms equal to each other.
P + 0.02 = 1/P
Step 14: Solve for P.
P^2 + 0.02P - 1 = 0
Step 15: Solve the quadratic equation to find the value of P.
Using the quadratic formula: P = (-0.02 ± √(0.02^2 - 4(1)(-1))) / (2(1))
P ≈ 0.9804 or P ≈ -1.0204 (ignore the negative value)
Step 16: Substitute the value of P back into the demand function to find the initial income.
Q ≈ (0.9804)^(-1.5)I^(0.3)
Step 17: Solve for I to find the initial income.
I ≈ (Q / (0.9804)^(-1.5))^(1/0.3)
Now, let's plug in the given constant value of Q and solve for I.
Please provide the value of Q to continue.