SOLUTION: A demand equation of gasoline is given as QD(G)= 7.5 - 0.5p(G) + 0.1(I) - 0.5P(A). Given that a consumer income (I) and price of automobiles (PA) are constant at$60 per year and $3

Question 1178454: A demand equation of gasoline is given as QD(G)= 7.5 - 0.5p(G) + 0.1(I) - 0.5P(A). Given that a consumer income (I) and price of automobiles (PA) are constant at$60 per year and $30 respectively.
(a) Express the equation as a relationship between quantity of gasoline demand and gasoline price.
(b) calculate the Y-intercept of the equation in (a) above.
(c) if per household annual income were to fall to$50 per year, while price of automobiles remain constant, show that, there will be an inward shift in the demand curve and not a movement along the demand curve.
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's break down this demand equation problem step-by-step:
**(a) Expressing the Equation as a Relationship Between QD(G) and p(G)**
Given:
* QD(G) = 7.5 - 0.5p(G) + 0.1(I) - 0.5P(A)
* I = $60
* P(A) = $30
Substitute the values of I and P(A) into the equation:
QD(G) = 7.5 - 0.5p(G) + 0.1(60) - 0.5(30)
QD(G) = 7.5 - 0.5p(G) + 6 - 15
QD(G) = 7.5 + 6 - 15 - 0.5p(G)
QD(G) = -1.5 - 0.5p(G)
So, the equation relating quantity demanded of gasoline (QD(G)) to its price (p(G)) is:
**QD(G) = -0.5p(G) - 1.5**
**(b) Calculating the Y-intercept**
The Y-intercept is the value of QD(G) when p(G) = 0.
Substitute p(G) = 0 into the equation:
QD(G) = -0.5(0) - 1.5
QD(G) = -1.5
Therefore, the Y-intercept is **-1.5**.
**(c) Showing an Inward Shift with Income Change**
Now, let's change the income (I) to $50, while keeping P(A) constant at $30.
Substitute I = $50 and P(A) = $30 into the original equation:
QD(G) = 7.5 - 0.5p(G) + 0.1(50) - 0.5(30)
QD(G) = 7.5 - 0.5p(G) + 5 - 15
QD(G) = 7.5 + 5 - 15 - 0.5p(G)
QD(G) = -2.5 - 0.5p(G)
So, the new equation is:
**QD(G) = -0.5p(G) - 2.5**
**Comparison:**
* Original equation (I = $60): QD(G) = -0.5p(G) - 1.5
* New equation (I = $50): QD(G) = -0.5p(G) - 2.5
**Explanation:**
1. **Parallel Shift:** Notice that the slope (-0.5) remains the same in both equations. This indicates that the new demand curve is parallel to the original demand curve.
2. **Y-intercept Change:** The Y-intercept has changed from -1.5 to -2.5. This means that for any given price, the quantity demanded is now lower.
3. **Inward Shift:** Since the Y-intercept has decreased, the entire demand curve has shifted downward (inward) on the graph. This shift represents a decrease in demand at every price level due to the decrease in income.
4. **Not a Movement Along the Curve:** A movement along the demand curve would only occur if the price of gasoline (p(G)) changed. Since the price of gasoline remains a variable in the equation and the income changed, the whole curve shifts.
**Conclusion:**
The decrease in household income from $60 to $50 results in an inward shift of the demand curve for gasoline. This shows that a change in income, a non-price determinant of demand, causes a shift of the demand curve, not a movement along it.

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