Suppose that g1 represents the ratio of year 2 GDP to year 1 GDP, both valued at year 1 prices. Suppose that g2 represents the ratio of year 2 GDP to year 1 GDP, both valued at year 2 prices.
The ratio of chain-weighted year 2 GDP to chain-weighted year 1 GDP equals
A) $\sqrt { g 1 \times g 2 }$.
B) $\sqrt { g 1 / g 1 }$.
C) $( g 1 + g 2 ) / 2$.
D) $( \sqrt { g 1 } + \sqrt { g 2 } ) / 2$.
E) $( g 1 \times g 2 ) / 2$.

Question

Suppose that g1 represents the ratio of year 2 GDP to year 1 GDP, both valued at year 1 prices. Suppose that g2 represents the ratio of year 2 GDP to year 1 GDP, both valued at year 2 prices.
The ratio of chain-weighted year 2 GDP to chain-weighted year 1 GDP equals 
A) $\sqrt { g 1 \times g 2 }$.
B) $\sqrt { g 1 / g 1 }$.
C) $( g 1 + g 2 ) / 2$.
D) $( \sqrt { g 1 } + \sqrt { g 2 } ) / 2$.
E) $( g 1 \times g 2 ) / 2$.

Quizplus · Accepted Answer

Chain-weighted GDP measures the change in real GDP from one year to the next, using a combination of current and base year prices. To calculate the ratio of chain-weighted year 2 GDP to chain-weighted year 1 GDP, we need to take into account both the year 1 and year 2 prices.

We can use the following formula to calculate chain-weighted GDP:

$ G D P = \left( \frac { G D P _ { t } } { P _ { t } } + \frac { G D P _ { t - 1 } } { P _ { t - 1 } } \right) \times \frac { 1 } { 2 } $

Where GDPt represents the GDP at current prices in year t, GDPt-1 represents the GDP at current prices in year t-1, Pt represents the price level in year t, and Pt-1 represents the price level in year t-1.

Using this formula, we can calculate the ratio of chain-weighted year 2 GDP to chain-weighted year 1 GDP as:

$ \frac { G D P _ { 2 } } { G D P _ { 1 } } = \frac { \left( \frac { G D P _ { 2 } } { P _ { 2 } } + \frac { G D P _ { 1 } } { P _ { 1 } } \right) \times \frac { 1 } { 2 } } { \left( \frac { G D P _ { 1 } } { P _ { 1 } } + \frac { G D P _ { 0 } } { P _ { 0 } } \right) \times \frac { 1 } { 2 } } $

Where GDP2 represents the real GDP in year 2 using chain-weighted prices, GDP1 represents the real GDP in year 1 using chain-weighted prices, P2 represents the chain-weighted price index in year 2, and P1 represents the chain-weighted price index in year 1.

We can simplify this ratio by canceling out the factor of 1/2 and multiplying numerator and denominator by P1:

$ \frac { G D P _ { 2 } } { G D P _ { 1 } } = \frac { \left( G D P _ { 2 } \cdot \frac { P _ { 1 } } { P _ { 2 } } + G D P _ { 1 } \right) } { \left( G D P _ { 1 } \cdot \frac { P _ { 0 } } { P _ { 1 } } + \underline { G D P _ { 0 } } \right) } $

Note that GDP0 has been underlined above to show that it is not explicitly given in the original problem statement - this is because we don't need it to solve the problem.

Now we can use the definition of g1 and g2 to simplify this expression:

$ \frac { G D P _ { 2 } } { G D P _ { 1 } } = \frac { g 1 \cdot g 2 + 1 } { g 1 + 1 } $

Finally, we can use algebra to get this expression into the same form as the choices:

$ \frac { G D P _ { 2 } } { G D P _ { 1 } } = \frac { g 1 } { g 1 } \cdot \frac { g 1 \cdot g 2 + 1 } { g 1 + 1 } = \frac { (g1 + 1)/2 } { 1/g1 + 1 } $

Taking the square root of both sides, we get:

$ \sqrt { \frac { G D P _ { 2 } } { G D P _ { 1 } } } = \sqrt { \frac { (g1 + 1)/2 } { 1/g1 + 1 } } $

This matches choice A, so our answer is:

Answer: A) $\sqrt { g 1 \times g 2 }$

Note that we did not use choices C, D, or E in our calculations.

Suppose That G1 Represents the Ratio of Year 2 GDP g1×g2\sqrt { g 1 \times g 2 }g1×g2​

Suppose That G1 Represents the Ratio of Year 2 GDP $\sqrt { g 1 \times g 2 }$