$M \cdot V=P \cdot Q$ $\newline$ The quantitative theory of money states that given $M$ dollars in circulation in a year, a monetary velocity of $V$ , a price level of $P$ , and a real output of $Q$ dollars, the given equation is correct. An economist considers the case where the dollars in circulation and the real output are known constants. Which of the following expressions is the change in monetary velocity as the price level increases by $1$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $M$ $\newline$ (B) $Q$ $\newline$ (C) $\frac{M}{Q}$ $\newline$ (D) $\frac{Q}{M}$

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Algebra 2

Quotient property of logarithms

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M \cdot V=P \cdot Q

\newline

The quantitative theory of money states that given

M

dollars in circulation in a year, a monetary velocity of

V

, a price level of

P

, and a real output of

Q

dollars, the given equation is correct. An economist considers the case where the dollars in circulation and the real output are known constants. Which of the following expressions is the change in monetary velocity as the price level increases by

1

\newline

Choose

1

answer:

\newline

(A)

M

\newline

(B)

Q

\newline

(C)

\frac{M}{Q}

\newline

(D)

\frac{Q}{M}

Start with equation: We start with the equation $M \cdot V = P \cdot Q$ . To find how $V$ changes with $P$ , we need to solve for $V$ . $\newline$ $V = \frac{P \cdot Q}{M}$
Find new price level: Now, we need to find the change in $V$ when $P$ increases by $1$ . Let's call the new price level $P + 1$ . The new $V$ , which we'll call $V_{\text{new}}$ , is: $V_{\text{new}} = \frac{(P + 1) \cdot Q}{M}$
Calculate change in V: To find the change in V, we subtract the original $V$ from $V_{\text{new}}$ .
Change in $V = V_{\text{new}} - V$
Change in V = \left(\frac{(P + $1$) \cdot Q}{M}\right) - \left(\frac{P \cdot Q}{M}\right)
Simplify expression: Simplify the expression by combining the terms over a common denominator.\(\newlineChange in $V = \frac{P \cdot Q + Q}{M} - \frac{P \cdot Q}{M}$ $\newline$ Change in $V = \frac{P \cdot Q + Q - P \cdot Q}{M}$
Cancel out terms: Cancel out the $P*Q$ terms. $\newline$ Change in $V = \frac{Q}{M}$
Final answer: The change in monetary velocity $V$ as the price level $P$ increases by $1$ is $\frac{Q}{M}$ . $\newline$ So, the correct answer is (D) $\frac{Q}{M}$ .