Final answer:
The total money supply (M) can be expressed as a function of high-powered money (H) using the money multiplier concept, which involves the reserve ratio and the cash-deposit ratio. The formula derived is M = (1 + c) * (H / (c + r)), where c is the cash-deposit ratio, and r is the reserve ratio.
Step-by-step explanation:
The student has asked how to express the total money supply (M) as a function of high-powered money (H). To answer this, we use the concept of the money multiplier, which is defined as the quantity of money that the banking system can create from each dollar of bank reserves. The money multiplier formula is 1/reserve ratio (r), where the reserve ratio is the fraction of deposits that banks hold as reserves.
When the reserve ratio is given, the money multiplier, m, can be defined as M = m * H, where H is the high-powered money. High-powered money includes both cash holdings by the public and reserves held by banks. As per the given question's definition, the money supply, M, is the sum of bank deposits, D, and cash holdings, E. Using the provided ratios, we can state M = D + E and because the cash-deposit ratio (c/d) is constant, E can be written as c * D, where c is the cash-deposit ratio. Therefore, M can be written as:
M = D + c * D (since E = c * D)
Because bank reserves are a fraction r of bank deposits, the high-powered money (H) is the sum of cash holdings (c * D) and bank reserves (r * D).
Hence, H = c * D + r * D
D = H / (c + r)
Substituting D back into the equation for M, we get:
M = (1 + c) * (H / (c + r)) which is the money supply as a function of high-powered money.