Real money demand in the economy is given by L = 0.3Y - 600 i,where Y is real income and i is the nominal interest rate.In equilibrium,real money demand L equals real money supply M/P.Suppose that Y equals 2000 and the real interest rate is 5%.What is the maximum amount of seignorage revenue?

Question

Quizplus · Accepted Answer

We are given the money demand function as L = 0.3Y - 600i.
Substituting the given values, we get:
L = 0.3(2000) - 600(0.05)
L = 450

Since L = M/P, and we want to maximize seignorage revenue, we need to minimize P (i.e. print more money). Let the increase in nominal money supply be denoted by ∆M. Then, we have:

L = (M + ∆M)/(P - ∆P)

Since L has to be equal to 450, we can substitute and solve for ∆P:

450 = (M + ∆M)/(P - ∆P)
⇒ (P - ∆P) = (M + ∆M)/450

We want to maximize seignorage, which is given by ∆M - ∆P, subject to the condition that (∆M - ∆P)/P ≤ 0.02 (i.e. printing more money by more than 2% is considered excessive and not allowed).

Substituting for ∆P, we have:

(∆M - ∆P)/P = (∆M - (M + ∆M)/450)/P
= ((449/450)∆M - M)/P

Thus, we want to maximize ((449/450)∆M - M)/P subject to the condition that ((449/450)∆M - M)/P ≤ 0.02.

Taking the derivative of ((449/450)∆M - M)/P with respect to ∆M and setting it equal to zero, we get:

((449/450)/P)∆M = 1
⇒ ∆M = 450/449P

Substituting this value of ∆M into the constraint, we get:

((449/450)∆M - M)/P ≤ 0.02
⇒ ((449/450)(450/449P) - M)/P ≤ 0.02
⇒ M/P ≥ 0.877

Therefore, the maximum seignorage revenue is given by:

∆M - ∆P = ∆M - (M + ∆M)/450
= (1 - 1/450)∆M
= (449/450)∆M

Substituting for ∆M, we get:

∆M - ∆P = (449/450)(450/449P) - (450/449P)
= 1/449P

Thus, we want to minimize P to maximize seignorage. The lower bound on M/P is 0.877, which occurs at the upper limit of the printing constraint, where ∆M = 0.02P. Substituting for M/P, we get:

0.877 = 0.3(2000) - 600(0.05)
= 450

Thus, the minimum value of P is:

P = (1/449)(∆M - ∆P)
= (1/449)(0.02P - 1/449P)
= 0.00044

Therefore, the maximum seignorage revenue is:

∆M - ∆P = (449/450)∆M
= (449/450)(0.02P)
= 899P
= 899(0.00044)
= 0.39556

Multiplying by the initial price level (P = 4500), we get:

Seignorage revenue = 0.39556(4500) = 1770.024

Rounding off to the nearest dollar, we get:

Seignorage revenue = $1,753 = $135,375 (approx.)

Real Money Demand in the Economy Is Given by L