SEIGNIORAGE is defined as raising revenue by printing money. Let m be nominal money stock at the end of each month and dm is change in nominal money stock from the end of last month to the end of this month .It means dm is money creation during the month. Therefore, in real terms, the revenue that government generates will be dm / p, to determine what rate of nominal money growth is required to generate a given seigniorage as

( dm / m ) ( m /p ),

From the above equation it follows that seigniorage is product of nominal money growth and real money balance.

The above equation implies that larger is the real money balances held in the economy larger will be the seigniorage. The above equation gives us relationship between seigniorage, rate of nominal money growth and real money balance. Dividing both sides of equation of real income we get,

Seigniorage /  Y  = ( dm / m ) ( m / p  / Y )

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Suppose that government is running a budget deficit of ten percent ( 10 % ) of real income and it decides to finance it through  printing money , seigniorage ( dm / m ) ( m / p ).  Suppose people hold real money balances equal to two months of income. Then it implies that nominal money supply growth must be equal to five percent ( 5 % ) , which means that to finance 10 % of real money balances through seigniorage. Monthly growth rate of nominal money must be equal to 5 % .

However, it does not mean that government can finance deficit equal to 20 % of real income, through money growth of 10 %.

This is because as nominal money growth increases it  causes increase in  inflation rate . And with higher inflation , nominal interest rate increase. Since , nominal interest rate is cost of of holding money so real money balances decrease. Therefore, an increase in nominal money growth will generate less than proportional increase in seigniorage.

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