The influence of an increase in ATM's on the demand for central bank money

This is a question posed in Blanchard's book on Macroeconomics.

The question is simple. Consider an economy with a central bank (CB) and private banks. The demand for CB money is then determined by both the demand for currency $$CU^d = cM^d$$ by people where $$c\in[0,1]$$, and the demand for reserve by the private banks $$R^d=\theta D^d=\theta(1-c)M^d$$ where $$D^d$$ is the demand for deposits by people so given by the fraction $$1-c$$ of the total demand for money. Now call $$H^d$$ the demand for CB money. Then $$H^d=[c+\theta(1-c)]YL(i)$$.

Suppose now that the number of ATM's provided by the private banks increases which makes it easier for people to deposit money. What happens to the demand for CB money?

My solution. The dependence of $$H^d$$ on the fraction of money in deposits is $$\dfrac{\partial H^d}{\partial (1-c)} = (\theta-1)YL(i)$$ which is a constant in $$c$$. So we can assume the demand for CB money is linear in $$c$$. If we assume that $$\theta\in[0,1]$$ then $$\theta-1\in[-1,0]$$ so $$H^d$$ depends negatively on $$c$$. If $$c$$ goes up, then $$H^d$$ goes down. (We also assumed $$M^d>0$$.)

• Are you looking for a solution verification? Nov 23 '21 at 18:15
• @HerrK. yes, I am. That, or if possible/necessary, comments on the solution. Nov 23 '21 at 23:10