Why can't price level be negatively correlated with money demand?

Question

Source: p 585, Economics, 3 Ed, 2014, by NG Mankiw, MP Taylor

Although many variables affect the demand for money, one variable stands out in importance: the average level of prices in the economy. People hold money because it is the medium of exchange. Unlike other assets, such as bonds or stocks, people can use money to buy the goods and services on their shopping lists. How much money they choose to hold for this purpose depends on the prices of those goods and services. The higher prices are,
the more money the typical transaction requires,
[1.] and the more money people will choose to hold in their pockets and bank accounts.
[2.] That is, a higher price level (a lower value of money) increases the quantity of money demanded.

I doubt 1 and 2 that feels too absolutely certain. (They're also restated, but rephrased, on pp 683 and 705.) Why must 1 and 2 be true?

Abbreviate Prior Money as PM, i.e. consumers' money held 'in their pockets and bank accounts' before the price rise.

Suppose that higher prices $\implies$ more money needed for each typical transaction. But even facing a higher price level, consumers may already possess enough PM to transact, and so needn't more money.

Moreover, even with enough PM for their regular purchases, the price rise may cause consumers to spend less. Then money demand would DEcrease.

Answer 1

You're supposing the amount of money held for transactions by players is arbitrary. This may be true if you look at an individual - I typically withdraw fixed amounts of money when my purse starts feeling empty - but is not true on average and in the long run.

But cash (in the wider sense, money on regular bank accounts count toward this) is also held by companies and foreign countries.

Even if you feel like discussing the rationality of the average person, you cannot discuss the rationality of the average money holder: the opportunity cost of holding too much money may be negligible for an employee with low savings. But it it isn't for a big multinational or a multi-millionaire that wants to stay richer than his neighbours in Beverly Hills.

Consumers are also retailers, at some point, so you cannot suppose the price rise affects them in the way they buy less goods. A potato producer still produces 1kg of potatoes after a price rise, only they are worth more and he need more money to buy carrots. The money demand increases because there is less money available for the same goods. This is a very different situation than a productivity drop, in which case the money demand would decrease.