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Let's say, before dollarization, I owned 1 million Zimbabwean dollars in cash and the next day dollarization went into effect.

So in dollarization, the current Zimbabwean dollars will be invalid and only USD will work in the economy. So what happened to my 1 million Zimbabwean dollars that I had in cash.

Did the banks exchange it for USD or did something else happened?

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Normally in order to full dollarization to occur there will be some conversion rate at which debt denominated in old currency is converted into new one. For example, according to the World Bank definition of full dollarization:

All government and private debt under full dollarization is denominated in dollars, and both public and private accounts must be converted to dollars. To make the conversion, countries must set the rate at which old debts, contracts, and financial assets will be converted to dollars.

However, I suppose technically the rate 1 Zimbabwean dollar = 0 U.S. dollar would be possible. This seems to be the case according to Mpofu (2015) [emphasis is mine]:

Zimbabwe is viewed as a dollarized economy given the dominance of the currency among the other currencies used in the country and the fact that the Zimbabwean Government conducts all its business using the US dollar. In the same month that the multi-currency regime was introduced in 2009, the Zimbabwean government suspended the use of the Zimbabwe dollar as legal tender and decreed that all wages, prices of all goods and services and transactions be done in U.S. dollars. They further decreed that all Zimbabwe dollars be removed from circulation with no compensation provided to currency holders, a decision that was viewed as indicating that the currency had no store of value.

This suggest that in the case of Zimbabwe simply the value of Zimbabwean dollar vis-a-vis US dollar was set to zero. However, I briefly looked over several papers discussing this episode (Noko (2011)]3, Kairiza (2009), Kramarenko, Engstrom, Verdier,Fernandez, Oppers, Hughes, ... & Coats,(2010), because this question caught my interest, but none of them seems to explicitly state what happened to private debt. This issue aside the sources mention that US dollars were already widely circulating in the Zimbabwean economy even without government sanction, so my guess is that most likely already most debts were denominated in US dollars already.

PS: An interesting tidbit is that actually Zimbabwe did not just dollarize they adopted multicurrency regime where besides dollar, the rand, euro, pound, metical, and kwacha also became official tenders but dollar was without doubt the most prominent one.

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