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I understand that aggregate demand is the demand for all goods and services in an economy. However, I later found out that aggregate demand and real GDP are calculated using the same formula, so does that mean they are the same thing?

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No they are not the same thing. Following Blanchard et al Macroeconomics:

Aggregate Demand:

Aggregate demand is the demand for all goods and services in the economy.

Gross Domestic Product

A measure of aggregate output, and aggregate output is just the total amount of output produced in the economy.

Why are they equal?

However, while these are two different concepts it is correct to say that in equilibrium $AD=Y$ where AD is aggregate demand and $Y$ is output, which you then can replace by its constituents. So depending on whether you want to use income or spending approach $Y=C+I+G$ or $Y= i+ \pi+r+w$ (in closed economy) and so given that $AD=Y$, $AD$ will be equal to those other measures as well.

This is because in equilibrium aggregate demand should be equal to the production of the economy. The reason why firms produce goods and services is to satisfy demand of consumers not just to waste resources on something that is unwanted. So $AD=Y$ just expresses the idea that firms produce goods and services that people demand (at least in equilibrium they do).

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