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I am reading Monetary policy, inflation, and the business cycle: an introduction to the new Keynesian framework and its applications by Gali.

In section 3.1 he presents/defines the consumption as $$ C_t:= \left(\int^1_0C_t(i)^{\frac{\epsilon-1}{\epsilon}}di\right)^{\frac{\epsilon}{\epsilon-1}} $$

$\epsilon$ is the elasticity of substitution and $i$ indexes the goods available for consumption. It is related to the New Keynesian Model.

Can someone give me verbal explanation of why it make sense to define the consumption this way?

$C_t(i):$ How should interpret the $i$-line?

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For some reason Ubiquitous edited this information into the question, rather posting it as an answer:

$i$ indexes the goods available for consumption

Integrals are just sums with a lot of components (technically infinite). Here the space of goods is assumed to have large dimension, rather than just consisting of two or three types of goods.

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