Intuition for the CES consumption index in New-Keynesian DSGE models

Question

I don’t understand, from an intuitive point of view, the CES aggregator in the context of a New Keynsian DSGE model. I understand that the consumption index Ct is the sum of all consumption goods and we conceive of a continuum of goods over the interval (0,1) and, as such, have;

I understand that we are making assumptions here about the imperfect substitutability of goods and that this is what gives firms market power in the monopolistic setting of the DSGE models in question.

What I do not understand is the intuition of how levels of consumption of all different goods maps onto the expression for Ct above. How can we represent this using an integral over a range? I do not understand the intuition, and nowhere seems to provide a decent explanation. I would be very grateful if one of you could possibly explain it to me?

Many thanks in advance.

Answer 1

Heuristically, you can think of the integral as just a sum:

$$ \bar{C} = \left( \sum_{i=1}^n C_i^{1-\frac{1}{\epsilon}} \right)^{\frac{\epsilon}{\epsilon - 1}} $$

where $\bar{C}$ is an index of aggregate consumption, and utility is given by $u \left( \bar{C} \right)$.

It's easy to check that the marginal rate of substitution between goods $j$ and $k$ is given by

$$ \text{MRS}_{jk} = \left( \frac{C_j}{C_k} \right)^{-\frac{1}{\epsilon}} $$

which means that the (absolute value of the) elasticity of substitution is $\epsilon$.

With a continuum of goods, the analogue of summation is integration, and you can (loosely) think of the MRS between goods $j$ and $k$ as

$$ \left( \frac{C_t (j)}{C_t (k)} \right)^{- \frac{1}{\epsilon}} $$