Constant Elasticity of Substitution

Macro

$C_t={\left[{\int }_0^1{C}_{i,t}^{\frac{ϵ-1}{ϵ}}di\right]}^{\frac{ϵ}{ϵ-1}}$

elasticity $ϵ>1$.

It defines the total consumption when there are many differentiated varieties. It defines how households value diversity.

This gives us intuition that households don’t just care about the total expenditure $C_t$, but also the allocations of goods.

• If $ϵ\to \infty$, The goods are perfect substitutes, the household only cares about total quantity and would consume the cheapest good.
• If $ϵ\to 1$, The goods are very complementary, the household strongly prefers a balanced basket. (like Cobb-Douglas Production Function)

Generally the household faces an cost-minimization problem, suppose the target is $C_t$, how to reach to it as cheaply as possible?

$\mathfrak{L}={\int }_0^1{P}_{i,t}{C}_{i,t}di-\lambda \left({\left[{\int }_0^1{C}_{i,t}^{1-1/ϵ}di\right]}^{\frac{ϵ}{ϵ-1}}-C_t\right)$

$\begin{array}{l}\frac{\partial \mathfrak{L}}{\partial C_{i,t}}=P_{i,t}-\lambda \left(\frac{ϵ-1}{ϵ}\frac{ϵ}{ϵ-1}{C}_{i,t}^{-1/ϵ}\right){\left[{\int }_0^1{C}_{i,t}^{1-1/ϵ}di\right]}^{\frac{ϵ}{ϵ-1}-1}=0\\ \Rightarrow C_{i,t}={\lambda }^{ϵ}{p}_{i,t}^{-ϵ}{C}_t\end{array}$

See MN4 p29 for derivation

${\int }_0^1{P}_{i,t}{C}_{i,t}di={\int }_0^1{P}_{i,t}{\left(\frac{P_{i,t}}{P_{j,t}}\right)}^{-ϵ}{C}_{j,t}di\Rightarrow C_{jt}=\frac{P_{j,t}^{-ϵ}{\int }_0^1{P}_{i,t}{C}_{i,t}di}{{\int }_0^1{P}_{i,t}^{1-ϵ}di}$