# Elasticity of substitution, marginal rate of substitution and price ratio

so there is a problem I do not know how to tackle:

The is an utility maximisation problem and the solution to the Lagrangian leads to:$$\frac{C_1}{C_2}=(1+\rho)^\frac{1}{\sigma}(\frac{P_1}{P_2})^\frac{1}{\sigma}C_2.\qquad(1)$$ And taking ln

$$ln(\frac{C_1}{C_2})=(\frac{1}{\sigma})ln(1+\rho)-\frac{1}{\sigma}ln(\frac{P_1}{P_2}).$$ We take the derivative in order to get the elasticity of substitution:$$-\frac{\delta ln(C_1/C_2)}{\delta ln(P_1/P_2)}=\frac{1}{\sigma}.$$ Note that (1) could also be written as: $$\frac{C_2}{C_1}=(\frac{P_1/P_2}{1+\rho})^{\frac{1}{\sigma}}.$$

Now , we have to recall that the marginal rate of substitution between two goods is equal to the price ratio: $$MRS_{C_2,C_1}=\frac {P_2}{P_1}$$Hence,$$MRS_{C_2,C_1}=\frac {1}{1+\rho}(\frac{C_1}{C_2})^\sigma.$$

The question is now: How do $$\frac{1}{\sigma}$$ and $$\rho$$ change the indifference curves between $$C_1$$ and $$C_2$$?

Can anyone give an answer with the intuition?

• If you are looking for intuition, why not plot some indifference curves and observe how they change when you alter the paramaters? – Giskard Apr 6 at 10:13