I am trying to calculate how to determine the interest rate ( = risk free rate + premium) within the following model where a consumer decides to invest in a safe asset or in a risky asset. The utility is a CRRA function: $U(C) = \frac{c^{1-\frac{1}{\sigma}}}{1-\frac{1}{\sigma}}$ where $\sigma$ is the intertemporal elasticity of substitution. Hence, the consumer decides to maximize its utility by the following max problem where $1-p$ is the probability of default (consumer) does not get its savings + return back and $p$ is the probability of no default.

$$max \: U = u(c_1) + \beta E [(1-p)u(c_2(D))] + \beta E [(p) u(c_2(s))]$$

and he faces the following budget constraints:

$$C_1 = Y_1 - T_1 - S^1 - S^2$$

for period 2 consumption in case of default and no default:

$$C_2 (D) = Y_2-T_2+(1+r^1)S^1$$ $$C_2 (S) = Y_2 - T_2 + (1+r^1)S^1 + (1+r^2)S^2$$

So, I substituted these two budget constraints in the maximization problem and differentiated with respect to savings (1) and savings (2) and got the following Euler equations:

$$u^\prime(C_1) = \beta (1+r^1) E [(1-p)u^\prime (C_2(D))] + \beta (1+r^1) E[p u^\prime(C_2(S))]$$ $$u^\prime(C_1) = \beta E[p (1+r^2) u^\prime(C_2(S))]$$

The risk free rate, $r^1$ is exogenously determined. I am trying to calculate $r^2$ by dividing the first Euler equation by the other, since the marginal utility of period 1 consumption is equal to each other. However, I can not find a solution, am I doing this corrrect?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.