Intertemporal Utility Maximization Suppose an economic agent’s life is divided into two periods, the
first period constitutes her youth and the second her old age. There
is a single consumption good, C, available in both periods. The
agent’s utility function is given by

$$U(C_{1},C_{2}) = \frac{C_{1}^{1-\theta }-1}{1-\theta }+\frac{1}{1+\rho}\frac{C_{2}^{1-\theta }-1}{1-\theta }$$

with $0&lt;θ &lt; 1,ρ &gt; 0$, and where the first term represents utility from consumption during youth. The second term represents discounted utility from consumption in old age, $1/(1+ ρ )$ being the discount factor.

During the period, the agent has a unit of labour which she supplies
inelastically for a wage rate $w$ . Any savings (i.e., income minus
consumption during the first period) earns a rate of interest $r$,
the proceeds from which are available in old age in units of the only
consumption good available in the economy. Denote savings by $s$. The
agent maximizes utility subjects to her budget constraint.

$i)$ Show that $θ$ represents the elasticity of marginal utility with
respect to consumption in each period.

$ii)$ Write down the agent’s optimization problem, i.e., her problem
of maximizing utility subject to the budget constraint.

$iii)$ Find an expression for s as a function of $w$ and $r$.

$(iv)$ How does s change in response to a change in $r$? In
particular, show that this change depends on whether $θ$ exceeds or
falls short of unity.

$(v)$ Give an intuitive explanation of your finding in $(iv)$



I am not being able to solve this problem. I proved the first part $i$.

$$C_{2}= (w-C_{1})(1+r)$$

So, $ii)$ Optimization Problem = 

$$\frac{C_{1}^{1-\theta }-1}{1-\theta }+\frac{1}{1+\rho}\frac{C_{2}^{1-\theta }-1}{1-\theta }+\lambda (w-C_{1}-(w-C_{1})(1+r))$$

$iii) s=w-C_{1}$ How do we express this int terms of $w$ and $r$?

Any help with this sum will be appreciated.
What you have is basically the Fisher Two-Period Optimization Problem. (Fisher)

For iii)
You first need to find the Euler equation, which tells you how to optimally trade off first-period and second-period consumption.

For starters, your optimization problem is set up incorrectly: 
$$\frac{C_{1}^{1-\theta }-1}{1-\theta }+\frac{1}{1+\rho}\frac{C_{2}^{1-\theta }-1}{1-\theta }+\lambda (w-C_{1}-\frac{C_{2}}{1+r})$$

You find the Euler equation by deriving and equating the first-order conditions of both consumption terms.

$$\frac{1}{1+r} C_{1}^{-\theta }= \frac{1}{1+\rho}C_{2}^{-\theta }$$

Then just solve for $C_{1}$, substitute $C_{2}=s(1+r)$, and plug into $s=w-C_{1}$, which gives:

$$
s=w-(1+r)^{\frac{\theta-1}{\theta}}(1+\rho)^{\frac{1}{\theta}}s \\
s=\frac{w}{1+(1+r)^{\frac{\theta-1}{\theta}}(1+\rho)^{\frac{1}{\theta}}}
$$

The derivative, giving the response to a change in $r$, is:
$$
\frac{\partial s}{\partial r}= -\frac{(\theta -1) w ( 1+\rho)^{1/\theta } (1+r)^{1/\theta }}{\theta  \left((\rho +1)^{1/\theta }+r (\rho +1)^{1/\theta }+(r+1)^{1/\theta }\right)^2}
$$

The derivative for $0&lt;\theta&lt;1$ is positive, indicating that a higher return on saving will lead to an increase in the amount saved. Preference for first period consumption is still low enough, so that the consumer is willing to save more to consume it later. Here the substitution effect dominates the income effect.

When $\theta&gt;1$, the derivative is negative, meaning that preference for first period consumption is so high that the consumer is willing to lower savings for current period consumption. Here the income effect dominates the substitution effect.