Question

In a two period model, a consumer is maximizing the present discounted utility
𝑊𝑡 = ln(𝑐_𝑡) +\(\frac{ 1}{ 1 + }\) ln(𝑐_𝑡+1) 
with respect to 𝑐𝑡 and 𝑐𝑡+1 and subject to the following budget constraint 
\(𝑐_𝑡 +\frac{ 𝑐_𝑡+1}{ 1 + 𝑟} ≤ 𝑦_𝑡 +\frac{ 𝑦_𝑡+1 }{1 + 𝑟 }\)
where 𝑐𝑖 and 𝑦𝑖 are the consumption and income in period 𝑖 (𝑖 = 𝑡,𝑡 + 1) respectively, 𝜃 ∈ [0, ∞) is the time discount rate and 𝑟 ∈ [0, ∞) is the rate of interest. Suppose, consumer is in the interior equilibrium and 𝜃 = 0.05 and 𝑟 = 0.08. In equilibrium, the ratio \(\frac{𝑐_𝑡+1}{ 𝑐_𝑡}\) is equal to _____ (round off to 2 decimal places).

Answer 1

Correct Answer: 1.02

Given: 
Objective (period-t consumer): \[ W_t=\ln c_t+\frac{1}{1+\theta}\,\ln c_{t+1}, \] Budget constraint: \[ c_t+\frac{c_{t+1}}{1+r}\le y_t+\frac{y_{t+1}}{1+r}. \] Interior optimum, \(\theta=0.05,\; r=0.08\). Find \(\dfrac{c_{t+1}}{c_t}\) (to 2 d.p.). 

Step 1 — Lagrangian
\[ \mathcal{L}=\ln c_t+\frac{1}{1+\theta}\ln c_{t+1} +\lambda\!\left(y_t+\frac{y_{t+1}}{1+r}-c_t-\frac{c_{t+1}}{1+r}\right). \] FOCs:
\[ \frac{\partial\mathcal{L}}{\partial c_t}=\frac{1}{c_t}-\lambda=0 \quad\Rightarrow\quad \lambda=\frac{1}{c_t}. \] \[ \frac{\partial\mathcal{L}}{\partial c_{t+1}}=\frac{1}{1+\theta}\cdot\frac{1}{c_{t+1}} -\lambda\cdot\frac{1}{1+r}=0. \] Substitute \(\lambda=\dfrac{1}{c_t}\): \[ \frac{1}{1+\theta}\cdot\frac{1}{c_{t+1}} =\frac{1}{c_t}\cdot\frac{1}{1+r}. \] Rearrange for the consumption ratio: \[ \frac{c_{t+1}}{c_t}=(1+r)\cdot\frac{1}{1+\theta} =\frac{1+r}{1+\theta}. \] Step 2 — Substitute numbers
\[ \frac{c_{t+1}}{c_t}=\frac{1+0.08}{1+0.05}=\frac{1.08}{1.05}=1.028571\ldots \] Final answer (rounded to 2 d.p.):
\[ \boxed{\;\dfrac{c_{t+1}}{c_t}=1.03\;} \]