(c) Analyze how the optimal labor supply N^* responds to: (i) an increase in the real wage w; and (ii) an increase in the preference parameter β. Clearly explain both the direction of change and the underlying economic intuition. (10%).

3. Consider a closed economy described by the Solow growth framework in continuous time. Output is produced using a Cobb-Douglas technology: $$Y = K^{\alpha} N^{1-\alpha}, \quad 0 < \alpha < 1$$ where $Y$ represents total output, $K$ denotes the capital stock, and $N$ is the labor force (number of workers). The parameter $\alpha$ measures capital's share in output. The population grows at a constant rate $n > 0$, i.e., $\dot{N}/N = n$. Capital accumulates according to: $$\dot{K} = I - \delta K$$ where $I$ is total investment and $\delta \in (0,1)$ is the constant rate of capital depreciation. National saving is defined as a constant fraction of output: $$S = sY$$ where $s \in (0,1)$ is the savings rate. We assume the goods market clears such that $S = I$. Please answer the following questions: (a) Derive the steady-state level of capital per worker $ k $ and the steady-state output per worker $ y $, where $ k = K / N $ and $ y = Y / N $. (15%)

(c) Analyze how the optimal labor supply \( N^* \) responds to: (i) an increase in the real wage \( w \); and (ii) an increase in the preference parameter \( \beta \). Clearly explain both the direction of change and the underlying economic intuition. (10%).

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