2. Consider a worker whose utility depends on consumption and leisure. The worker's income comes from labor income and non-labor income. The worker's utility maximization problem is given by:

$$\underset{C,N}{Max} \quad U = \ln C + \beta \ln(H - N)$$

$$s.t. \quad C = wN + v$$

where $C$ denotes consumption, $N$ denotes hours of labor supplied, $H$ is the total time endowment, $w$ is the real wage, $v > 0$ is non-labor income, and $\beta > 0$ measures the relative importance of leisure in utility. The worker takes $w$, $v$, $H$, and $\beta$ as given.


Complete the following tasks:

(a) Explain the income effect and substitution effect of an increase in the real wage on labor supply. (Please provide detailed economic intuition; no mathematical derivation is required.) (10%)

(b) Solve for the worker's optimal consumption \( C^* \) and optimal labor supply \( N^* \). (5%)

(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%).

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%)

(b) Determine the long-run growth rates of output per worker \( y \) and total output \( Y \). Based on your results, explain why the Solow model is classified as an exogenous growth model. (10%)

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