題組內容

4. Consider a closed economy with two sectors, a goods sector and a long-term care (LTC) sector. Let $L$ denote the total labor supply in this economy. The production function for the goods sector is:

$Y = A K^\alpha L_Y^{1-\alpha},$

where Y is total goods being produced, K is physical capital employed in the goods sector, $L_Y$ is labor allocated for goods production, A is the technology scaling factor in the goods sector, and $\alpha$ is the capital income share in the goods sector. The production function of the LTC sector is:

$X = B L_{LTC},$

where X is total LTC service provided, $L_{LTC}$ is labor inputs in the LTC sector, and B is the technology

scaling factor in the LTC sector. Labor is inelastically supplied, so $L \equiv L_Y + L_{LTC}$. Let goods be the price numeraire, $p$ the price of LTC service, and $w_Y$ and $w_{LTC}$ the wage rates in the goods sector and the LTC sector, respectively. There is no population growth. Workers are homogeneous, and all markets are competitive. Answer the following questions.

(b) Now assume workers live for two periods: young and old. Each worker is endowed with one unit of labor time when young, supplies labor to the market inelastically during their youth, retires and consumes LTC services when old. The lifetime utility of a worker born in period \( t \) is

$U^t = \ln(c_t^y) + \beta \ln(x_{t+1}),$

where $c_t^y$ is consumption when young, $x_{t+1}$ is LTC service purchased when old, and $\beta > 0$ is the discount factor or the preference for old-age utility. Denote the saving of a worker when young by $s_t$. A worker's budget constraint when young and when old are:

$c_t^y + s_t = w_t,$

$p_{t+1}x_{t+1} = R_{t+1}s_t.$

where $R_{t+1}$ is the gross return on savings, and $w_t \in \{w_{Y,t}, w_{LTC,t}\}$. Capital fully depreciates after one period of use. Let $k_t \equiv K_t/L_{Y,t}$ denote the capital intensity in the goods sector. Solve for the steady-state values for $k^*$, $p^*$ and the fractions of labor allocated to the goods sector ($L_Y/L$) and the LTC sector ($L_{LTC}/L$). Analyze how the steady-state values of $k^*$, $p^*$, $L_Y^*/L$ $L_{LTC}^*/L$ respond to an increase in $B$, $\beta$ and $\alpha$. Explain your results with economic intuition. (10%)