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.