(c) Prove that H is orthogonal to (5 %)

(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%)

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(c) Prove that H is orthogonal to (5 %)

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