Define the following set of feasible continuation allocations: \[\Gamma(k) \equiv \left\{ (c,n,l,k^\prime,x) \in \mathbb{R}_+^4 \times \mathbb{R} \middle| \begin{gather} c + x \leq f(k,n) \\ k^\prime \leq (1-\delta) k + x \\ l + n \leq 1 \\ \end{gather} \right\} \]
The sequential programming problem is \[ V(k_0) = \max_{\left\{c_t,n_t, l_t, k_{t+1}, x_t \right\}_t} \sum_{t=0}^\infty \beta^t u(c_t, l_t) \]
The Bellman's equation is: \[V(k) = \max_{(c,n,l,k^\prime,x)\in \Gamma(k)} \left[u(c,l) + \beta V(k^\prime) \right]\]
Assume that \(\beta\in(0,1)\). Assume that \(\sigma\in (0,1]\).
Assume that \(u\) is strictly increasing, and continuous.
Assume that f is strictly increasing, continuous, \(f(0)=0\), and there is some maximimum sustainable capital, \(\hat{k}\)
The assumptions on \(f\) and \(\delta\) allow us to conclude that capital is bounded (in \([0,\max{\hat{k},k_0]\)). This, along with the budget constraint, allows us to conclude that consumption is bounded. Then the assumptions on \(u\) allow us to conclude that per-period utility is bounded as well. Finally, with \(\beta\in(0,1)\), we can then conclude that \(v(k_0)\) is bounded.
The boundedness of \(v(k_0)\) allows us to safely apply the sup norm and so use Blackwell's sufficient conditions (the problem clearly satisfies monotonicity and discounting) to conclude that there is a unique fixed point \(v\)
The Theorem of the maximum establishes the continuousness.
With inelastic labor supply, \(n_t = \bar{n}\). For simplicity, let \(u(c)\equiv u(c,0\) and let \(f(k)\equiv f(k,\bar{n}\). Additionally, if we assume that the utility and production functions are strictly increasing in consumption and capital, respectively, then the resource and investment constraints will bind. Combining them yields: \[c_t = f(k_t) +(1-\delta)k_t - k_{t+1}\]
Redefine the set of feasible continuation allocations: \[\dot{\Gamma}(k) \equiv [0, f(k_t) +(1-\delta)k_t]\]
Now the functional equation is \[ V(k) = \max_{k^\prime \in \dot{\Gamma}(k)} \left[ u(f(k) +(1-\delta)k - k^\prime) + \beta V(k^\prime) \right] \]
Taking the First Order condition of this, we get that at the optimum: \[0 = {\partial \over \partial k^\prime} \left[ u(f(k) +(1-\delta)k - k^\prime) + \beta V(k^\prime) \right] = -u^\prime(f(k) +(1-\delta)k - k^\prime) + \beta V^\prime(k^\prime) \]
Let \(k^\prime(k)\) be the policy function for \(k^\prime\).Then the envelope condition is: \[ V^\prime(k) = {\partial \over \partial k} \left[ u(f(k) +(1-\delta)k - k^\prime(k)) + \beta V(k^\prime(k)) \right] = u'(f(k) +(1-\delta)k - k^\prime(k)) (f^\prime(k) + 1 - \delta ) + {\partial \left[ u(f(k) +(1-\delta)k - k^\prime(k)) + \beta V(k^\prime(k)) \right]\over \partial k^\prime}{\partial k^\prime \over \partial k} \] And in the optimum, the FOC tells us that the last term is equal to zero, so the Envelope condition is: \[ V^\prime(k) = u'(f(k) +(1-\delta)k - k^\prime(k)) (f^\prime(k) + 1 - \delta ) \]
TODO: characterize by combinniginingDefine \(✉(k,k^\prime) \equiv \left[ u(f(k) +(1-\delta)k - k^\prime) + \beta V(k^\prime) \right] \) Now the functional equation is \[ V(k) = \max_{k^\prime \in \dot{\Gamma}(k)} ✉(k,k^\prime) \] FOC: \[0 = {\partial \over \partial k^\prime}✉(k,k^\prime)\] Envelope: \[{\partial \over \partial k} V(k) = {\partial \over \partial k} ✉(k,k^\prime(k)) = {\partial \over \partial k} ✉(k,k^\prime) + {\partial \over \partial k^\prime} ✉(k,k^\prime) {\partial k^\prime \over \partial k} ✉(k,k^\prime) = {\partial \over \partial k} ✉(k,k^\prime) \]
Assume that \(\alpha, \beta \in (0,1)\), \(\delta_k, \delta_h \in (0,1]\), and \(\sigma,A,h_0,k_0 > 0\).
Because leisure does not enter into the utility function, we can safely assume that \(n_t=1 \; \forall t\), and because the utility and production function are strictly increasing, the other constraints will bind. This suggests some obvious simplifications to to make in the constraints via the following substitutions: \begin{align} z_t & = h_t \\ x_{kt} & = k_{t+1} - (1-\delta_k ) k_t \\ x_{ht} & = h_{t+1} - (1-\delta_h ) h_t \\ \end{align}
Then the value function for the planner's problem above is equivalent to the following: \[V(k_0, h_0) = \max_{\{c_t,k_{t+1},h_{t+1}\}_t} \sum_{t=0}^\infty \beta^t {c_t^{1-\sigma} \over 1-\sigma}\]
The first order conditions for this simplified problem are: \begin{align} 0 & = \beta^t c_t^{-\sigma} - \lambda_t \\ 0 & = - \lambda_t + (1-\delta_k )\lambda_{t+1} + Ak_{t+1}^{\alpha-1}h_{t+1}^{1-\alpha} \lambda_{t+1} \\ 0 & = - \lambda_t + (1-\delta_h )\lambda_{t+1} + Ak_{t+1}^{\alpha}h_{t+1}^{-\alpha} \lambda_{t+1} \\ \end{align}
The FOCs, together with the budget constraint, characterize the optimal sequence of allocations. Suppose that the sequence \(\{\hat{c}_t,\hat{k}_{t+1},\hat{h}_{t+1}\}_t\) is a solution to the above maximization problem for initial stocks \((\bar{k}_0,\bar{h}_0)\). Then there exist some lagrange multipliers \(\{\hat{\lambda}_t\}_t\) such that: \begin{align} 0 & = \beta^t \hat{c}_t^{-\sigma} - \hat{\lambda}_t \\ 0 & = - \hat{\lambda}_t + (1-\delta_k )\hat{\lambda}_{t+1} + A\hat{k}_{t+1}^{\alpha-1}\hat{h}_{t+1}^{1-\alpha} \hat{\lambda}_{t+1} \\ 0 & = - \hat{\lambda}_t + (1-\delta_h )\hat{\lambda}_{t+1} + A\hat{k}_{t+1}^{\alpha}\hat{h}_{t+1}^{-\alpha} \hat{\lambda}_{t+1} \\ \hat{c}_t + \hat{k}_{t+1} + \hat{h}_{t+1} & = A\hat{k}_t^\alpha \hat{h}_t^{1-\alpha} + (1-\delta_k ) \hat{k}_t + (1-\delta_h ) \hat{h}_t \\ \hat{c}_0 + \hat{k}_{1} + \hat{h}_{1} & = A\bar{k}_0^\alpha \bar{h}_0^{1-\alpha} + (1-\delta_k ) \bar{k}_0 + (1-\delta_h ) \bar{h}_0 \\ \end{align}
add term including initial condition, show multiopliers still work, badabingThis system can be summarized by the following diagram. (key)
The loop of being employed is on the right, and the loop of searching for a job is on the left. \(V(w)\) is the present value of having offer \(w\) in hand. \(V^U\) is the present value of starting a period unemployed.Consider two economies with different unemployment benefits \(b_1\) and \(b_2\), but that are otherwise identical.
Let \(b_2 > b_1\). Define: \[ \hat{b}(x) \equiv x\left({1-\beta F(x) \over 1-\beta}\right) - \beta \int_{x}^{B} {w \over 1-\beta} f(w) dw \] From above, we know that \(b_i = \hat{b}(\bar{w_i})\). Taking the derivative, \[ \hat{b}'(x) = \left({1-\beta F(x) \over 1-\beta}\right) - x{\beta\over 1-\beta}f(x) - {\beta\over 1-\beta} (-xf(x)) = \left({1-\beta F(x) \over 1-\beta}\right) > 0\] So because \(\hat{b}(x)\) is strictly increasing, \(b_2>b_1 \implies \hat{b}(\bar{w_2})>\hat{b}(\bar{w_1}) \implies \bar{w_2} > \bar{w_1} \)More intuitively, but a bit less rigourously, let \(V^U_i\) be the present value of starting a period unemployed in economy \(i\). We know that \({\bar{w_i} \over 1-\beta} = V^E(\bar{w_i})=b_i+\beta V^U_i\) and so \(\bar{w_i}=(1-\beta)(b_i+\beta V^U_i)\)
Note that \(V_2^U \geq V_1^U\). If the agents in each economy follow the same strategy, and witness the same sequence of random events, then the agent in economy 2 will always have per-period income at least as high as the agent in economy 1. So it can't possibly be that \(V_2^U < V_1^U\). And so: \[\bar{w_2}=(1-\beta)(b_2+\beta V^U_2) > (1-\beta)(b_1+\beta V^U_1) = \bar{w_1}\] This makes sense. Higher unemployment benefits make being unemployed a more attractive prospect, increasing the opportunity cost of accepting any given job offer, which means that fewer job offers will be accepted.Consider two economies with different distributions for job offers \(F_1\) and \(F_2\), but that are otherwise identical.
Let \(F_2\) be a mean-preserving spread of \(F_1\), and let both distributions have support within \([0,B]\). We will prove that the reservation wage is at least as high in economy 2: \(\bar{w_2}\geq\bar{w_1}\)
If you are comparing distributions of the form \(F(w,r)\) where \(r\) is a parameter such that riskiness increases as \(r\) increases, then you can take the derivative of both sides with respect to \(r\) and find that \(\bar{w_i}'(r) > 0\). Otherwise:
Recall that the properties of a mean preserving spread imply that \(E_1[w]=E_2[w]\) and that for any value of \(\bar{w}\), \(\int_{0}^{\bar{w}} F_2(w) dw \geq \int_{0}^{\bar{w}} F_1(w) dw\). Then: \begin{split} \bar{w_2} - \bar{w_1} & = (1-\beta)b + \beta E_2[w] + \beta \int_{0}^{\bar{w_2}} F_2(w) dw - (1-\beta)b - \beta E_1[w] - \beta \int_{0}^{\bar{w_1}} F_1(w) dw \\ & = \beta \int_{0}^{\bar{w_2}} F_2(w) dw - \beta \int_{0}^{\bar{w_1}} F_1(w) dw \\ & = \beta \int_{\bar{w_1}}^{\bar{w_2}} F_2(w) dw + \beta \int_{0}^{\bar{w_1}} \left[ F_2(w)-F_1(w)\right] dw \geq \beta \int_{\bar{w_1}}^{\bar{w_2}} F_2(w) dw \\ \end{split} Note that if \(\bar{w_2} \geq \bar{w_1}\), then \(\int_{\bar{w_1}}^{\bar{w_2}} F_2(w) dw \geq 0 \) and \(F_2(\bar{w_1}) \leq F(w)\) for all \(w \in [\bar{w_1}, \bar{w_2}]\).