王芸阁

经济学本科生

Ramsey model

preliminaries

\[H={1,2,...,M} \ with \ M=\infty\]

Time Separable Preferences

\[u_{i}(c_{i,t})\]

Infinite Horizon and the Representative Household

行为经济学 hyperbolic discounting

von Neumann-Morgenstern expected utility function:

\[E^{i}_{0}\sum^{T}_{t=0}\beta^{t}_{i}u_{i}(c_{i,t})\]

Preferences, Technology and Demographics

    • Representative household with instantaneous utility function $u({c_{t}})$ Assumption, u (c) is strictly increasing, concave, twice continuously differentiable with derivatives u′and u′′, and satisfies the following Inada type assumptions: $lim_{c→0 } u’(c) = ∞; lim_{c→∞} u’(c) = 0$
    • Suppose representative household represents set of identical households (normalized to 1).
    • Each household has an instantaneous utility function given by the above utility function.
    • $L {0}= 1 \ and \ L{t}= exp (nt)$
\[U_{0}≡ \int_0^{\infty} {e^{-\rho t}L (t)}u(C (t)) , dt=\int_0^{\infty} e^{(n-\rho ) t}u (C (t))\, dt\tag{1}\]

where $c_{t}$ consumption per capita at t, ρ is the subjective discount rate, and effective discount rate is ρ − n

  • Continues time analogue of $\sum^{\infty}{t=0}\beta^{t}{i}u_{i}(c_{i,t})$
  • Objective function (1) embeds
    • (a) Household is fully altruistic towards all of its future members, and makes allocations of consumption (among household members) cooperatively.
    • (b) strictly concave of u (·)
  • each household member $c_{t}≡\left( \frac{C_{t}}{L_{t}} \right)$
  • at time t, total of $L_{t}u(c_{t})=\exp(nt)u(c_{t})$
  • With discount at rate of $exp {(−ρt)}$, obtain (1)
  • assumption: ρ > n
  • Production possibilities set of the economy:
\[Y_{t}=F(K_{t},L_{t})\tag{2}\]
  1. 在 solow 模型里一样的
\[y_{t}=\frac{Y_{t}}{L_{t}}=F\left[ \frac{K_{t}}{L_{t},1} \right]=f(k_{t})\] \[R_{t}=F_{K} [K_{t},L_{t}]=f(k_{t})-k_{t}f'(k_{t})\] \[w_{k}=F_{L[K-t],L_{t}}=f(k_{t})=k_{t}f'(k_{t})\]
  1. Denote asset holdings of the representative household at time t by $A_{t}$
\[\dot{A_{t}}=r_{t}A_{t}+w_{t}L_{t}-c_{t}L_{t}\] \[a_{t}=\frac{A_{t}}{L_{t}}\] \[\dot{a_{t}}=(r_{t}-n)a_{t}+w_{t}-c_{t}\]
  1. The total net supply of bonds is zero in a closed economy without government
    • market clearing:
    \[a_{t}=k_{t}\]

    资产的无风险回报率：$r_{t}=R_{t}-\delta$

budget constraint

\[\dot{a_{t}}=(r_{t}-n)a_{t}+w_{t}-c_{t}\tag{3}\]

is a flow constraint

focused on No Ponzi Game Condition

The No Ponzi Game Condition

  • finite: $a_{T}=0$
  • infinite: 现值=0
\[\lim_{ t \to \infty } a_{t}e^{-\int_0^{t}(r_{s}-n)ds}\geq 0\] \[\rightarrow\lim_{ t \to \infty } a_{t}e^{-\int_0^{t}(r_{s}-n)ds}=0\] 
这个式子的意思是折现

household maximization

对课件第17页的个人理解

\[\max \ U_{0}= \int_0^{\infty} e^{-\rho t}L(t)u(c_{t})\,dt=\int_0^{\infty} e^{(n-\rho)t}u(c_{t})\,dt\] \[\text{s.t. } a' = w + (r-n)a - c\] \[a(0)=a_{0}\ is \ given\]

current-value Hamiltonian 方程:

\[H = u(c_{t})+m[w+(r-n)a-c]\] \[\text{s.t. } a' = w + (r-n)a - c\]

necessary FOCs:

\[\frac{\partial H}{\partial c}=0=u'(c_{t})-m\] \[\frac{\partial H}{\partial m}=a'\] \[\frac{\partial H}{\partial a}=-m'+(\rho-n)m=m(r-n) \tag{*}\]

$\tag{*}$

\[\lim_{ t\to \infty }e^{(n-\rho)t}m_{t}a_{t}=0\] \[\dot{a_{t}}=(r_{t}-n)a_{t}+w_{t}-c_{t}\] \[from (*):\frac{\dot{m_{t}}}{m_{t}}=-(r_{t}-\rho)\] \[u'(c_{t})=m\]

以上是证明过程，和讲义有出入 下面使用讲义表述

\[H(a,c,\mu)=u_{t}+\mu(r_{t}-n)a_{t}+w_{t}-c_{t}\]

得到

\[H_{c}=u'(c_{t})-\mu_{t}=0\tag{1}\] \[H_{a}=\mu_{t}(r_{t}-n)=-\dot{\mu_{t}+(\rho-n)\mu_{t}}\tag{2} \lim_{ t \to \infty } [e^{(n-\rho)t}\mu_{t}a_{t}]=0\tag{3}\] \[\dot{a_{t}}=(r_{t}-n)a_{t}+w_{t}-c_{t}\tag{4}\]

因此：

\[\frac{\dot{\mu_{t}}}{\mu_{t}}=\rho-r_{t}\] \[u'(c_{t})=\mu_{t}\]

凑：

\[{\frac{u''(c_{t})c_{t}}{u'(c_{t})} }\frac{\dot{c_{t}}}{c_{t}}=\frac{\dot{\mu_{t}}}{\mu_{t}}\]

得

\[\frac{\dot{c_{t}}}{c_{t}}=\frac{1}{ε(c_{t})}(r_{t}-\rho)\] \[r_{t}=f'(k_{t})-\delta\] \[\frac{\dot{c_{t}}}{c_{t}}=\frac{1}{ε(c_{t})}(f'(k_{t})-\delta-\rho)\]

Optimal growth

Steady-State EQUILIBRIUM

Transitional Dynamics

\[\dot{k_{t}}=f(k_{t})-(\delta+n)k_{t}-c_{t}\] \[\frac{\dot{c_{t}}}{c_{t}}=\frac{1}{ε(c_{t})}(f'(k_{t})-\delta-\rho)\]

references