$$ \begin{aligned}E_{t}[\Gamma_{t+s}]-E_{t-1}[\Gamma_{t+s}]&=\frac{\alpha(E_{t}[\sigma^{2}\xi_{t+s}]-E_{t-1}[\sigma^{2}\xi_{t+s}])}{2}\\&=\frac{\alpha\varphi_{s}\lambda_{t}}{2}\quad\text{with}\quad\sum_{j=0}^{\infty}\left(\frac{1}{1+r}\right)^{j}\varphi_{j}<\infty\end{aligned} $$

となる．これを(40)式と置き換えて同様の計算を行えば，(46)-(48)式が得られる．[QED]

7．「6．6．絶対的危険回避度遅減型効用関数と「緩衝在庫」モデル」における(60)(61)

(54)-(59)式を,それぞれ $ x_{t}=x^{p_{t}}=x^{p^{*}} $ , $ x^{t_{t}}=0 $ , $ e^{p_{t}}=e^{t_{t}}=1 $ , $ H_{t}=H^{*}=x^{p^{*}}/r $ , $ \sigma^{2}\xi_{t+1}=\sigma^{*2} $ , $ W_{t}=W^{*} $ , および $ c_{t}=c^{*} $ の近傍で線形近似すれば,次の6本の方程式体系

$$ \begin{aligned}&c_{t}-c^{*}=\beta_{W}(W_{t}-W^{*})+\beta_{H}(H_{t}-H^{*})+\beta_{\sigma}(\sigma^{2}\xi_{t+1}-\sigma^{*2})\\&W_{t+1}-W^{*}=(1+r)\{(W_{t}-W^{*})-(c_{t}-c^{*})\}+(x_{t+1}-x^{p^{*}})\\&x_{t}-x^{p^{*}}=x^{p_{t}}-x^{p^{*}}+x^{t}_{t}\\&(x^{p_{t+1}}-x^{p^{*}})-(x^{p_{t}}-x^{p^{*}})=(e^{p_{t+1}}-1)x^{p^{*}}\quad\quad(\because(1-1)(x^{p_{t}}-x^{p^{*}})=0)\\&x^{t}_{t}=(e^{t_{t}}-1)x^{p^{*}}\quad\quad(\because(1-1)x^{p_{t}}=0)\\&H_{t}-H^{*}=\frac{x^{p_{t}}-x^{p^{*}}}{r}\\ \end{aligned} $$

が得られる.この6式に新しい変数$c_{t}\equiv c_{t}-c^{*},W_{t}\equiv W_{t}-W^{*},H_{t}\equiv H_{t}-H^{*},\sigma^{2}\xi_{t+1}\equiv\sigma^{2}\xi_{t+1}-\sigma^{*2},x^{p_{t}}\equiv x^{p_{t}}-x^{p^{*}},\varepsilon^{p_{t+1}}\equiv(e^{p_{t+1}}-1)x^{p^{*}},\varepsilon^{t}\equiv(e^{t}-1)x^{p^{*}}(=x^{t_{t}})$,および$\beta_{xp}\equiv\beta_{H}/r$を代入し整理すれば,(60)(61)式が得られる.

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