# 进一步的计量经济学 Further Econometrics ECON60622T

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There is still another motivation available for the imperfect price level adjustment we are assuming. For reasons of simplicity, we here consider the case of a Cobb-Douglas production function, given by $Y=K^{\alpha} L^{1-\alpha}$. According to the above we have
$$p=w / F_{L}\left(K, L^{p}\right)=w /\left[(1-\alpha) K^{\alpha}\left(L^{p}\right)^{-\alpha}\right]$$
which for given wages and prices defines potential employment. Similarly, we define competitive prices as the level of prices $p_{c}$ such that
$$p_{c}=w / F_{L}\left(K, L^{d}\right)=w /\left[(1-\alpha) K^{\alpha}\left(L^{d}\right)^{-\alpha}\right]$$
From these definitions we get the relationship
$$\frac{p}{p_{c}}=\frac{(1-\alpha) K^{\alpha}\left(L^{d}\right)^{-\alpha}}{(1-\alpha) K^{\alpha}\left(L^{p}\right)^{-\alpha}}=\left(L^{p} / L^{d}\right)^{\alpha}$$

As far as consumption is concerned we assume Kaldorian differentiated saving habits of the classical type $\left(s_{w}=1-c_{w}=1-c \geq 0, s_{c}=1\right)$, i.e., real consumption is given by:
$$C=c v Y=c \omega L^{d}, \quad v=\omega / x, \omega=w / p \text { the real wage }$$
and thus solely dependent on the wage share $v$ and economic activity $Y$. For the investment behavior of firms we assume
\begin{aligned} \frac{I}{K} &=i_{1}((1-v) y-(i-\pi))+n \ y &=\frac{Y}{K}, \quad n=\hat{L}+\hat{x}=n+n_{x} \text { trend growth. } \end{aligned}