Time-varying elasticity of input factors

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Production technogy based on a unit-elasticity (Cobb-Douglas) production function

\[ y_t = F(a_t, b_t) = a_t{}^\gamma \ b_t{}^{1-\gamma} \]

Period profits are given by

\[ \Pi_t \equiv py_t\, y_t - pa_t\, a_t - pb_t\,b_t - \Xi_{y,t} \]

and include the cost of changing the input factor proportions

\[ \Xi_{y,t} \equiv \tfrac{1}{2} \xi_y \left[ pa_t \, \extern{a}_t \left( \bDelta \blog \frac{a_t}{\extern{y}_t} \right)^2 + pb_t \, \extern{b}_t \left( \bDelta \blog \frac{b_t}{\extern{y}_t} \right)^2 \right] \]

Optimization problem with a possibly heavier discounting, \(\beta_y\in[0,1]\), to incorporate higher uncertainty of future profit flows

\[ \bmax_{\{a_t, b_t\}}\ \E_0 \sum_{t=0}^\infty \left( \beta\, \beta_{y} \right)^t \, vh_t \, \Pi_{y,t} \]

Optimal choice of input factors (omitting higher-order terms from the adjustment costs)

\[ \gamma \, py_t \, y_t \approx pa_t \, a_t \left[ 1 + \xi_t \left( \bDelta \blog \frac{a_t}{y_t} - \beta \, \beta_{y} \, \bDelta \blog \frac{a_{t+1}}{y_{t+1}} \right) \right] \]

\[ (1- \gamma) \, py_t \, y_t \approx pb_t \, b_t \left[ 1 + \xi_t \left( \bDelta \blog \frac{b_t}{y_t} - \beta \, \beta_{y} \, \bDelta \blog \frac{b_{t+1}}{y_{t+1}} \right) \right] \]