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Production math

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Productivity

Global productivity component

\[ \bDelta \blog a_t^{gg} = \rho_a \ \Delta \blog a_{t-1}^{gg} + (1-\rho_a) \left[ \blog \roc{a}^{gg}_\ss + \epsilon_{a,t} \right] \]

Area-specific relative productivity component

\[ \blog ar_t = \rho_{ar} \ \blog ar_{t-1} + (1-\rho_ar) \ \blog ar_\ss + \epsilon_{ar,t} \]


Total area productivity

\[ a_t = a^{gg}_t\, ar_t \]

Production stages

Stage \(T-4\): Combine imports from the rest of the world

Production function with time-varying elasticity of substitution

\[ mm_t = F_4\left( mm_t^1, \dots, mm_t^A \right) \]
\[ mm = my_t + mx_t \]

where

  • \(my_t\) is the intermediate import inputs into local production

  • \(mx_t\) is the intermediate import inputs into export production (re-exports)

Stage \(T-3\): Combine non-commodity variable factors

Production function with time-varying elasticity of substitution

\[ \begin{gathered} y_{3,t} = F_3\bigl( mm_t, nv_t\bigr) \newl nv_t \equiv \left( nh_t - \gamma_{nv} nh_\ss \right) \, nf_t \end{gathered} \]

where

  • \(nv_t\) is the variable labor input with \(\gamma_{nv} nh_\ss\) being the overhead labor needed to maintain production regardless of the output actually produced

Stage \(T-2\): Combine variable factors with business capital

\[ \begin{gathered} y_{2,t} = F_2\bigl( uk_t\, y_{3,t} \bigr) \newl uk_t = u_t \, k_t \end{gathered} \]

Stage \(T-1\): Add dependence on commodity inputs

  • Short-term: No elasticity of substitution (Leontief)
\[ y_{1,t} = \min \left\{ \frac{y_{2,t}}{\alpha_{y1}},\ \frac{mq_t}{\alpha_{mq}} \right\} \]
  • Long-term: Unit elasticity of substitution (Cobb-Douglas)
\[ y_{1,t} = a_{y1} \cdot \kappa_{y1} \cdot y_{2,t}{}^{1-\gamma_{mq}} \cdot mq_t{}^{\gamma_{mq}} \]

Stage \(T-0\): Sticky prices

Sticky price setting

Total profits

Total profits summed up across all production stages are given by

\[ \begin{multline} \Pi_{y,t} \equiv py_t \, y_{0,t} - pmm_t \, my_t - w_t \, nh_t \, \xnf_t - pu_t \, u_t\, k_t - pq_t \, mq_t \ \cdots \\[10pt] \cdots -\ \Xi_{y4,t} - \Xi_{y3,t} - \Xi_{y2,t} - \Xi_{y1,t} - \Xi_{py,t} \end{multline} \]

Final goods

The final goods produced domestically are demanded as one of the following types of goods

  • Private consumption (by households), \(ch_t\)

  • Government consumption, \(cg_t\)

  • Private investment (by households), \(ih_t\)

  • Inputs into export production, \(yx_t\)

The market clearing conditions is therefore given by

\[ y_t = ch_t + cg_t + ih_t + yx_t \]