Translog Production Function

2 years ago
Atiqur Rahman
4 minutes

The translog production function can be expressed as:

π‘Œ = 𝑓(π‘₯1, π‘₯2 ….. , π‘₯𝑛)

Where,

Y: Output

𝛼0: Efficiency Parameter

𝛼𝑖 π‘Žπ‘›π‘‘ 𝛽𝑖𝑗: π‘ƒπ‘Žπ‘Ÿπ‘Žπ‘šπ‘’π‘‘π‘’π‘Ÿπ‘  (π‘ˆπ‘›π‘˜π‘›π‘œπ‘€π‘›)

π‘₯𝑗: 𝐼𝑛𝑝𝑒𝑑 𝑗

Logarithm Form:

Taking log both sides, we obtain:

The translog production function is a generalisation of the Cobb-Douglas production function. For 𝛽𝑖𝑗 = 0, the log form of the translog function reduces to a Cobb-Douglas production function. It is linear in parameters, so it can be estimated using the least squares method.

Monotonicity and Translog Production Function:

The marginal product, i.e. addition to the total product due to the addition of one more factor, is-

𝑀𝑃𝑖 = βˆ‚Y/βˆ‚xi = βˆ‚lnY/βˆ‚lnxi . π‘Œ/π‘₯𝑖

Where βˆ‚lnY/βˆ‚lnxi is the production elasticity which can be calculated from log form.

βˆ‚lnY/βˆ‚lnxi =

  • MP of xi can be positive for a range in values of xj but can be negative if 𝛽𝑖𝑗 > 0
  • (all i,j) and xj β†’ 0. If there exists at least one 𝛽𝑖𝑗 < 0, then 𝑀𝑃𝑖 < 0 π‘Žπ‘  xj β†’ ∞. Thus, the translog function is not monotonic.

Are Isoquants under Translog Function Convex?

The isoquants are strictly quasi-convex if the Bordered Hessian matrix is negative definite. In order to construct a Bordered Hessian matrix, we need to derive second direct and cross-partial derivatives using the chain rule.

Here the values of the first and second partial derivatives vary with input levels, and there is no guarantee that the isoquants are globally convex.

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