Consider the 2 × 2 production model. Production functions are given by: f1(z11, z21) = (z11)^α1...

Consider the 2 × 2 production model. Production functions are given by: f1(z11, z21) = (z11)^α1 · (z21)^1−α1 f2(z12, z22) = (z12)^α2 · (z22)^1−α2 where αj ∈ (0, 1) for all j = 1, 2 and zij ≥ 0 for all i = 1, 2 and j = 1, 2. Show mathematically that if the set of technologically efficient factor allocations (the Pareto set) ever cuts the diagonal of the Edgeworth Box, then it must coincide with the diagonal.

Expert Solution

Technical efficiency is the effectiveness with which a given set of inputs is used to produce an output. A firm is said to be technically efficient if a firm is producing the maximum output from the minimum quantity of inputs, such as labour, capital, and technology.

Technical efficiency requires no unemployment of resources.

Given a certain quantity of inputs (natural resources) – technical efficiency is achieved when we produce the maximum output possible. Note, we could produce all guns or all butter.

Technical efficiency rate

Example: Suppose a firm produces 300 cars a week from its current workforce and quantity of robots. If the maximum potential output was 320.

The technical efficiency rate of the firm would be (300/320) = 93.75%

In this case, some workers and factor inputs are underemployed, and there is a degree of potential output missed.

Related concepts

Productive efficiency. The concept of technical efficiency is related to productive efficiency. Productive efficiency is concerned with producing at the lowest point on the short run average cost curve. Productive efficiency requires technical efficiency.

X-inefficiency. The concept of technical efficiency is also related to X-inefficiency. X-inefficiency is said to occur when a firm fails to be technically efficient because of an absence of competitive pressures. e.g. a monopoly employs inefficient working practices because it has no incentive to cut costs.

Here since (z11)^α1 · (z21)^1−α1 f2(z12, z22) = (z12)^α2 · (z22)^1−α2 where αj ∈ (0, 1) for all j = 1, 2 and zij ≥ 0

si z21=z22 which depends on its variables and the factors

Rahul Sunny answered 1 month ago

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