Time and Production. Production in the Short-Run

Production Function

A production function is a model (usually mathematical) that relates possible levels of physical outputs to various sets of inputs. It shows the maximum amount of the good that can be produced using alternative combinations of capital (K) and labor (L):

Q = f (L, K, technology, ...).

Here we will use a Cobb-Douglas production function that usually takes the form: Q = AL^a K^b

Talking about production function it is important to note that it shows as well theproduction technology which means the quantitative relationship between inputs and outputs. There are two kinds of technologies:

Labor-intensive technology relies heavily on human labor instead of capital.

Capital-intensive technology relies heavily on capital instead of human labor.

And to find out which is more effective we use such terms as:

1) Technical efficiency defined as a ratio of output to input:

Efficiency Technical = Output/ Input

2) Economical efficiency defined as a ratio of value of output to value of input:

Efficiency Economical = Output value / Input value

In this simplified version, each production function or process is limited to increasing, constant or decreasing returns to scale over the range of production. In more complex production processes, "economies of scale" (increasing returns) may initially occur. As the plant becomes larger (a larger fixed input in each successive short-run period), constant returns may be expected. Eventually, decreasing returns or "diseconomies of scale" may be expected when the plant size (fixed input) becomes "too large." This more complex production function is characterized by a long run average cost (cost per unit of output) that at first declines (increasing returns), then is horizontal (constant returns) and then rises (decreasing returns).

This production function can exhibit any returns to scale:

f(mK,mL) = A(mK)^a(mL) ^b = Am^a+b K^aL^b = m ^a+b f(K,L)

– if a + b = 1 – constant returns to scale

– if a + b > 1 – increasing returns to scale

– if a + b < 1 – decreasing returns to scale

As the period of time is changed, producers have more opportunities to alter inputs and technology. Generally, four time periods are used in the analysis of production:

"Market period" – a period of time in which the producer cannot change any inputs nor technology can be altered. Even output (Q) is fixed.

"Short-run" – a period in which technology is constant, at least one input is fixed and at least one input is variable.

"Long-run" – a period in which all inputs are variable but technology is constant.

"The very Long-run" – during the very long-run, all inputs and technology change.

Cost analysis in accounting, finance and economics is either long run or short-run.