Question

What kind of returns to scale does a perfect complements production have? Explain your answer.

Answer 1

The production functions of firms for a particular good represents the maximum amount of good which can be produced using alternative combinations of factor inputs. They may be of the form y = f(Z_1,Z_{2, ....,}Z_n) where z = factor inputs like capital , labour, land etc.

Isoquants are quite similar to indifference curvesand shows the various combinations of the factor inputs which can produce a given level of output. Just like indifference curves, two isoquants can not cross. Each isoquant represents different levels of output. As we move onto higher isoquants, the output increases. The relationship between output and the factor inputs is given by the production function.

The slope of the isoquant shows the rates at which the two factor inputs can be substituted for one another. It is equal to MRTS or Marginal Rate of Technical Substitution which is the number of factor inputs one needs to give up in order to obtain one unit of the other factor , holding the output constant.

Now, perfect complements production is a kind of fixed proportions production function. It is not possible to do substitution among these inputs and each output requires a certain combination of inputs. Both the factor inputs need to be increased in order to increase the output. The isoquants of a perfect complements production function are L-shaped, with a kink at factor 1 , say L = factor 2, say K. Take a look at fig 1. MRTS = slope of isoquants = MP_L/MP_K

Now, what is Returns to Scale? It is the effect of a proportionate change in the factor inputs on the output. For example, if all the inputs are doubled, will the output double?

If the output increases in the same proportion, i.e. it doubles if inputs are doubled, it is said to have cconstant returns to scale. If the output is lesser than doubled than it is decreasing returns to scale. And if the outputs are more than doubled when inputs are doubled, it shows increasing returns to scale.

Now, the generalized perfect complements production function can be written as:

q = f(z₁, z₂) = {min(az₁ + bz₂)}

Where z₁, z₂ are the two factor inputs, this production function will show:

^{= 1, Constant returns to scale}

^{> 1 , Increasing returns to scale}

^{< 1, Decreasing returns to scale.}

So, a perfect complements can have any of the three, CRS, DRS or IRS.

If the perfect complements function : F(z_1,z₂) = min{z₁, 2z₂} , in this case, 2 units of z₁ and 1 unit of z₂ is needed to produce every unit of output. This coupled with CRS production fucntion can be seen in fig 2. This is a perfect 2 with 1 complement.

Or, in case that the two inputs are perfect 1 with 1 complement, i.e. the firm always requires to combine each unit of z₁ with 1 unit of z₂. So, the example of such a production function with a decreasing returns to scale can be given by F(z_1,z₂) = min{z₁, 2z₂} ^0.8. Take a look at fig 3.