## Examples and exercises on isoquants and the marginal rate of technical substitition

### Isoquants for a fixed proportions production function

Consider the fixed proportions production function F (z1, z2) = min{z1,z2}. The 1-isoquant is the set of all pairs (z1z2) for which F (z1, z2) = 1, or min{z1,z2} = 1. That is, the 1-isoquant is the set of all pairs of numbers whose smallest member is 1: the set of all pairs (1,z2) for z2 1 and all pairs (z1,1) for z1 1. This set is shown in the following figure, together with the isoquant for the output 2.

Now consider the fixed proportions production function F (z1, z2) = min{z1/2,z2}, which models a technology in which 2 units of input 1 and 1 units of input 2 are required to produce every unit of output. The 1-isoquant for this technology is the set of all pairs (z1, z2) for which min{z1/2,z2} = 1. This isoquant, together with the 2-isoquant is shown in the following figure.

For a general fixed proportions production function F (z1, z2) = min{az1,bz2}, the isoquants take the form shown in the following figure.

### Isoquants for a technology in which there are two possible techniques

Consider a technology in which there are two possible techniques. In each technique there is no possibility of substituting one input for another, but various mixes of the two techniques may be used by the firm. For example, perhaps machines can be operated at two possible speeds, fast and slow. If they run fast, then a relatively small amount of labor is used together with a relatively large amount of raw material (since some is wasted). If they run slowly, then a relatively large amount of labor is used together with a relatively small amount of raw material. The firm can run some of its machines fast, and some slowly. An isoquant for such a technology has the form shown in the following figure. (I am considering only raw material and labor as inputs, ignoring the machine.)

The two corners of the isoquant correspond to the case in which all the machines in the factory run slowly, and the case in which they all run fast. The points in between, on the downward sloping section, correspond to cases in which the firm runs some of its machines fast and some slowly.

### Isoquants for a production function in which the inputs are perfect substitutes

If the production function models a technology in which the inputs are perfect substitutes, then it takes the form
F (z1, z2) = az1 + bz2.
In this case the y-isoquant is the set of all pairs (z1, z2) for which
az1 + bz2 = y,
a straight line with slope a/b. Thus the isoquants are parallel straight lines:

### Isoquants for a Cobb-Douglas production function

Consider the production function
F (z1, z2) = z11/2z21/2.
The y-isoquant is the set of all pairs (z1, z2) such that
y = z11/2z21/2,
or
y2 = z1z2.
Thus the y-isoquant is a rectangular hyperbola for every value of y.

For a more general Cobb-Douglas production function the isoquants are not necessarily hyperbolae, but have the same general shape. For the production function

F (z1, z2) = Az1uz2v
the y-isoquant is given by
y = Az1uz2v.

### Exercise

Draw some isoquants for the production function
F (z1, z2) = z11/2 + z21/2.
[Solution]

### Exercise

Draw some isoquants for the production function
F (z1, z2) = z12 + z22.
[Solution]

### Marginal rate of technical substitution for a fixed proportions production function

The isoquants of a production function with fixed proportions are L-shaped, so that the MRTS is either 0 or , depending on the relative magnitude of z1 and z2.

For the specific case

F (z1, z2) = min{z1,z2},
we have
 MRTS(z1, z2) = if z1 < z2 0 if z1 > z2
and the MRTS is not defined if z1 = z2. (Along the line z1 = z2 the isoquants are kinked.)

### Marginal rate of technical substitution when the inputs are perfect substitutes

The isoquants of a production function for which the inputs are perfect substitutes are straight lines, so the MRTS is constant, equal to the slope of the lines, independent of z1 and z2.

For the specific case

F (z1, z2) = z1 + z2,
the slope of every isoquant is 1, so that
MRTS(z1, z2) = 1
for all values of (z1, z2).

In the general case

F (z1, z2) = az1 + bz2,
the slope of every isoquant is a/b, so that
MRTS(z1, z2) = a/b
for all values of (z1, z2).

### Marginal rate of technical substitution for a Cobb-Douglas production function

Consider the production function
F (z1, z2) = z11/2z21/2.
The y-isoquant of this production function is defined by the condition
y2 = z1z2,
or
z2 = y2/z1.
Differentiating, we obtain
dz2/dz1 = y2/z12 = z2/z1.
Thus
MRTS(z1, z2) = z2/z1.

NOTE: This is NOT a general formula for the MRTS! It is specific to this example.

### Exercise

Find the MRTS for the production function
F (z1, z2) = z11/2 + z21/2.

[Solution]

### Exercise

Which of the following production functions has a diminishing marginal rate of technical substitution?
• F (z1, z2) = z1 + z2.
• F (z1, z2) = z11/2z21/2.
• F (z1, z2) = z12 + z22.
[Solution]