See above graph.
20 - -
reaches max. at L
here multiplies output by 4 (compare a and c). Hence the
function exhibits increasing returns to scale.
To calculate the elasticity
of substitution, it's best to express the MRTS as some
of the capital-labor ratio (K1L). SOMETIMES, this isn't possible (as
in the case where the MRTS is NOT well-defined, and other times, the MRTS
may be constant and therefore NOT depend on K1L. Let's then proceed and see
where it takes us:
Here, we know that the MRTS is NOT well-defined because the production
function is not differentiable. HOWEVER, we know that for perfect complement
(Leontief) technologies, the finn will ALWAYS use K and L in the same FIXED
proportion (here 2L for every 5K, so a K1L ratio of2/5)! Thus, because the
percentage change in K1L is ALWAYS 0, we know that the elasticity
substitution will be
as well. This result makes sense in that NO MATTER
WHAT happens to the MRTS (as it goes around the isoquant and moves from
infinity to zero) the effect on the K1L ratio is always O. Thus, with fixed
proportion technologies, K and L are NOT AT ALL substitutable, so the EASE
this substitution (and therefore the value
of cr) is O.
Here, we know that the MRTS is CONSTANT (and equal to 16/37 for those
you following along at home)! Thus, the denominator in my elasticity expression
is now 0, because it (the MRTS) NEVER changes. One can certainly have
different K1L ratios here as we move around an individual isoquant, so this
situation implies an elasticity
of substitution that's infinite! Make sense? Sure,
when one realizes that with perfect substitutability, the ease with which K can be
substituted for L is invariant to where one is on the isoquant. .
...thus, that ease is
absolute (and infinite in tenns
of cr ).
Now, perhaps, the FUN begins. Find an expression for the MRTS here, and
convince yourself that:
as one might expect with a Cobb-Douglas looking production function. So, given
the fact that the K1L ratio is log-linear in the MRTS, we have that this elasticity
substitution is 1 . Why? Well, as we move around the isoquant, the MRTS
changes proportionally to the change in the K1L ratio used. This implies EASIER
substitutability than in (a) but NOT as easy substitutability as in (b). Reflect on
the fact that cr is essentially tracking how movements in the MRTS drive
movements in the cost-minimizing K1L ratio to produce a given quantity.