Unformatted text preview: s production twice as expensive.
The cost function is homogeneous of degree one in the prices of inputs: doubling prices implies doubling
the total cost of producing Q=30.
(d) Find the marginal product of capital and labor: ( ) ( ) ( , ) ( ) (e) Find the average product of capital and labor:
( ) ( ) (f) Suppose that the actual production function is in reality given by:
( ) where denotes the decade we’re talking about -beginning in the 20th century. This is:
from 1911 to 1920,…,
from 1981 to 1990..etc. What is the total cost of producing
20 units of output in 1906? And in 1994?
Notice that the scaling factor (time period) doesn’t distort the optimality condition; this is, at an
optimum, the relationship between and will not be affected. However, when substituting the values
for or into the constraint, the SIZE of these two ( and ) will indeed change depending on the
( ) ( ) The F.O.C. are given by:
() ( ) () ( ) () ( ) ( ) =0 ( ) =0 From (1) and (2) we obtain the optimality condition:
Which we can replace into the constraint: (
Or ( )
( , so that (
) If )
and ( )( ) and using the optimality condition we obtain ( )( ) The total cost of producing
( ) ) ( ) units is now:
( ) ( )( ) the cost of producing 20 units in 1906( ( )( )
) is ()
( ) The cost of producing the same number of units in 1994 is ( ) ()...
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