Microeconomics: Short-Run Production Function, Returns to Scale, MRTS, Marginal and Average Cost, Cost Minimization

Economics 1
Microeconomics
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Institution
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Economics 2
Microeconomics
Question 1
a. 𝑓(𝐾, 𝐿) = 50LK0.5 + L2K2 –L3K3 ........................................ 1
A short-run production function shows the maximum amount of a single quantity that the
function can generate given by the sets of capital and labor, assuming that one input is fixed.
From the function above, capital is fixed at 4 units thus short run function will be calculated as
follows;
( 𝐿) = 50L(40.5) + L242 –L343
( 𝐿) =100L +16L2 -64L3
b. Marginal production of labor (MPL) is the extra output produced due to an additional unit
of labor to the production process with fixed inputs of other units. On the other hand,
Average Production of labor (APL) is the output per the labor unit, also referred to as
labor productivity. From equation (1) above, we can calculate the Marginal production of
labor (MPL) as follows;
MPL =Change∈total Production ¿ ¿
MPL =50*1L0K0.5 +2LK2 -3L2K3
MPL = 50K0.5 +2LK2 -3L2K3
And, APL will be calculated as follow with respect to equation (1) above;
APL= f (K , L)
Labor
APL = (50LK0.5 + L2K2 –L3K3)/L

Economics 3
APL =50K0.5 +LK2 –L2K3
c. The elasticity of labor will be represented with the symbol Ę L thus calculated as;
Ę L = Marginal production of labor (MP L)
Average Production of Labor ( AP L)
(50K0.5 +2LK2 -3L2K3)/ 50K0.5 +LK2 –L2K3
Question 2
a. Q =M0.5K0.5L0.5
The function above exhibits an increasing return to scale. For example, if we have M =2, K=2,
and L=2, then the value of Q=2.828. However, if the value of inputs is doubled, say M=4, K=2,
and L=4, then the value of Q=8 which shows that the output, Q will increase by more than the
double. It, therefore, shows that doubling the value of the material, capital, and labor will lead to
an increased value of output, Q with more than double.
b. Q= L +0.5K
The function above shows an exhibit constant return to scale. For example, if L=2, and K=2,
then the total value of Q=3. Again, doubling the value of inputs, i.e. L=4 and K=4, then the
output will be Q=6. It, therefore, shows that when the doubling the value of inputs leads to an
increased value of output, Q with the same double value.
c. Q =0.5LK0.25
The function above shows exhibit increasing return to scale. For example, if L=2, and K=2, then
the total value of Q=1.189. Again, doubling the value of inputs, i.e. L=4 and K=4, then the