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

This article covers the concepts of short-run production function, returns to scale, marginal and average cost, cost minimization, and MRTS in microeconomics. It also includes solved examples and equations for better understanding.

Economics 1
Microeconomics
By [Name]
Institution
Date

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

Economics 4
output will be Q=2.828 which shows that the output, Q will increase by more than the double
increase in the value of inputs.
d. Q=4L0.5+4K
The function above shows an exhibit decreasing return to scale. For example, if L=2 and K=2,
then the value of Q=13.66. However, doubling L=4 and K=4, output Q will be 24 thus increasing
by less than double. As a result, Marginal Production of labor will decrease as the value of
Marginal Production of capital remains constant. As a result, for any value of labor that is given,
when capital increases by a unit, the output will have to increase by 4 units.
Question 3
Marginal Rate of Technical Substitution (MRTS) is the rate at which inputs are substituted with
a constant output. It is therefore defined as;
MRTS = |-FL / FK | = FL / FK
Thus, from the definition above, MRTS, therefore, measures an additional value of capital that is
required to replace one unit of labor with fixed output, Q.
a. Q =L0.5K0.5
PML =0.5(K/L)0.5
PMK =0.5(L/K)0.5
MRTS = |-FL / FK | = FL / FK
K/L

Economics 5
b. Q =L0.5 +K0.5
PML =0.5L-0.5
PMK =0.5K-0.5
MRTS = |-FL / FK | = FL / FK =(K/L)0.5
c. Q =L +K
PML =15
PMK =1
MRTS = |-FL / FK | = FL / FK = 1 which is constant
Question 4
By definition, Marginal cost (MC) is the extra cost incurred due to the production of an extra
output while average cost (AC) is the total cost per unit of the total output. A decrease in the
Marginal cost leads to a decrease in the value of Average cost because at any point that an extra
unit is going to be cheap than the average cost, then the curve of the Average Cost will be pulled
down. Likely, an increase in the value of MC leads to a higher value of the Average cost. Thus,
the point of intersection between Marginal Cost and Average Cost is the minimum point for the
Average Cost.
From the functions given i.e. where the Total Cost (TC) is 2q3 -16q2 +90q thus Average cost will
be given by dividing the total output by the unit, q. This is therefore given as;
AC=2q2-16q +90

Economics 6
Marginal Cost is therefore found by taking the first derivative off the Total Cost with respect to
Q i.e.
dTC/dq =6q2 -32q +90
However, to find the minimum point where MC intersects the AV at the minimum, we will take
the derivative of AC and equate it to zero.
dAC/dq = 4q – 16 = 0. Finding the value of q;
q*=4 showing that MC=AC at the point where output is 4 units
Question 5
a. False since the firm must weigh both its implicit and explicit costs before making a
decision on whether to borrow or not to borrow. As a result, the interest rate that the firm
will receive on its cash reserve is known as implicit opportunity cost whiles the interest
rate that the firm must pay when it borrows funds to the lender is known as the explicit
cost. It will now depend on the management on how best to measure its costs of either
borrowing or lending the cash. If the interest rate received from the cash reserve is greater
than the rate of borrowing the cash for the equipment, then the firm is well off to borrow
the cash to finance its equipment
b. True. This is because the university must weigh the opportunity cost of buying the parcel
of land away from its current campuses, or to just set up a building with no rental interest
rate. The interest rate accrued due to the yearly rental of a premise will, therefore, be
higher than setting the building at the downtown thus helps to minimize the cost showing
that it is cost effective.

Economics 7
Question 6
a. Q=15L0.25K
Introducing Lagrangian, we get the following equation
L= 15L0.25K + ک( 10,000-10 L -50K) …………………………….1
Finding the first order condition with respect to L, K, and ک
LL =0.25(15)L-0.75K-10 = ک0..…………………………………2
LK =15L0.25 -50 = ک0…………………………………………3
L = ک10,000-10 L-50K=0 ……………………………………4
Using equation 2 and equation 3, we can derive the optimal ratio of capital to labor i.e.
K/4L =10/50
If we rearrange, we, therefore, get K= 4L/5
Substitute K= 4L/5 into the 4th equation i.e.
10,000 =10L +40L =50L
L*=200 and K*=160, Q*=9025.40
b. Yes. The first two conditions i.e. equation 2 and equation 3 generated the MRTS
condition where w/r is equated to the cost minimization problem. As a result, if the firm
is in the processes of producing maximum output at a particular cost, then it must adhere

Economics 8
to produce output at a lower cost in order to maximize profit. This is known as the
theorem of duality in the cost production process.
Question 7
a. Min rL +wK
Introducing Lagrangian, we get the following equation
L=rL +wK – ک (L0.25K0.75 –C)
b. First Order condition with respect to Labor and Capital
LL = r- ) ک0.25( L-0.75K0.75 ………………………………………1
LK = w- ) ک0.75( L0.25 K-0.25 …………………………………….2
c. Equating the first equation in (b) and the second equation
r/(0.25)L-0.75K0.75 = ک = w/(0.75)L0.25 K-0.25
If we rearrange, we get;
(0.75)L0.25 K-0.25 / (0.25)L-0.75K0.75 =w/r
Showing that 3L=w/r which is also the same as L=wK/3r
It, therefore, shows that the cost minimization of capital and labor is wK/3r hours of labor for
every unit of capital invested. As a result;
C= L0.25K0.75 = (wK/3r)0.25K0.75

Economics 9