PhD Macroeconomics question
Fire 2
ThoeryofProduction.docxTheory of Production
In microeconomics, production is the act of making things, in particular the act of making products that will be traded or sold commercially. Production decisions concentrate on what goods to produce, how to produce them, the costs of producing them, and optimizing the mix of resource inputs used in their production. This production information can then be combined with market information (like demand and marginal revenue) to determine the quantity of products to produce and the optimum pricing.
Production is the process of combining and coordinating inputs (resources or factors of production) to create goods or services. Production is a process whereby certain goods and/or services are used to create goods and/or services of different nature. Production is the name given to the process of conversion of inputs into output. Farm production refers to the producing of food fiber and livestock by using several different kinds of inputs. Farmers use land as a factory that helps them to produce desired crops. To this manufacturing plant (land) labor and capital are added to cultivate plants and harvest the crop. The crops produced are in turn, consumed by the population, fed to animals which produce meat, milk, eggs and many other livestock and poultry products through complex biological processes.
Technical transformation
Inputs (Xi)
{Land, Labor, Capital & Management}
Output (Qi)
{Say wheat grain, straw}
Fig.1. Production process
What is Production?
In economics, Production is a process of transforming tangible and intangible inputs into goods or services. Raw materials, land, labor and capital are the tangible inputs, whereas ideas, information and knowledge are the intangible inputs. These inputs are also known as factors of production. Production in economics is sometimes defined as the creation of utility. Utilities are created in three forms:
· Form utility
· Time utility
· Place utility
What is the production function?
The production function is a mathematical representation (equation) that explains the relationship between the factors of production (inputs) and the outputs. Production function answers the queries related to marginal productivity, level of production, and the least-cost mode of production of goods and services. Symbolically;
Q= f(X)
Q= f(Xi) where X is input and i = 1………n
Q = f (Land, Labor, Capita, Management,……..)
That is the quantity of output is a function of the inputs listed inside the parentheses. There may be a different combination of different inputs to produce the desired amount of output.
Fixed Inputs vs Variable Inputs
Fixed or variable is an attribute of inputs with respect to the planning horizon for the production process. A fixed input is one whose quantity a manager cannot change during a given time. A variable input is one whose quantity a manager can change during a given time.
Q = f(X1, X2/X3,……..,Xn)…… production function with fixed inputs (X1 and X2 are fixed)
Q = f(X1, X2, X3,……..,Xn)…… production function with no fixed inputs (all variable)
There are two types of production function, namely long run, and short run, depending on the nature of the input variable. The first one represents the short-run and the later one represents the long-run production function.
Total, Average and Marginal Product
Total product is the output derived through the completion of inputs transformation into what we call something produced.
X Q
Transformation/black box
Total Product (TP or Q) = f(X)
Average Product (AP) = TP/Q; Marginal Product (MP) = . These relationships can be shown in the tabular form:
|
Table 1. A simple production function[footnoteRef:1] [1: In this example, the underlying equation showing Total, Average and Marginal products as a function of the amount of labor, L (with the level of capital assumed constant), are TP = 10L + 4.5L2 – 0.3333L3 AP = 10+4.5L -0.333L2 MP = = 10 + 9L – 1.0L2 Adapted from: Farnham, Paul, G.2014. Economics for Managers. 3rd Edition. Pearson.] |
|||||
|
Quantity of Capital (K) |
Quantity of Labor (L) |
Total Product (TP or Q) |
Average Product (AP) |
Marginal Product (MP) ( |
Marginal Product (MP)
|
|
10 |
1 |
14 |
14 |
14 |
18 |
|
10 |
2 |
35 |
17.5 |
21 |
24 |
|
10 |
3 |
62 |
20.7 |
27 |
28 |
|
10 |
4 |
91 |
22.8 |
29 |
30 |
|
10 |
4.5 |
106 |
23.6 |
30 |
30.25 |
|
10 |
5 |
121 |
24.2 |
30 |
30 |
|
10 |
6 |
150 |
25 |
29 |
28 |
|
10 |
6.75 |
170 |
25.1875 |
26.67 |
25.1875 |
|
10 |
7 |
175 |
25 |
25 |
24 |
|
10 |
8 |
197 |
24.6 |
22 |
18 |
|
10 |
9 |
212 |
23.6 |
15 |
10 |
|
10 |
10 |
217 |
21.7 |
5 |
0 |
|
10 |
11 |
211 |
19.2 |
-6 |
-12 |
Table 1 presents a numerical example of simple production function. MP can be calculated either for discrete changes in labor input (column 5) or for infinitesimal changes in labor input using the specific MP equation in the table (column 6). Column 5 shows the MP between units of input (column 2), whereas column 6 shows the MP calculated precisely at a given unit of input. Column 6 gives the exact mathematical relationships.
Production relationship
There are three types of production relationships.
· What to produce (Product- Product Relationship)
· How much to produce (Factor- Product Relationship)
· How to produce (Factor-Factor Relationship)
Economic optimum: in factor-product relationship
· Value marginal product (VMP)[footnoteRef:2] equals marginal input cost (MIC) or marginal physical product equals to price ratio. [2: VMP = MP* Price of output. MIC is price of input used.]
· Marginal cost (MC) cuts Marginal Revenue (MR) from below. In a perfectly competitive market, MR=AR = Price of unit product(P)
Economic optimum: in factor- factor relationship
For the profit maximization, there should be:
· Marginal rate of technical substitution (MRSx 1x 2) equals to inverse price ratio.
· The slope of the isoquant is equal to the slope of the isocost line.
Economic optimum: in product-product relationship
MRTS,C = - (ignore the negative sign)
The negative sign reflects that the MRT of two products is generally negative, since increasing output of one requires production of the other to be reduced.
Soyabean (S)
S*
Tangent point
Production Possibility frontier
Isorevenue line
C*
Corn (C)
Fig. Economic optimum in product-product relationship
For the profit maximization, there should be:
· MRT equals the inverse price ratio.
· The slope of the production possibility frontier equals the slope of the isorevenue line.
Relationship Among Total, Average and Marginal Product (Graphical)
If we plot the TP, AP and MP data from the table above, the following figure would be established.
Fig. TP, AP and MP: short-run production function.
Most important message from this graph is to understand the behavior of marginal product curve. The MP curve increases up to labor input level L1. We call this the region of increasing marginal returns. Once we have employed L1 units of labor, the marginal product of labor begins to decline and keeps decreasing until it becomes zero, when L3 units of labor are utilized. This portion of the marginal product curve illustrates what is known as the law of diminishing marginal returns (or the law of diminishing marginal product). All short-run marginal product curves will eventually have a downward slopping portion and exhibit this law. Beyond L3 units of labor, the marginal product of labor is negative. This is the region of negative marginal returns. Economic optimization occurs when the labor units are employed between L2 and L3. It is irrational to stop production at L1 because MP is increasing and there is still a venue to utilize labor productivity. Similarly, production after L3 would be irrational because MP is zero and hence there is no room to further harvesting labor productivity.
Production and Cost Analysis
Production and cost are two building blocks on the supply side of the market. Just as consumer behavior forms the basis for demand curves, producer behavior lies behind the supply curve. The prices of the inputs of production and the state of technology are two factors held constant when defining a market supply curve. Production functions (process) and corresponding cost functions, which show how costs vary with the level of output produced, are also very important when we analyze the behavior and strategy of individual firms and industries.
Producer Equilibrium
Production is a very important economic activity. The survival of any firm in a competitive market depends upon its ability to produce goods and services at a competitive cost. One of the principal concerns of business managers is the achievement of optimum efficiency in production by minimizing the cost of production. Here comes the concept of Effectiveness and Efficiency[footnoteRef:5] to ensure the objective of value optimization[footnoteRef:6] of firm. It is possible to determine the optimum amount of production possible considering different combinations of these inputs. Such a determination is called the producer’s equilibrium. [5: Manager is a person while management is a process of accomplishing multivariate tasks to achieve an objective of an organization. Manager is deemed to be effective and efficient. Effectiveness is “do the right things” and efficient is “do things well”.] [6: Economic optimization is the single most goal of any organization. Value of the firm = present value of expected future profit = = Economic Value = π = Profit, TR = Total revenue and TC = Total Cost So, in managerial economics, the primary objective of management is assumed to be “maximization of the value of the firm.]
This optimum level of production, also called producer’s equilibrium, is achieved when the maximum output is derived from minimum costs. In order to achieve this, producers first have to classify their resources into different combinations. Each combination would provide production in different quantities. The combination that provides the highest amount of produce at the least amount of costs is the optimum level of production. To find out producer’s equilibrium, we first need to understand isoquant curves and iso-cost. These two concepts help us calculate optimum production.
Isoquant (Iso-product): The Isoquant curves show the different combinations of two different inputs with which a firm can produce equal amount of output (product). We can also call them equal-product curves or production indifference curves. Just like indifference curves, isoquants are also negatively sloping and convex to the origin. They never intersect with each other. The curve on the right represents greater output and curves on the left show less output.
Isocost Lines: Isocost lines represent combinations of two factors that can be bought with different outlays. In other words, it shows how we can spend money on two different factors to produce maximum output.
An isocost line is a curve which shows various combinations of two different inputs that can be purchased with the same amount of cost. For the two production inputs say labor and capital, with fixed unit costs of the inputs, the isocost curve is a straight line. This line is also called budget line or budget constraint lines to the producer. Graphically,
Fig. Isocost line
This is obtained where the slope of the isoquant is equal to the slope of the isocost line. The slope of the isoquant is called Marginal Rate of Technical substitution[footnoteRef:7]. If this MRTS is equated with inverse price ratios (here price ratio of Capital and Labor inputs) of two inputs, there is the cost minimizing point or least cost combination of inputs to produce the predetermined level of output. [7: The marginal rate of technical substitution (MRTS) is an economic theory that illustrates the rate at which one factor must decrease so that the same level of productivity can be maintained when another factor is increased. The MRTS reflects the give-and-take between factors, such as capital and labor, that allow a firm to maintain a constant output. MRTS differs from the marginal rate of substitution (MRS) because MRTS is focused on producer equilibrium and MRS is focused on consumer equilibrium. MRTS is the slope of an Isoquant curve. ]
(where L)
Numerical Example for Least Cost Combination of 2 Inputs
Problem.1. From the given data set of labor (hours) and nitrogen usage (lb) for producing 3500 lb of Soyabean in the northern region of Missouri. Calculate the least cost combination of labor hours and nitrogen fertilizer. The per hour wage rate and per lb nitrogen can be purchased from the given market in $4 and $6.64 respectively.
|
Labor (x1) |
26.6 |
20 |
15 |
10 |
5 |
0 |
|
Nitrogen (x2) |
0 |
2.5 |
5 |
8 |
13 |
22 |
Solution: Following 3 steps can solve this type of problem.
Note: (Where dx1 refers to number of units replaced of input x1 (here labor) and dx2 refers to number of units added of input x2 (here nitrogen).
Step 2. Calculate the inverse price ratio i.e. Inverse price ratio = Px2/Px1
Note: Where Px2 refers to marginal input cost (MIC) of added input (here nitrogen) and Px1 refers to MIC of replaced input (here labor).
Step 3. Equate step 1 with step 2 i.e.
(-) dx1/dx2 = Px2/Px1 ………………………………the required condition
|
Labor(x1) |
Nitrogen(x2) |
dx1 |
dx2 |
-(dx1/dx2) |
Inverse price ratio (Px2/Px1) |
|
26.6 |
0 |
- |
- |
- |
- |
|
20 |
2.5 |
6.6 |
2.5 |
2.46 |
1.66 |
|
15 |
5 |
5 |
2.5 |
2 |
1.66 |
|
10** |
8** |
5 |
3 |
1.66 |
1.66 |
|
5 |
13 |
5 |
5 |
1 |
1.66 |
|
0 |
22 |
5 |
9 |
0.55 |
1.66 |
|
** indicates the least cost combination of Labor hours and kg. of nitrogen for the required yield. |
Numerical example for solving optimal variable inputs, TP, AP and MP
Problem 2. Find out the optimum level of a variable input from the given production function,
Q = 15 + 6.32x – 0.2 x2. The per unit price of input(x) and output(Q) is $3.2 and 4.0 respectively. Also calculate the AP, TP and MP.
Solution:
Step 1. Take the first order derivative of given equation with respect to x
dQ/dx = MP = 6.32-0.4x .………………………………….I
Step 2. Convert MP into value term by multiplying it by per unit price of the output (Q) to obtain value marginal product (VMP)
VMP = P * MP
= 4 (6.32-0.4x)
= 25.28-1.6x ………………………………….II
Step 3. Equate VMP with marginal input price (price of input)
25.28-1.6x = 3.2
Then, x = 13.8 (optimum level of a variable input)
Step 4. Substitute the value of x in original equation for total product (TP).
Q = 15 + 6.32x – 0.2 x2
Q = 15 + 6.32* 13.8 – 0.2 *(13.8) 2
Therefore, Total Product (TP) = Q = 64.128
Step 5. Divide total output by (subtracting intercept value) optimum level of input to find AP.
AP = (64.128 – 15)/ 13.8
= 3.56 units
Step 6. Substitute the value of x in equation I.
MP = 6.32-0.4x
= 6.32- 0.4 * 13.8
= 0.80 unit
Graphically , producer is in equilibrium when the point of tangency of the isocost line with the highest possible isoquant curve.
Fig: Producer’s equilibrium
In the figure shown above, the isoquant curve represents targeted output, i.e., 200 units. Isocost lines EF, GH and KP show three different combinations in which we can utilize the total outlay of input, i.e. capital and labor. The isoquant curve crosses all three isocost lines on points R, M and T. These points show how much costs we will incur in producing 200 units. All three combinations produce the same output of 200 units, but the least costly for the producer will be point M, where isocost line GH is tangent to the isoquant curve.
Points R and T also cross the isoquant curve and equally produce 200 units, but they will be more expensive because they are on the higher isocost line of KP. At point R the producer will spend more on capital, and labor will be more expensive on point T. Thus, point M is the producer’s equilibrium. It will produce the same output of 200 units but will be a more profitable combination as it will cost less. The producer must, therefore, spend the OC amount on capital and OL amount on labor.
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