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A Mathematic Model for Supply-Demand Equilibrium and the Optimal Solution for Labor Assignment
Supply function and Production Possibility Frontier (PPF) are basic concepts in Economics. We present a model that can give mathematic formula for PPF and the supply as a function of the price vector and capability parameters. With this Supply function, the generic Supply-Demand equilibrium problems can be solved numerically. We apply the supply-demand equilibrium to give optimal solutions for team work management problems or labor assignment problems. Concrete examples are given for managing an engineer team in Boeing Corporation.
Keywords: Resource Allocation, Labor Assignment, Teamwork Management, Supply Function, Production Possibility Frontier, Supply-Demand equilibrium
Cite this paper: Gang Liu , A Mathematic Model for Supply-Demand Equilibrium and the Optimal Solution for Labor Assignment, American Journal of Economics, Vol. 4 No. 2, 2014, pp. 83-98. doi: 10.5923/j.economics.20140402.01.
- Production Possibility Frontier (PPF) is one of the basic concepts in Economics. PPF can be simply defined as the boundary of frontier of the economy’s production capabilities. Most of Economic books normally have a whole chapter to discuss PPF and its applications [1, 4, 10, 11]. Most of these text books discuss the PPF and supply function for two products and in qualitative format, as shown in Figure 1. It is well-known that the PPF is convex and the supply-price curve is an upward-sloping curve. However, no one has given the PPF nor supply function in mathematic formats. The law of supply and the law of supply-demand equilibrium are basic laws in modern economics. However, without knowing the supply function in mathematic format, these laws can be only discussed quantitatively, and the general supply-demand equilibrium cannot be solved numerically.In early 2002, we have given the PPF and supply function in mathematic format for any number of products . In this paper, we first introduce the model that can formulate the PPF in mathematic format, and then derive the supply as a function of the prices of all products. Some of the basic economic theorems can be derived from the PPF formula, such as the convexity of the PPF curve, the Law of Supply, as well as supply elasticity. We also proposed methods for solving the general supply-demand equilibrium numerically.There are wide applications of the proposed PPF functions. As an example, we apply the PPF function and the Supply-Demand equilibrium equation to give optimal solution for team work management and labor assignment problems. Our analysis and test results show that the proposed model can improve production efficiency by 40% for most of the team work management problems.The rest of the paper is organized as follows. Section 2 introduces some useful notations and definitions. Section 3 formulates the PPF and the supply function for a micro system. Section 4 further formulates the PPF and the supply function for a macro system. Section 5 discusses the lower Production Possibility Frontier. Section 6 discusses the Law of Supply and the Elasticity of Supply. Section 7 introduces two methods for solving the supply-demand equilibrium. Section 8 introduces two special production possibility curves that can be used to show how efficiency of the PPF. Section 9 applies our supply function to a dummy system to show concrete PPF curves and supply curves. Section 10 discusses the best scenario and the worst scenario. Section 11 applies our model to solve the labor assignment problem for a labor oriented team. Section 12 shows a concrete example of labor assignment problem for project oriented team. In Section 13 and 14, we analyze the efficiency improvements by the optimal solution. Section 15 introduces two management methods that can reach the optimal solution for team work management. Conclusions are presented in Section 16.
2. Notations and Definitions
- In order to describe our model and the PPF curve, let us first introduce some notations and definitions.
- : An economic system that contains some sub systems. It can be as small as a family, a firm, and can be as large as a country or the whole world. is also called a macro system compared with its sub systems.: An integer represents the total number of the sub systems in .: An integer represents the total number of the products that are produced in .: The ith unit or sub system in system . It could be an individual member, or a group of people, such as a firm or another economic system in . However, there are no overlaps between different Units. For example, if an individual has been included in one unit, the same individual can’t be included in another unit. is also called a micro system compared with its parent system.: The kth product that can be produced in system . Actually, the product here means any labor activities that can be done by any unit in . It can be a real product, a project, a job, a task, and anything that need to be done or produced by any unit in .: A time span that the above mentioned products are produced in .: , which is the set of non-negative M-dimensional real vectors and with the original point excluded.: The same as . However, we use it to particularly represent the Production Space with its kth Cartesian coordinate representing the amount of a parameter related to product.: The demand amount of product that requested by .: The maximum amount of product that can be produced by within time . Normally can be produced by when it uses all of its resources to work on product . All the make up an matrix, which is called the micro capability matrix of system .: The amount of product that are actually produced by . All the make up a vector in , and is called micro supply vector.: The amount of product produced by the macro system . It is also called the macro supply of system . All the make up a vector in , and is called a macro supply vector of system .; : The price of the product in system .
- Definition: Micro System and Macro System: An economic system is called a macro system if it contains some sub-systems, such as the system introduced above. An economic system is called a micro system if it is contained in a macro system, such as the sub-system described above.Definition: System Parameter: An economic parameter is called a system parameter if it is associated with an economic system. For example, the supply amount, demand amount, and price are all examples of system parameters.Definition: Micro Parameter and Macro Parameter: A system parameter is called a micro parameter if it is associated with a micro system, and is called a macro parameter if it is associated with a macro system. Some parameters may be associated with both micro system and macro system. As long as there is no confusion, we use the same alphabet character to represent a system parameter: if denotes a macro parameter associated with a macro system , then will be used to denote the same micro parameter associated with micro system .For example, and represent the macro and micro supply amounts of product , and denote the macro PPF of system and the micro PPF of the micro system correspondingly.Definition: Vector Parameter and Scalar Parameter: a system parameter may also be associated with each of the production, for example, supply amount normally means the amount of a product that can be provided by an economic system. In a system with products, a system parameter could be -dimensional with each component representing the amount associated with one product. These -dimensional parameters make up a vector in the production space , thus is called a Vector Parameter. A vector parameter can be represented by a vector in . Examples of vector parameters are: Supply, Demand, and Price.A system parameter is called a scalar parameter if it is one dimensional. Examples of scalar parameters are: income, GDP, revenue, etc.Additive Parameter: A system parameter is called additive parameter if the corresponding macro parameter can be derived by aggregating the same parameter over all of its micro systems. Or precisely, system parameter X is additive if
2.3. Known Parameters and Unknowns
- Using the above notations, we can list the input data in table format. Basically, we assume the following data are known input data:, the production capability matrix for each micro system and for each product.: The macro demand of product that needs to be accomplished by system .We assume the following parameters are unknowns and need to be resolved: Price, micro supply, macro supply, micro income, macro revenue.
3. Micro PPF and Micro Supply Function of a Micro System
- For any micro system , let us assume that it can arrange its resources to make any requested products. However, its production amount for product should be limited to or bounded by due to its limited resources and capabilities. Any of its feasible production state will be a point or a vector in the production space . All of its feasible production vectors should be a bounded range in the production space . We call this bounded range as its micro Production Possibility Range (PPR) and denoted as . The upper boundary of is called the micro PPF, denoted as . Normally, as long as the micro system is small enough compared with macro system , each of the micro PPF can be approximated by the linear plane curve that passing the axis at in production space .Given the micro capability matrix, the micro PPF can be formulated as:
4. Macro PPF and Macro Supply Function of a Macro System
- We have formulated the micro PPF and the micro supply as a function of the price vector and the capability parameters, as shown in Equation (16). As each of the micro system has a maximum production frontier and a limited range as its PPR, the macro system should also have a macro PPR in the production space , and must have a macro PPF. As the supply parameter is an additive parameter, we can get the macro supply vector by aggregating all the micro supply vectors, that is:
5. Macro Worst Production Possibility Frontier of the Macro System
- Similarly, we can formulate the Worst Production Possibility Frontier (WPPF) for the micro as well as the macro system. Let denote the micro WPPF of micro system , and denote the macro WPPF of the macro system . is defined as the lower boundary under the conditions that all of its micro systems have reached their micro PPF . The WPPF is defined as the set of micro supply vectors that are on the and results a macro supply vector on the macro WPPF if aggregated for all micro systems. and can be formulated using the same Equations as shown in Equation (16) and (17), as long as we set . We will not rewrite these Equations here.Let be the region bounded by and , and be the region bounded by and . is called the micro Maximum Production Possibility Range (MPPR), and is called the macro MPPR.Given a price vector , through Equations (16) and (17),we can get a macro supply vector on , a macro supply vector on , a micro supply vector on , and a micro supply vector on . It can be shown that the price vector is the normal vector of those curves at the corresponding points as indicated by these supply vectors .
6. The Supply Elasticity and the Law of Supply
- The Supply Elasticity is defined as the absolute value of the ratio of the percentage change in quantity supplied to the percentage change in price, which brings about the change in supply. Let denote the supply elasticity for the product. By applying partial differentiation to Equation (16) and (17), we can easily have:
7. Supply-Demand Equilibrium
- Supply-Demand Equilibrium is one of the most important theorems in Economics. It states that Equilibrium is defined to be the price-quantity pair where the quantity demanded is equal to the quantity supplied, represented by the intersection of the demand and supply curves. The general format for the Supply-Demand Equilibrium can be expressed as:
7.1. Solving the General Equilibrium as an Equation Problem
- By combining Equation (23) and Equation (17), the General Equilibrium Equation can be formulated as the following equation problem:
7.2. Solving the General Equilibrium as an Linear Programming Problem
- The General Equilibrium Equation (23) can also be formulated as the following optimization problem:
8. Some Special Production Possibility Curves
- In this section, we introduce two special scenarios with special production possibility curves. These curves can be used to compare with the PPF and can show how efficiency of the production states on the PPF.
8.1. Linear Production Possibility Frontier
- We have introduced the micro vertex points as shown in Equation (5). A plane curve can be formulated by passing through these vertex points as following:
8.2. Self Sufficient Scenario
- Another special scenario is the so called Self Sufficient case, in which every micro system just simply works on itself to provide all products with the amounts to be proportional to the requested demands. Within this scenario, there is no cooperation among micro systems. Let us call the PPF Curve for this scenario as SPPF. Let and be the micro SPPF and macro SPPF for the Self Sufficient scenario. is the same as the plane curve as shown in Equation (26). However, the macro SPPF is different from the Macro LPPF .By definition, given the requested demand as , the micro SPPF can be expressed as:
9. A Dummy System with Dummy Data
- In order to give concrete examples to demonstrate our model and methodology, let us create a dummy system with dummy data. The dummy system is called the carpenter system, which contains units or members, and need to make products. The carpenter system needs to produce 4 table legs, 1 table top, and 6 chairs, which make up the 3 components of the demand vector. The micro capability matrix and demand vector are listed in Table 1. The requested amounts or demands are listed in row 3, and the production capabilities are listed in row 5 to 14.
|Figure 2. The micro PPF, the micro WPPF , the micro LPPF , and the Micro SPPF are all shown as the same plane curve in the production space|
10. The Best and the Worst Scenarios
- Suppose each of the micro system has reached its micro PPF, then the aggregated macro state of the macro system must be in the macro MPPR. Also, suppose the macro system is requested to produce the demand vector. The demand vector is as shown as the line in Figure 3. Figure 3. also shows the four points, , , and, which are the intersection points of the line with the curves, , , and.For all possible production states in, the Line gives all possible states that are proportional to the requested demand. Any other states in may have some products wasted or not needed compared with the requested demand. So, for all possible states in, only the states on line can best meet the requested demand. We are only interested in the states that are on the line.For all states on the line, the point is the best scenario, because it gives the maximum possible amount for all supplies, is the worst scenario, whereas and are somewhere in the middle. Point can be treated as the average state for all points in the macro MPPR.Let us assume that point is the average state that the macro system reached without using any optimal management method. Now, let denote the improved production efficiency by comparing the best scenario with the average scenario, then can be formulated as:
|Figure 4. Supply Curves. When price rises and all other prices remain unchanged, the supply rises, whereas the supply for other products decreases, for example, Supply decreases. When is large, the supply curves and become step curves|
11. Application 1: Labor Assignment for Labor Oriented Team
- The mathematic formula for the Macro PPF and the supply function may have lot of applications and can improve the production efficiency significantly. Here we just introduce one of the applications of the macro PPF and the supply function: finding the optimal solution for team work management problems.We have developed a software tool that can find optimal solution for labor assignment problems by solving equation problem listed in Equation (24), or by solving linear programming problem as listed in Equation (25).By applying this software tool to the Carpenter team with the input data as listed in Table 1, we find the optimal solution as shown in Table 2.The solution given in Table 2 tells us at least the following information:
12. Application 2: Labor Assignment for Project Oriented Team
- Application 1 gives an example of a team working on products, which is more applicable to a labor oriented team.A product oriented team is basically a team working on products and the demands can be simply formulated as the amounts of the products. A project oriented team is a team working on projects, while each project is required to be delivered in a given deadline. For a project oriented team, the capability parameter can be explained as the delivery ratio (or amount) of the kth project by team member . The demand for each project is 1 within required delivery time.Let denote the required delivery time for the kth project , and denote the fastest delivery time of team member to deliver the kth project under the condition that works only on project . Then the equivalent demand vector can be expressed as:
13. Production Efficiency Improvement for the Carpenter Team
- Note that, even with the worst scenario case, all team members have worked hard to reach their micro PPF states. In other words, although all team members have exhausted all of their resources to work on the requested products, the production output are very different for the macro system. The inefficiency for the worst scenario is completely due to improper management. With proper arrangement, we can have the team to improve production efficiency by 148% compared with the worst scenario.Table 7 lists the macro supplies for the optimal solution, the worst scenario solution, the Linear Macro Supply scenario, as well as the Self-Sufficient scenario. It also lists the improved production efficiency ratios by comparing with the Self-Sufficient scenario as well as the Linear Macro Supply scenario.
14. Estimate the Improved Production Efficiency for Generic Cases
- As discussed in the previous sections, by applying our optimal management method, the Carpenter team can improve its production efficiency by 43%. That is amazing! However, people might argue that the Carpenter team is just a special case. In this section, we estimate the production efficiency improvement ratio that can be brought by the optimal solution for generic cases. Our analysis shows that the optimal solution can really improve production efficiency by 40% for most of the team work management problems.To simplify our analysis, we assume the demand vector as in this section. Let be the intersection point of the demand vector (or its extension) with the curve , and denote the length of the line . Then can be easily found as:
|Figure 5. can be used to approximate the PPF and LPPF . Here all the curves are shown in 2-dimensional. You need to imagine it in M-dimensional|
15. Micro and Macro Management
- By applying the proposed method and the software tool, we can find the optimal solution for the team work management, or labor assignment problems. Suppose you are the team manager, how are you going to manage your team to reach the optimal management? Here we propose two management methods: Micro Management and Macro Management.
15.1. Micro Management
- Our optimal solution for team work management problems, as shown in Table 2 and Table 4, lists all the micro information for each team member, including micro supply amount , and the micro income amount . Using these micro information, a team manager can tell each team member to work on the with the amount as indicated in the optimal solution. Note that most of the team members may work on only one product. The team manager can also tell each team member about how much he should be paid, which would be . So, with micro management, a team manager manages each of the team members in micro details, including what and how much need to be done, as well as how much should be paid as return. That is why we call this management method as Micro Management.
15.2. Macro Management
- Our optimal solution for team work management problems, as shown in Table 2 and Table 4, also gives the intrinsic prices . Given this intrinsic price vector, each team member will reach his maximum income by working on the product with the amount as requested by the optimal solution. Suppose each team member is targeting at maximizing his income, the team manager needs only declare the intrinsic prices, then, every team member will automatically to work on the product with the amount such that the whole team will reach its optimal solution as indicated in the optimal solution.The team manager doesn’t manage the team members in micro details, but manage it in macro level, which simply tells how much will be paid for product or project to be delivered. That is why we call this management method as macro management.
- The main contributions of this paper include a new model that can formulate the Supply function and the Production Possibility Frontier as a function of the capability parameters and the price vector. The Law of Supply, the supply elasticity can be easily derived from the proposed model. Another contribution is the methods for solving the Supply-Demand equilibrium. Lastly, we present a model and method that can give optimal solution for labor assignment and team work management problems based on the Supply-Demand equilibrium. Our test results and generic analysis show that the proposed management method can improve the team work efficiency significantly. For most cases, it can improve the production efficiency by 40%!