Production Optimization Problem

A production optimization problem is an essential economic activity in all production processes that involve utilizing scarce resources in an economy to produce goods and services using a production technology. The ability to optimize production is a crucial aspect of any economy. By efficiently using resources, businesses can produce goods and services that meet the needs and demands of consumers. Therefore, solving production optimization problems is essential for the success and growth of any industry. In this article, you shall learn the basics of the production optimization problem in profit-maximizing firms in a microeconomic setting.

1. Introduction: Production Optimization Problem

Small, medium, and even large businesses (firms) use production technology to produce goods meant to satisfy the needs of consumers locally and in international markets via trade, assuming the rational behavior of firms. How do firms decide how much to produce during social interactions with other economic agents? From an economic perspective, the production process should be efficient by optimizing the production of goods and services in society. In general, the efficiency of the production process influences not only the use of resources but also affects the competitiveness of businesses in their respective markets.

In economic theory, there are several ways to analyze the production optimization problem using economic models, e.g.;

• A one-input-one-output production model is a simple model of production showing the most basic approach to production optimization problem but does not deliver a realistic foundation of analysis,
• A two-input-one-output production model that extends the first model by introducing an input factor as a complement or substitute to the other resource (input) is preferred since it delivers a more realistic approach to real-life situations of production processes.
• A two-input-two-output production model adds output to the analysis of the production optimization process and allows for studying resource allocation in producing different outputs, an economic problem.
• The general production model extends the notation of production functions for the case where economists assume that there are many inputs and outputs in production processes in an economy using vectors and matrices. General models derive optimal production for a whole economy, generalizing the outcomes of the first three preceding models.

2. Basic Models of Production

In this section, let us focus on the characteristics of each production model and identify the mechanism of production optimization.

2.1 One-Input-One-Output Production Model

In this one-input-one-output production model, economists assume that the world economy exists in an environment with one scarce resource, e.g., labor, capital, or natural resources, and one consumer good needed to satisfy the world population’s needs. Though such an assumption leads to an oversimplification of the world problem of efficiently utilizing the technology, it is vital to learn some basic approaches to describing a production technology.

2.1.1 Model Assumptions

In this case, let us assume a production technology that only uses labor $l$ to produce a service $x$ that satisfies the need of the population in an economy using the following production function:

x(l)=a\cdot l^{b}

Where $a>0$ and $b>0$ are positive coefficients of the production function and the output level $x>0$ is positive for positive levels of the labor input. Further, assume that one labor $l$ unit costs $w$ monetary units (money) called wages in a competitive labor market. In contrast, one unit of the service $x$ sells at a competitive market price of $p>0$ monetary units. The revenue function quantifies the revenues generated by selling the services depending on the prices of output $p$ and the output level $x(l)$ as follows:

R(x)=p\cdot x(l)=p \cdot a\cdot l^{b}

The revenue increases when the price increases or the output and sales level rise. On the other hand, the following cost function quantifies the costs of production of the service depending on the level of inputs used in the production process:

C(l)=w\cdot l

2.1.2 Solving the Production Optimization Problem

To find the optimal level of production, the firm aims to maximize its profits, the difference between the revenues from sales and costs of production as follows:

\Pi(x)=R(p,x(l))-C(l)

The profits of the firm depend on the market price $p$ in the service market, the wage level $w$ in the labor market, and the output level $x$, which depends the input level of labor $l$:

\max_{l} \Pi(x)=p⋅x(l)-w\cdot l

2.1.3 First and Second Order Conditions

To find the optimal production level, we need to identify the production level that offers a maximum level of profits, assuming that the price level and wages are constant and the production technology is known. At the same time, let us assume that any output could be sold in the service market without delay. Using derivative rules of local extrema (first and second order conditions), we should find that optimal output level where the marginal profit $\Pi^{\prime}(x)$ is zero concerning the input factor labor $l$ (see derivation rules):

\Pi^{\prime}(x)=R^{\prime}(p,x(l))-C^{\prime}(l)=0

That takes place where the marginal revenues $R^{\prime}(p,x(l))=p⋅x^{\prime}(l)$ are equivalent to the marginal costs of labor $C^{\prime}(l)=w$. Whereby the marginal revenues depend on the unit price $p$ and the marginal product of labor $x^{\prime}(l)=ab\cdot l^{b-1}$. Marginal labor costs are equivalent to the wage level in the labor market.

\Pi^{\prime}(x)=p⋅x^{\prime}(l)-w=0

The marginal profit function, as the first derivative of the profit function, delivers the first-order condition for our production optimization process in a profit-making firm:

\Rightarrow R^{\prime}(p,x(l))=C^{\prime}(l)

Inserting the derivatives of marginal revenue and marginal costs leads to the following statement:

\Rightarrow p⋅x^{\prime}(l)=w

Dividing both sides with the price of the services leads to the statement that the marginal product of labor $x^{\prime}(l)$ must be equivalent to the real wage of labor $\frac{w}{p}$:

\Rightarrow x^{\prime}(l)=\frac{w}{p}

Insert the derivative of the production function to the get the following:

\Rightarrow ab\cdot l^{b-1}=\frac{w}{p}

Solve for the optimal labor input demand of the firm that depends on the real wage, nominal wage in the labor market relative to the price of the service, and the parameters of the production function $a$ and $b$, which characterize the production technology used by the firm:

\Rightarrow  l^{*}=\Big(\frac{1}{ab} \cdot \frac{w}{p} \Big)^{\frac{1}{(b-1)}}

The optimal production level is the output level at the optimal level of input:

\Rightarrow  x^{*}(l^{*})=a \cdot l^{b}=a \cdot \Big(\frac{1}{ab} \cdot \frac{w}{p} \Big)^{\frac{b}{(b-1)}}

After locating the solutions to the production optimization problem, we must verify using the second-order condition that the outcomes reveal a profit-maximizing output level using the optimal factor demand of labor. The second-order condition of the profit-maximizing problem of the firm is that the second derivative concerning labor must be negative so that the profit outcome is identified as a maximum profit:

\Pi^{\prime \prime}(x)=R^{\prime \prime}(p,x(l))-C^{\prime \prime}(l) < 0

The second-order condition indicates that the change in marginal profits $\Pi^{\prime \prime}(x)<0$ should be negative, which depends on how the change in marginal revenue $R^{\prime \prime}(p,x(l))=ab\cdot (b-1)\cdot l^{b-2}$ relates to the change in marginal costs $C^{\prime \prime}(l)=0$. In the case of the model setting here, the change in marginal costs is zero.

\Rightarrow \Pi^{\prime \prime}(x)=ab\cdot (b-1)\cdot l^{b-2}-0 < 0

\Rightarrow \Pi^{\prime \prime}(x)=ab\cdot (b-1)\cdot l^{b-2} < 0

In order to satisfy the second-order condition, the coefficient $0<b<1$ must be confined to a value between 1 and 0. This condition implies that the production technology must exhibit diminishing marginal returns to scale for labor. This means that even though increasing the amount of labor input should lead to an increase in output, it will not increase as much as previous lower levels of labor input. This model could be recalculated by first deriving the invest production function and inserting it in the cost function $C(l)$, which leads to the cost function $C(x)$ depending on the output level

x(l)=a\cdot l^{b}  \Rightarrow l (x)=\Big( \frac{x}{a} \Big)^{(1/b)}

C(l)=w\cdot l \Rightarrow C(x)=w \cdot\Big( \frac{x}{a} \Big)^{(1/b)}

R(x)=p\cdot x(l) \Rightarrow R(x)=p \cdot x

 \Rightarrow \Pi(x)=R(x)- C(x)

 \Rightarrow \Pi(x)=p \cdot x- w \cdot\Big( \frac{x}{a} \Big)^{(1/b)}

Show that the optimal profit is attained at the optimal production level using the optimal input level of labor:

 \Rightarrow \Pi(x^{*})=p \cdot x^{*}- w \cdot l^{*}

2.2 Two-Input-One-Output Production Model

In a two-input-one-output production model, the first model in 2.1 can now be extended by introducing an input factor as a complement or substitute to the other resource (input). Such an extension is preferred since it delivers a more realistic approach to real-life situations of production processes, where resources compete and complement each other. Economic models in this field use two inputs in their analysis: labor $l$ and capital $k$ (see scarce resources). How can firms solve the production optimization problem in a two-input-one-output model?

2.2.1 Inputs as Substitutes and Compliments in a Production Optimization Problem

To understand the analogy, let us review the production process of producing one unit of a four-legged table, e.g. in Leontief and Cobb-Douglas production processes. The production of goods and services requires different types of resources, e.g., a carpenter constructing a table will use his time as labor input but also his human capital (knowledge, talent, and health), capital (his woodwork tools, and machines), natural resources (wood and other raw materials). One economic question arises: is a three-legged table equivalent to a four-legged one if it is assumed that both production processes (technology) are similar? Both tables might offer the same functional satisfaction to their users at the practical level but differ in their input levels. Some economists may prefer to say that the outputs are heterogeneous.

If we look closely at the carpenter’s work, we will identify the difference in the production process (how inputs are combined and transformed into outputs). The carpenter can choose between producing 3 four-legged tables or getting an additional tabletop and producing 4 three-legged tables. If the time used to create each table, regardless of the number of legs, is the same, then the carpenter must invest additional hours and other resources to produce the fourth table. We agree that the carpenter could construct a two or one-legged table under certain natural restrictions (e.g., gravity).

Looking closely at this example, we can say that inputs are partially substitutes and complements simultaneously. Similarly, the production technologies are heterogeneous in how many inputs are needed to produce each table type.

2.2.2 Mathematical Approach

The mathematical approach of the two-input-one-output production model will be discussed in class.

2.3 Two-Input-Two-Output Production Model

The two-input-two-output-model will also be discussed in class.

3. Literature