In our earlier post, we discussed the profit-maximizing conditions of a single-product firm. But in the real world, the business firm produces more than one product. In this post, we will discuss the profit-maximizing conditions of a multi-product firm. Also, note that the equilibrium of a firm equally means profit maximization of that firm and vice versa.

Before we delve into the core topic, let us first go through the very important tools from the economic profession – Edgeworth Box diagram, production possibility curve, and iso-revenue line. These tools are the fundamentals to understand the equilibrium of multi-product firms. Then after, we will elaborate on the necessary and sufficient conditions for the equilibrium of a multi-product firm.

- Assumptions of Theory
- Edgeworth Box diagram and Contract Curve
- What does The Production Possibility Curve Show?
- What is the Iso-revenue Line?
- How does a Multi-product firm Reach equilibrium?

## Assumption of theory

The exposition of theory rests upon the following assumptions:

- Objective of a firm is profit maximization.
- Firm is multi-product firm and produces at least two product X & Y.
- Each production function uses capital and labor to produce one single good i.e. production functions to individual good take the form of X =f (K,L) and Y =f(K,L).
- Prices of factors inputs and prices of products remain constant at least during the analysis.
- Technology of production remains same throughout the analysis.

Based on the above assumptions, we can say that firm produces only two goods – X and Y – using both of the factor inputs capital, K, and labor, L. Also note that there are two types of production techniques – X =f (K,L) and Y =f(K,L) to produce goods X and Y respectively. Each production technique can be represented by the corresponding sets of isoquants. To say, we can assume set of isoquants X’s to represent technique to produce X good, and set of isoquants Y’s to represent technique to produce Y good. At this backdrop, let us present the Edgeworth Box diagram in the following section.

## What does Edgeworth Box diagram show?

It is very important tool in economics and shows how economic efficiency one can obtain with the best usages of limited productive resources. Not every combination of resources – capital and labor- is efficient within the Edgeworth Box, but some points that form contract curve are efficient. Edgeworth Box diagram depicts how to mobilize resources from inefficient usages to efficient usages.

Also observe the following facts in Edgeworth Box:

$latex \Rightarrow$ Set of isoquants X’s originating to Ox denotes production function X=f(K,L) and set of isoquants Y’s originating to Oy denotes production function Y=f(K,L).

$latex \Rightarrow$ The higher the isoquants X’s, the larger the output of X it shows. Similarly, the further down the isoquants Y’s, the larger the output of Y it shows.

$latex \Rightarrow$ Total amount of capital K – shown in vertical axis, and total amount of labor L – shown in horizontal axis, are the total available resources.

### Explanation of Edgeworth Box diagram

We can perceive isoquants X’s and Y’s, facing to each others, will be tangent at points somewhere within the Edgeworth Box. We can expect very large number of isoquants X’s and Y’s despite we have shown few of them, and thus, large number of such tangecy points. If we connect such points of tangency, we get what is called a contract curve (curve OxOy in picture). This curve is the locus of points representing all the economically efficient methods of producing X and Y goods. Any point off this curve implies the inefficient use of limited resources.

For instance, suppose firm produces at point T (in above figure), where it uses OxK1 of total capital K and OxL1 of total labor L to produce X3 output. Similarly, it uses remaining resources – K1K of total capital K and L1L of total labor L – to produce Y3 output.

However, moving towards points P and Q on contract curve while using the same amount of resources, firm can produce more of at least one good without reducing the output of other good. At point P, firm can produce Y4 without reducing X3. Similarly, at point Q, firm can produce X4 without losing Y3. If the firm moves to points in between P and Q, say point m, then it can produce more of both goods using the same resources. Thus, only points on contract curve represent the efficient method of production; any points off this curve indicate inefficient method of production.

## What is production possibility curve?

It is the very important tool in economics which enables producers or firms to efficiently choose from the alternative products. We can derive production possibility curve (PPC) directly from contract curve of Edgeworth Box by projecting it (contract curve) into X,Y plane of goods.

Points P, m and Q on contract curve are the respective points P’, m’ and Q’ on PPC curve. Also note the corresponding amount of goods X’s and Y’s in PPC curve. Any points above the PPC curve YP’m’Q’X are desirable but unattainable given the state of technology and factor inputs. Likewise, any points below the PPC curve are attainable but undesirable because they represent the inefficient usages of resources.

The PPC curve is also called product transformation curve because moving from one point to other on PPC implies the transformation of one product to another. However, the transformation of product takes place via the diversion of resources from one product to another.

### What is marginal rate of product transformation?

It is the rate at which one product is transferred to another product, holding constant the resources. In other words, it is the amount of Y good forgone for the additional one unit of X good, and is denoted by MRPTx,y. The more we transfer Y to X, the lesser the X we get from transformation; and thus, MRPTx,y increases. The concave shape of PPC curve is the result of increasing marginal rate of product transformation. Also note that marginal rate of product transformation (also a slope of PPC at that point) at any point on PPC curve is given by the slope of tangent at that point.

## What is iso-revenue line?

It is a locus of combinations of products that gives the same revenue. In other words, it represents the various combinations of X and Y goods whose sales give the same amount of revenue to firm. That means to say, any points on iso-revenue curve give the constant level of revenue to firm. The higher the iso-revenue lines from the origin, the greater the revenue it depicts.

Let us present iso-revenue line mathematically,

R = P_{x }. X + Py . Y

where P_{x} is price of good X, Py is price of good Y, and R is revenue. Also note that the slope of iso-revenue line is given by the ratio of P_{x} to Py i.e. P_{x} / Py.

## How a multi-product firm reaches equilibrium?

After having introduced with PPC and iso-revenue line, it is time to explore how a multi-product firm reaches equilibrium (maximizes revenue for the given prices of products). Firm has to maximize revenue constrained by given prices of factor inputs and products, given PPC, and given limited factor inputs. So, this analysis is a constrained profit maximization.

There are two conditions for the profit maximization:

- Slope of PPC must equal to slope of iso-revenue line i.e. MTPT
_{X,Y}= P_{X}/P_{Y }. - PPC must be concave to origin.

In the following figure, firm reaches equilibrium at point E fulfilling the both conditions, and produces X* amount of X good and Y* amount of Y good. Given the prices of products X and Y goods, and thus given P_{X} /P_{Y}, iso-revenue line MN is the highest possible line to represent highest amount of revenue. Any lines above MN line are desirable but unattainable. Similarly, any lines below MN line are attainable but undesirable and inefficient because, they do not maximize profit. Thus, only MN line maximizes profit fulfilling the required conditions.

What happen if PPC becomes convex to origin? The answer is, it ends with the corner solutions producing one of either goods, and resulting in still larger amount of revenue than otherwise. In such a situation, thus, we have to rule out the assumption of multi-product firm. This is on contradiction to our basic premise, so we consider only concave PPC.