Expansion Path
In economics, an expansion route (also known as a scale line) is a graph with amounts of two inputs displayed on the axes, often physical capital and labour. As the size of production grows, the route links the best input combinations. When a producer wants to make a certain number of units of a product for the least amount of money, he or she picks the point on the expansion route that is also on the isoquant for that output level.
“Expansion route,” according to economists Alfred Stonier and Douglas Hague, is “the line that depicts the least–cost technique of creating various amounts of production, assuming factor prices stay constant.”
The points on an expansion route are formed when the firm’s isocost curves, which each indicate a set total input cost, and its isoquants, which each show a certain amount of output, are tangent; each tangency point specifies the firm’s conditional factor needs. The company goes from one of these tangency points to the next as its production rises; the curve connecting the tangency points is known as the expansion route.
The manufacturing technique is homothetic if an extension route follows a straight line from the origin (or homoethetic). In this situation, regardless of the amount of output, the ratio of input usages remains constant, and the inputs may be enlarged proportionally to preserve this ideal ratio as the level of output increases. A Cobb–Douglas production function is an example of a production function with a straight line through the origin as its expansion route.
Expansion Path’s Meaning:
We know that the firm’s production function
f = q (x,y)
Gives us the firm’s isoquant map, one isoquant (IQ) for each specific level of production, and the firm’s cost equation.
rXx + rYy = C
Given the costs of the inputs rX and rY, we get a family of parallel iso-cost lines (ICLs), one ICL for each cost level. Figure 1 shows the IQ-map as well as the ICL family. If we draw a curve connecting the point of origin 0 and the points of tangency E1, E2, E3, etc. between the IQs and the ICLs, this curve will tell us the firm’s growth route.
The growth road is named from the fact that if the company intends to extend its activities, it must follow this path. It’s worth noting that the company may grow in two ways.
First, it could wish to grow by gradually raising its cost or spending on the inputs X and Y, i.e., by consuming more and more inputs and, as a result, creating more output.
Second, the company could seek to grow by raising its production every quarter. This may be accomplished by increasing input expenditures, i.e., consuming more and more of them.
Both routes to growth seem to be similar in that they both entail an increase in spending. There is, however, a significant distinction. In the first situation, the choice is made first and foremost on the basis of cost. Costs are raised incrementally, and then attempts are made to optimise production while adhering to the cost limit.
In the second situation, on the other hand, decision-making takes place first and foremost at the point of production. The corporation chooses to generate more product first, and then makes attempts to produce the output at the lowest feasible cost.
Expansion Path Types
Expansion by an increase in the level of input expenditure
Assume that the firm’s starting cost level is such that its ICL is L1M1, and that output maximisation under cost constraints happens at the point of tangency, E1, between the ICL, L1M1, and an IQ, IQ1. The company at E1 utilises the first input’s X1 and the second input’s y1 to create the greatest feasible output, q1, which is represented by IQ1.
If the business chooses to grow by raising its cost level from L1M1 to L2M2, it will be in output-maximizing equilibrium at the point of tangency E2 (x2, y2), on IQ2, utilising more inputs, x2 > x1 and y2 > y1, and creating an output level, say, q2, q2 > q1, since IQ2 is a higher isoquant than IQ1.
Similarly, if the firm decides to expand further, it will raise its cost level from L2M2 to L3M3, and it will produce the maximum output subject to the cost constraint at the point of tangency E3 (x3, y3) on IQ3 by using more inputs, x3 > x2 and y3 > y2, and producing a higher level of output, say, q3, q3 > q2, because IQ3 is a higher IQ
The process of expanding a corporation’s operations via cost increases may continue indefinitely as long as the firm intends to do so. We may now derive the firm’s growth route by connecting the point of origin O and the points E1, E2, E3, and so on with a path.
To put it another way, if the business grows by raising its cost level, it will have to shift from one equilibrium point to the next along this growth route.
We connected the path through the equilibrium points E1, E2, etc. with the point of origin O because if the firm moves backward along the expansion path by lowering costs, it will move from the initial equilibrium point, say, E3 to E2, then from E2 to E) and eventually reach point O, which will be the process’ limiting point.
The input and output quantities would all decline and trend to zero as the firm’s cost level falls and approaches to zero, and therefore the point of origin O would be the limiting point.
Expansion Path via an increase in output level
Assume, in Fig., that the business first chooses to create q1 of output, which may be produced at any point on the isoquant, IQ1. At point E1, which is the point of tangency between IQ1 and an iso-cost line such as ICL1, the company would be in cost-minimizing equilibrium. The company would utilise Xi and y] quantities of the two inputs at point E1, and its cost would be C1, which is the lowest achievable.
The company may now choose to grow by boosting production from q1 to q2 on IQ2. If the firm makes this decision, its cost-minimizing equilibrium will be found at the point of tangency E2 (x2, y2) on L2M2 by using more of the inputs, x2 > x1 and y2 > y1, and incurring a cost level C2 on L2M2, which is the bare minimum required to produce the output of q2, and incurring a cost level C2 on L2M2, which is the bare minimum required to produce the output of C2 > C1 because L2M2 has a greater ICL than L2M2.
Similarly, the corporation may opt to boost its production from q2 to q3 on IQ3 as well. The point of tangency E3 (x3, y3) on the ICL, L3M3, would be the firm’s equilibrium point in this scenario. At E3, the company would employ even more inputs, x3 > x2 and y3 > y2, resulting in a cost level C3 on L3M3, which is the bare minimum for creating q3 of output. C3 > C2 because L3M3 has a greater ICL than L2M2.
The firm’s growth process may continue in this manner as long as it chooses to do so. The expansion route, which would begin at the origin O and travel via the locations E1, E2, E3, and so on, would be acceptable.
If the business chose to shrink and create less output, the point of origin O, where the firm’s utilisation of inputs, cost level, and output all trend to zero, would be the limiting point of the contraction process.
The Expansion Path Equation
A point of tangency between an isoquant and an iso-cost line exists at each point on the expansion path. As a result, we obtain numerical slope of the IQ = numerical slope of the ICL at each location along the growth route.
MRTSX,Y = rX/rY MRTSX,Y = rX/rY MRTSX,Y = rX/
rX/rY = constant fX/fY= [… rX and rY are constant and provided]
As a result, we get the expansion route equation.