**Expansion Path in the Long Run**

In the long run, the firm can change its old machines, equipment and plants, scale of production, organization and management in order to expand its output. The firm’s objective is the choice of optimal expansion path in order to minimize its costs or maximize its profits. The expansion path is the locus of different points of firm’s equilibrium when it changes its total outlay to expand output while relative factor prices remain constant.

In other words, the expansion path shows how factor proportions change when output changes, relative factor prices remaining constant. “With given factor prices (w, r) and given production function, the optimal expansion path is determined by the points of tangency of successive isocost lines and successive isoquants.”

**Assumptions**

(i) There are two factors of production, labour and capital, which are variable.

(ii) All units of labour and capital are homogeneous.

(iii) The price of labour (w) is constant.

(iv) The price of capital (r) is constant.

(v) The firm increases its total outlay in order to expand its output.

**Explanation**

Given these assumptions, in order to maximize its profits or to have the least cost combination, the firm combines labour and capital in such a way that the ratio of their MP is equal to the ratio of their prices, i.e., MP_{L}/MP_{K} = w/ r. This equality occurs at the point of tangency between an isocost line and an isoquant curve.

This is explained in Figure 1, where С_{1}L_{1} C_{2}L_{2} and C_{3}L_{3}are the different isocost lines. The line C_{2}L_{2}shows higher total outlay than the line C_{1}L_{1} and С_{3}L_{3} still higher total outlay than the line C_{2}L_{2}. They are shown parallel to each other thereby reflecting constant factor prices. There are three isoquants 100, 200 and 300 representing successively higher levels of output.

The firm is in equilibrium at point P where the isoquant 100 is tangent to its corresponding isocost line С_{1}L_{1} and similarly the other two isoquants 200 and 300 are tangent to isocost lines С_{2}L_{2}and C_{3}L_{3} respectively at points Q and R. Each point of tangency implies optimal combination of labour and capital that produces an optimal output level. The line OS joining these equilibrium points P, Q and R through the origin is the expansion path of the firm. The firm expands its output along this line keeping factor prices as constant.

The straight line expansion path through the origin, OS, implies a homogeneous production function (or constant returns to scale). Such an ex- pension path is called an isocline which is the locus of points о along which MRTS_{LK} = MP_{L}/MP_{K} = w/r. Thus OS is the optimal expansion path for the firm in the long run.

But the choice of the expansion path depends on the ratio of factor prices. If the ratio of factor prices increases, the isocost lines become flatter, as shown in figure 2, and the optimal expansion path will be ОТ. If initially the slope of the isocost lines is steep and the expansion path is OS, with the increase in the ratio of factor prices the optimal expansion path of the firm changes to ОТ. Both the expansion paths show homogeneous production function.

In case the production function is non- homogeneous the optimal expansion path will not be a straight line from the origin. Rather, it will be a zigzag line OS, as shown in Figure 3. It is a curved isocline which is the optimal expansion path of the firm because at the points of tangency L, M and N, the slopes of the isocost lines (w/r) and isoquants (MRTS_{LK} ) are equal.

**Expansion Path in the Short Run**

In the short run, the firm can increase only the variable factors and not the fixed factors in order to increase its output, while relative factor prices remain constant. Suppose capital is the fixed factor and labour is the variable factor, other assumptions remaining the same. The firm cannot choose the optimal expansion path OS. It can expand its output only along the line С С’, as shown in Figure 4. But this is not the optimal expansion path because points P, S and T are not on the isocline.

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