In the short-run, total quantity of at least one factor of production remains fixed and quantities of other inputs can be changed. Suppose, in a two input framework, capital is the fixed input and OK amount of capital is employed (see figure 9.8).
To show the relation between short run Total Cost curve and the long run Total Cost curve we use a fixed set of isocost lines (represented by a2b2, a3b3 and a5b5) and a particular isoquant map (comprising of isoquants I, II, III only) is used.
First, consider isocost line ‘a2b2‘. Given this cost constraints and prefixed amount of capital OK, maximum amount of labour to be employed is determined by the point of intersection between KK line and isocost a2b2 (i.e., point ‘n’).
This point of intersection shows that amount of labour input required is ‘ Kn’. The maximum possible output with ‘OK’ capital and ‘ Kn’ labour is determined by the isoquant that passes through isoquant I.
If it were the case of long-run, the same quantity of output could be produced at a lower cost. To explain this point, we introduce another isocost line a1b1, which is tangent to the isoquant I. Since isoquant I can be attained by isocost line a1b1, a1b1 < a2b2. It is clear that the same quantity of output can be produced with lower cost.
In case of long-run, where quantity of both labour and capital can be varied, it would be possible to produce the same quantity of output (as represented by isoquant I) at a lower cost represented by isocost line a1b1, since the said isoquant is tangent to it.
Now, consider that the isocost line shifts to a3b3. In this situation, the maximum output to be produced by OK quantity of capital is determined by isoquant II. To attain isoquant II with the given capital of OK, the amount of labour that can be used within the cost constraint ‘a3b3‘ is determined by the point of intersection between KK and the isocost line a3b3. This point of intersection is denoted by point p and the quantity of labour input is identified as Kp.
The maximum output that can be produced with this input combination is represented by isoquant II. It is in this particular case, that in the long-run the optimum input combination to attain isoquant II is OK and KP which is also same as this particular case of short-run.
Thus, for isoquant II, quantities of inputs to be employed to obtain maximum output are same for short-run as well as for long-run. As a result, total costs of production in the short-run and in the long-run are same.
Take another case, where isocost line shifts to a5b5. For the given quantity of capital i.e., OK total labour required to maximize output within the cost constraint a5b5 is determined as Ks, represented by the point s, where KK intersects the isoquant III.
Clearly, had it been the case of long-run, the same quantity of output could be produced represented by isoquant III could be attained at a smaller cost, as denoted by the point of tangency Y on the isocost line a4b4.
From the above analysis we can show the relation between SRTC curve and the LRTC curve. If we compare the Short-Run Total Cost (SRTC) with Long-Run Total Cost (LRTC), we observe that SRTC touches the LRTC curve at an output level (say point ‘g’) represented by isoquant II (see figure 9.9).
Let us assume that isoquant II represents OQ level of output. Thus, when production exceeds the output level represented by isoquant II, SRTC lies above LRTC. It implies that beyond output level OQ, SRTC lies above LRTC.
For similar reasons, SRTC also lies above LRTC for output levels below OQ. Only at output level OQ, both SRTC and LRTC coincide. Relation between SRTC and LRTC is graphically represented in figure 9.9.
SRAC can be obtained from the SRTC curve.
We have established the S-like shape of total cost curves – both for short run and long run. Now, if you look at the LRTC (Figure 9.10), we will find that there exists a point of inflection at P. So, up to output level Q1, LRTC Output/ gradually falls (i.a, α > β > ϴ).
Since LRTC / Output is nothing but long run average cost (LRAC), LRAC will monotonically decline upto output level Q1. Beyond Q1 output level, LRTC / Output gradually increases (i.e. ϴ < δ < γ), leading to monotonic increase in LRAC. Thus we can obtain LRAC curve from LRTC curve.
Similarly, using the same method we can justify the relationship between SRAC curve and SRTC curve. But since SRTC lies above LRTC excepting one point (refer figure 9.11), SRAC will also lie above LRAC excepting one point (Figure 9.11).