## Monday, 8 April 2013

### Firms and Isoquant Maps

If you read the previous post about indifference analysis then you'll notice a lot of similarities when studying this topic. Isoquant analysis is essentially the same as indifference analysis but from the point of view of a firm. Each isoquant measures the combinations of capital and labour a firm would need to produce a constant output. They follow the same shape as indifference curves, sloping downwards, as you can see below.

The downwards sloping shape is due to the diminishing marginal rate of technical substitution (MRTS). It's the rate at which we can substitute capital for labour and still end up with the same level of output. We have to give up capital to add more labour, hence why there is a negative slope.

An isoquant map is a series of isoquants showing combinations of capital and labour that give different levels of output. It looks very similar to an indifference map.

Each isoquant represents a different level of output. The further up and to the right you go, the higher the production. From these maps we can see what returns to scale the firm in question is facing. By returns to scale we are talking about the increase in output given an increase in capital and labour. If we doubled both capital and labour and saw a doubling of output then the firm would be facing constant returns to scale. On the isoquant map this is shown by the isoquants being evenly spaced. If we doubled the inputs and received more than double the output then we'd say the firm is facing increasing returns to scale. The isoquants would get closer together when increasing returns to scale was present. Finally, if the firm doubles the inputs and receives less than double the output then the firm is facing decreasing returns to scale. On an isoquant map this would be shown by the isoquants getting further apart.

We move on now to isoquants and marginal returns for firms. A marginal return measures the change in output the firm gets when one variables is changed and the other is held constant. Look at the diagram below this paragraph - we'll hold capital constant at 25.

So, to achieve output of 5000 with capital held at 25 we need 10 units of labour. To get from 5000 to 10000 production we need to add an additional 20 units of labour (30-10). To get from 10000 to 15000 production we need to add an additional 35 units of labour (65-30). We can see that the more labour we add the less productive they become - this shows the principle of diminishing marginal returns. Each additional worker will add less production than the previous one.

Right, now for firms to choose their optimal level of production we need to include their budget into the analysis. This works the same as a budget line. Anywhere along the line gives us combinations of the two inputs with equal costs.

The dotted line above shows an example of changing factor costs and what would happen to the isocost line. Here, the price of labour (wages) have fallen and therefore the firm can afford more of them with a given budget. The line swings out, pivoting around the point on the y axis. If the price of labour rose the line would swing in. If the firm's overall budget increased/decreased then the whole isocost line would shift out/in.

Now, firms choose their production in one of two ways. They either go down the route of getting the least cost combination of factors for a given level of output or they aim to maximise output for a given production cost. The two examples can be seen in the diagram below.

Time to get a bit mathematical now. We are going to work out the equilibrium point of production and what occurs at this point. So, the slope of an isoquant is as follows: If we reduce capital (K) then the loss of output will be: (MPP being the marginal physical product).

And if we increase labour at the same time, the gain in output will be :

Now, at any point on the isoquant the change in quantity is 0, therefore these two terms must equal each other:

A simple rearrangement and we are left with the following formula for the slope of the isoquant,which equals the marginal rate of technical substitution:

The slope of an isocost now. The reduction in cost should we reduce capital would be: (- the price of capital times the change in capital).

The rise in cost if we increase labour will be:

Once again, the change in cost along the line is 0 therefore these two will equal each other at all times. Equating these two together and rearranging we get the slope of an isocost as:

In equilibrium, the slope of the isoquant will equal the slope of the isocost:

The final rearrangement now, I promise. We can derive this sneaky formula:

What's interesting about this is that it tells us that money spent on each factor at the margin should yield the same level of additional output for the firm. Interesting.

We can map out the firm's costs in the long run on an isoquant map. It is called the expansion path as you can see below.

Typically, in the long run the firm will experience a varying level of costs. At low levels of output they will experience economies of scale. Then as output increases there will come a time when costs become constant. As output increase further still they will eventually reach a point of diseconomies of scale - where being a mass producer actually makes things costlier. In the short run costs are always higher than in the long run, always. Why? Because capital stock is fixed, we can only vary the amount of labour we employ.

That's it, the firm and isoquants covered. As usual post a comment if something doesn't make sense or you need further clarification - I'd be happy to help. Share the blog if you find it helpful, please. Cheer guys.
Sam.