## Sunday, October 28, 2018

### Demand and Supply: Part 2

In the previous post, there was a table of coefficients for average cost (AC), total cost (TC) and marginal cost (MC). Let's decompose TC into two components: one that doesn't vary as quantity (Q) varies and another that does. We'll call the components total fixed cost (TFC) and total variable cost (TVC). We can also decompose AC into average fixed cost (AFC) and average variable cost (AVC)

AC AFC AVC TC TFC TVC MC
Q-1 a a 0       0
Q0 b 0 b a a 0 b
Q1 c 0 c b 0 b 2c
Q2 d 0 d c 0 c 3d
Q3       d 0 d

AFC: $$p = \frac{a}{q}$$
AVC: $$p = b + cq + dq^2$$
TFC: $$p = a$$
TVC: $$p = bq + cq^2 + dq^3$$

Note how $$AC = AFC + AVC$$ and $$TC = TFC + TVC$$
Note too how MR is unrelated to TFC, but has the same relation to both TC and TVC.

### Demand and Supply: Part 1

Let's use a little bit of mathematics to assist our understanding of demand and supply, focusing here on the slightly more complicated concept of supply (demand functions identically, but we don't often decompose revenue into fixed and variable components). In the study of economics we often encounter three "curves" that describe how the price of an item is related to the quantity (Q) a firm is willing to supply: average cost (AC), total cost (TC) and marginal cost (MC). These three functions are mathematically related by two equations: $$AC = \frac{TC}{Q}$$ (which also means that $$TC = ACxQ$$) and $$\frac{dTC}{dQ} = MC$$ (which also means that $$TC_q = \int_0^q MC dQ$$). It's too soon to explain, but these are not quantities the firm is willing to supply, but rather the quantities the firm is able to supply.

A third degree polynomial function should be sufficient for this demonstration. In the table below, I've displayed the coefficients of each power of Q, highlighting the "fixed" component of TC (at Q0) and AC (at Q-1) noting that MC has no such "fixed" component.

AC TC MC
Q-1 a 0
Q0 b a b
Q1 c b 2c
Q2 d c 3d
Q3 d

AC: $$p = \frac{a}{q} + b + cq + dq^2$$
TC: $$p = a + bq + cq^2 + dq^3$$
MC: $$p = b + 2cq + 3dq^2$$

If the firm has any fixed costs (e.g. monthly rental payments on commercial property), they would be represented by the coefficient $$a$$; if it has variable costs (e.g. monthly wages payable to labour) they would be represented by the coefficients $$b$$,$$c$$ and $$d$$ (these are the costs that vary as Q is varied.