# Demand function for partially subsumable products

I am struggling with this question that should be simple for economists (I am not an economist at all):

There is a market with a limited number of (heterogenous) consumers with two firms, each offering a product. The two products are ‘partially’ substitutable. I am looking for demand functions to show how the two products interplay in terms of price and demand.

I remember there were some traditional models in microeconomics showing something like this:

d1= a1 – b1.p1 + c1.p2

d2= a2 + b2.p1 – c2.p2

where d and d2 show the demands and p1 and p2 denote prices of products 1 and 2 respectively.

The issues with these functions are that:

• They include many unknown parameters (i.e., a1, b1, c1, a2, b2, c2);
• They do not directly consider ‘partial’ substitutability;
• They include prices identified by the firms, but ideally, I would also like to include ‘values’ of products (v1 and v2) identified by consumers regardless of their prices (perhaps, this is similar to what utility functions consider).

Is there any way to solve these issues by using stronger functions?

Thank you

If we consider a framework of two goods and if we excluding all other variables, except for the prices this gives two demand equations: $$q_1 = d_1(p_1, p_2),\\ q_2 = d_2(p_1, p_2).$$ Usually, it is assumed that the law of demand holds, which means that demand is decreasing in own prices. If so, $$\frac{\partial q_1}{\partial p_1} < 0$$ and $$\frac{\partial q_2}{\partial p_2} < 0$$. An interesting question is how the demand for good 1 changes if the price of good 2 changes. In a first setting, it might be that the two goods are subsitutes (e.g. apples and pears), in this case, we should have that $$\frac{\partial q_1}{\partial p_2} > 0.$$ the reasoning is the following. If the price of good 2 increases, its demand decreases and, in turn, the demand for good 1 increases. The idea is that consumers will buy less of the more expensive good and will (at least partially) compensate this by buying more of good 1.
A second setting is where the two goods are complements (i.e. consumed together like coffee and milk). In this case, one can expect that $$\frac{\partial q_1}{\partial p_2} < 0$$. If good $$2$$ becomes more expensive then the consumer will not only buy less of good 2 (due to its higher price) but also less of good 1.
One particular specification that is frequently used (for example in theoretical work) is the linear specification: $$q_1 = a_1 + b_1 p_1 + c_1 p_2\\ q_2 = a_2 + b_2 p_1 + c_2 p_2.$$ If $$b_1 < 0, c_1 > 0, b_2 > 0$$ and $$c_2 < 0$$, this gives the correct signs to have substitutes: $$\frac{\partial q_1}{\partial p_1} = b_1 < 0 \text{ and }\frac{\partial q_1}{\partial p_2} = c_1 > 0\\ \frac{\partial q_2}{\partial p_1} = b_2 > 0 \text{ and } \frac{\partial q_2}{\partial p_2} = c_2 < 0$$
• In practice we are not really interested in the exact values of the parameters $$a_1, a_2, b_1, b_2, c_1$$ and $$c_2$$, but only their signs, because this determines whether the law of demand holds and whether the goods are substitutes or complements. You can think of the values of $$|c_1|$$ and $$|b_2|$$ as the degree of substitutability (of complementarity)