Question

Suppose that one individual’s demand curve is D1(p) = 20−p and another individual’s is D2(p) = 10−2p. What is the market demand function? We have to be a little careful here about what we mean by “linear” demand functions. Since a negative amount of a good usually has no meaning, we really mean that the individual demand functions have the form D1(p) = max{20 − p, 0} D2(p) = max{10 − 2p, 0}. What economists call “linear” demand curves actually aren’t linear functions! The sum of the two demand curves looks like the curve depicted in Figure 15.2 (intermediate economics hal varian 8th edition). Note the kink at p = 5.

1) Consider the example of "Adding up Linear Demand Curves" discussed in Ch 15. Choose new numeric values of the two parameters in the individual demand functions, and solve for the resulting market demand curve. Choose numeric values for parameters "a" and "b" in an inverse linear demand curve of the following form: P=a-bQ. If you were in charge of setting the price for a product your company produces, and you had a good estimate of the demand for your product, you could determine an estimate of the total revenue you would make at each price. If you total costs are zero, what price would you set? What is the equation of the marginal revenue curve?

Answer 1

Given D1(p) = 20−p and  D2(p) = 10−2p be the individual demand functions.

Market Demand function represent the total demand of the market i.e. sum of individual demand, hence Market Demand (MD) can be represented as

MD(p) = D1(p) + D2(p)

MD(p) = 20 - p + 10 - 2p

MD(p) = 30 - 3p ..........(1)

Thus above equation 1 represent market demand function.

Demand curve D(p) = a - bP

considering some other values of parameters for demand function, let

D1(p) = 20 - 4p and D2(p) = 35 - 5p

Thus market demand function will be D1(p) + D2(p) i.e. (20 - 4p + 35 - 5p) = 55 - 9p

Diagrammatical representation shows that

In above diagram 1st and 2nd graph has straight downward sloping demand curve as they represent individual demand curve but 3rd graph has point K where line breaks and a little kink appears. This is because of the summition of 2 individual demand curves and their different demands.

Now as we know inverse demand curve equation can be represented as P=a-bQ thus choosing different values for a and b gives us different inverse demand curves. For say let P = 60 - 3Q where a = 60 and b = 3, now

we know Total Revenue (TR) = P.Q

Price	Quantity	Total Revenue	Marginal Revenue
60	0	0	-
57	1	57	57
54	2	108	51
51	3	153	45
48	4	192	39
45	5	225	33
42	6	252	27
39	7	273	21
36	8	288	15
33	9	297	9
30	10	300	3
27	11	297	-3
24	12	288	-9
21	13	273	-15
18	14	252	-21
15	15	225	-27
12	16	192	-33
9	17	153	-39
6	18	108	-45
3	19	57	-51
0	20	0	-57

Given Total Cost = 0

As per the equilibrium condition when Profit is Maximum i.e. difference between Total Revenenue(TR) and Total Cost is maximum i.e. as per the schedule at price 30 when profit is 300. before this price profit is increasing while after this point profit starts decreasing thus it will be the price determined in the market.

Equilibrium condition also states that euilibrium point would be the one where Marginal Cost = Marginal Revenue and Marginal Cost cuts Marginal Revenue from bottom. As in above case Marginal Revenue is approximately equal to Marginal Cost at price 30 only thus it would be the equilibrium price.

we know TR = P.Q and P = 60 - 3Q

thus from both these conditions we get

TR = (60 - 3Q). Q

TR = 60Q - 3Q²

Marginal Revenue (MR) refers to additional amount of revenue derived by selling one more uniy thus it can be derived by differentiating TR w.r.t.Q

dTR/dQ = 60(dQ/dQ) - 3(dQ²/dQ)

MR = 60 - 3.2.Q

MR = 60 - 6Q

Above equation shows Marginal Revenue Curve Equation MR = 60 - 6Q and price will be set at 30