# User:TDang/Sandbox

## Contents

### Consumer surplus

A demand curve for a good with network externalities shows marginal willingness-to-pay for each potential quantity sold. In this way it is like a typical demand curve. However, because the demand curve for the product with network externalities shows demand equilibria, the meaning is a little different. This difference is significant enough that reading consumer surplus off the demand curve becomes impossible.

It may help to consider a discrete example. We'll consider 6 possible consumers of a product with network externalities. Suppose each of them values the good even without network externalities, and that they all get the same benefit from network externalities, with the benefit equal Q, the size of the network.

Their values without network externalities are 7, 5, 7, 3, 7, and 1. We'll follow the traditional idea for constructing a demand schedule or "curve" (although since it's discrete it's not really a curve in this case), and order the consumers from those who value the good most highly to those who value it least. Then we can plot their WTP on a graph.

Consumer Value w/o network externalities Q Value including Q
1 7 1 7+1 = 8
2 7 2 7+2 = 9
3 7 3 7+3 = 10
4 5 4 5+4 = 9
5 3 5 3+5 = 8
6 1 6 1+6 = 7

The graph shows the value (WTP) for each of the six consumers. If we could read willingness-to-pay from this graph in the same way as a typical demand graph (which we can't), we could see that at a price of 6, the first consumer has a surplus of 8-6, the second of 9-6, etc. Adding up the differences for all the consumers, we would find a consumer surplus of 15.

This is wrong, though. The actual consumer surplus (which we can figure out from our original description, but not from the graph) is 30. The reason can be figured out from the first consumer. Their value for the good is 7+Q, where Q is the number of other people purchasing the good. When we draw the demand curve, each point on the curve was a demand equilibrium, so that expected quantity demanded and actual quantity demanded match. This is useful for drawing the demand curve, but it means the willingness-to-pay is only correct for that exact quantity.

Q WTP
1 7+1 = 8
2 7+2 = 9
3 7+3 = 10
4 7+4 = 11
5 7+5 = 12
6 7+6 = 13

This table shows the willingness-to-pay for the good depending on how many people actually purchased it. If the price was 6, and all 6 consumers decided to purchase the good, the WTP of the first consumer would be 13, instead of the 8 shown on the graph.

The green areas on the updated graph show the WTP if every consumer assumes that 6 people will be purchasing the product. Now we can find the consumer surplus by summing the difference between consumer's values and the price they pay. However, this is only correct if 6 consumers actually purchase the product. If fewer or more purchased the product, this would also give an incorrect consumer surplus value.

### Efficiency in competitive equilibrium

If a product has no externalities, we generally find that the competitive equilibrium is efficient. There are a number of ways economists talk about this. One was is to point out that, in a competitive equilibrium, the marginal willingness-to-pay (from the demand curve) is the same as the marginal cost. Since usually the marginal willingness-to-pay (WTP) is falling and/or the marginal cost is rising with increasing quantity, this means that ...

1. ... selling any less than the competitive equilibrium is inefficient, because at lower quantities the marginal willingness-to-pay is higher than the marginal cost, and
2. ... selling any more than the competitive equilibrium is inefficient, because at higher quantities the marginal willingness-to-pay is lower than the marginal cost.

The part of that which remains true with (positive) network externalities is that in competitive equilibrium, the marginal willingness-to-pay equals the marginal cost. The part which is different is what happens when the quantity sold increases. If the competitive quantity sold is Q1, where WTP = MC, and then the quantity sold increases to Q2, then the WTP for the Q1st unit increases due to the larger network benefits.

As an example, let's work with the demand from the consumer surplus section, and now suppose that the marginal cost is 9. In a competitive equilibrium, this would mean that 4 units would be sold at a price of 9.

Consumer Value w/o network externalities Q Value including Q WTP if Q=4 Surplus if Q=4 WTP if Q=5 Surplus if Q=5 WTP if Q=6 Surplus if Q=6
1 7 1 7+1 = 8 7+4 = 11 11-9 = 2 7+5 = 12 12-9 = 3 7+6 = 13 13-9=4
2 7 2 7+2 = 9 7+4 = 11 11-9 = 2 7+5 = 12 12-9 = 3 7+6 = 13 13-9=4
3 7 3 7+3 = 10 7+4 = 11 11-9 = 2 7+5 = 12 12-9 = 3 7+6 = 13 13-9=4
4 5 4 5+4 = 9 5+4 = 9 9-9 = 0 5+5 = 10 10-9 = 1 5+6 = 11 11-9 = 2
5 3 5 3+5 = 8 3+4 =7 not sold 3+5 = 8 8-9 = -1 3+6 = 9 9-9 = 0
6 1 6 1+6 = 7 1+4 = 5 not sold 1+5 = 6 not sold 1+6 = 7 7-9 = -2
Total Surplus 6 9 12

## Some Variations

### Stand-alone value

This graph represents a slightly different network. As you can see there is only one equilibrium point and this point is stable. The curve, Qx starts above the 45 degree line, even when Q=0, and remains so until point c. Should Q become greater than point c, then people would leave the network until Q retreated to point c, where Qx = Q.

On the other end, when Q=0, Qx is still positive. This illustrates the idea that even when there is no current demand, or no one currently in the network. There is still value in the network and people will want to join, ultimately reaching point c. This is an alternative to above, in that there is no critical mass point to achieve, because of this expectations play a much smaller role, if at all, in reaching that desirable equilibrium point c.