17.8 Exercises

Posted: August 25th, 2021

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17.8 Exercises

Question 1

1. Let’s suppose that this good is sold at cost 14 to any consumer who wants to buy a unit. What are the possible equilibrium numbers of purchasers of the good?

r(x)f(z)=(1-x) z

r(x)f(z)=r(z)f(z) =(1-z) z =z-z^ =0.25 =p*

The value of z that satisfies: z=0.5=x

z=x=0, no consumer buys anything

f (0) =0, value of product is 0

This would not be an equilibrium if a negative number of consumers could buy the product. Since our model is bounded by this non-negativity constraint, z = x = 0 is an equilibrium. This gives us a total of 2 equilibriums. 

• Suppose that the cost falls to 29 and that the good is sold at this cost to any consumer who wants to buy a unit. What are the possible equilibrium numbers of purchasers of the good?

r(x)f(z)=r(z)f(z) = (1-z) z = z-z^ = 2/9 = p*

The value of z that satisfies: z=1/3    z=2/3=x

Equilibrium z=x=0, total of 3 equilibriums

• Briefly explain why the answers to parts (a) and (b) are qualitatively different.

The properties of these three equilibriums are not the same. When we only have three equilibriums like in (b), the second equilibrium represents the tipping point, the minimum level of expectations you need to reach for your product to converge towards a non- zero equilibrium. And the third equilibrium represents where your sales will be in the long term if you have reached the tipping point. In example (b), It means that if you want to launch the product, you need to build expectations around a minimum level of 1/3 of the market to be successful, and that if you are successful you will eventually sell your product to 2/3 of the market. However, if you only have two equilibriums like in (a), then the tipping point and the long-term point are the same.

d) Which of the equilibria you found in parts (a) and (b) are stable? Explain your answer.

(a): Equilibrium z=0=x is stable: any upper deviation from the equilibrium is automatically compensated by a downward pressure and we will converge again towards the same equilibrium.

(b): Equilibrium z=0=x is stable: any upper deviation from the equilibrium is automatically compensated by a downward pressure and we will converge again towards the same equilibrium.

Equilibrium z=2/3=x is stable: any upper deviation from the equilibrium is automatically compensated by a downward pressure and we will converge again towards the same equilibrium.

Question 2

1. How many equilibria are there? Why? [You do not need to solve for the number(s) of users; a graph and explanation are fine.]

Individual x reservation price, r (x) = 1 – x before network effect

Network effect is given by f (z) = z for z = ¼ and by f (z) = (1/2) – z for z >= ¼

Equilibria is attained at r (x) = f (x)=1

Then r (x) * f (z) = 1

But r (x) = 1 – x and f (z) = z

Hence f (x) = z = 1/(1-x)

When z <= ¼, x = -3

When z >= ¼, e.g. ½, x = -1, z = 0, x = 0

The point will not be equilibria if a negative number of consumers could buy the product. Since the model is bounded by this non-negativity constraint, z = x = 1 is an equilibrium. This gives us a total of 3 equilibriums.

• Which equilibria are stable? Why?

Equilibria (1,1) is the most stable. The reason is that changes in the level of consumption are compensated by equivalent changes in supply thus returning the consumer to the original level of satisfaction as before.

• Consider an equilibrium in which someone is using the good. Is social welfare maximized at this number of users, or would it go up if there were more users, or would it go up if there were fewer users? Explain

Adding an extra use of the commodity maximizes social welfare as the individual’s level of utility is increased by making the consumer’s position in the society better off. However, it would reduce with an increase in the users of the commodity given that the available quantity remains the same. It is because an increase in users lowers the marginal utility of each consumer.

Question 3: How Would You Attempt to Convince Users to Switch to Your Product?

Since the two commodities are network effect products having similar production costs, to ensure that customers are attracted will involve the following strategies;

1. Sponsoring technology-related events that give the product an avenue of exposure before the potential customers. The strategy helps create awareness besides luring customers that are unsatisfied by the quality of services being given in the competitor product.
2. Expanding advertising avenues, coupled with offers to first-time customers. The strategy helps communicate information about the product across a range of consumers, thus expanding the potential market share.

The above strategies are aimed at ensuring product performance reaches a tipping point to remain competitive against the established one.

Question 4: Case of Two Competing Products with Network Effects

The equilibrium configuration of consumers when there is no expected use of the product and hence no positive value is (0,0), in case half of the consumers are expected to use the product and half of the consumers buy the product, then equilibrium configuration is (0, ½). At the point when all consumers use the product, the equilibrium configuration is (1,1). A stable equilibrium configuration for the competing products is (1,1). The point (1,1) is attained when all substitution factors influencing customer choices all return to this pointi.e., the pressure of supply of competing factors in either product is compensated by the pressure of demand factors affecting consumption of the individual.