Consider a market in which consumption of the good being traded generates a positive externality.

There are 100 identical consumers, each with a utility function given by 1/2√q+m+√G, where G denotes the total level of consumption in the market.

The good is sold by competitive firms that produce with a constant marginal cost of 1 $/unit.
now assume that the good is sold by a monopolist that produces using the same technology.

QUESTION: In this case, what is the difference between the optimal level of total consumption and the level of total consumption in equilibrium?

To understand the difference between the optimal level of total consumption and the level of total consumption in equilibrium, we need to analyze the situation under both competitive and monopolistic market structures.

In a competitive market, firms maximize profit by producing at a quantity where marginal cost equals the market price. Since the marginal cost of production is constant at $1/unit, the equilibrium quantity produced will be determined by the intersection of the market demand curve and the marginal cost curve. However, in this case, we have a positive externality associated with consumption.

Given that the utility function for each consumer includes the total level of consumption in the market (denoted as G), we can deduce that as G increases, the utility of each consumer will also increase. This positive externality means that consumers value the good more when others consume more of it.

In the competitive equilibrium, the market price is determined by the intersection of the market demand and supply curves. The demand curve reflects the marginal social benefit (MSB) because it incorporates the positive externality. However, the supply curve represents the marginal private cost (MPC) to the firms, which does not take into account the positive externality.

Due to this difference between the MPC and MSB, the competitive equilibrium quantity of consumption is lower than the socially optimal level of consumption. The competitive market tends to underproduce the good because firms only consider their private costs, which leads to a suboptimal outcome in terms of total consumption.

On the other hand, when a monopolist controls the production and sale of the good, the monopolist is the price-setter and faces the entire market demand curve. The monopolist maximizes its profit by producing at the quantity where marginal cost equals marginal revenue, which is given by the intersection of the marginal cost and marginal revenue curves.

Under this monopolistic scenario, the monopolist will take into account the positive externality associated with consumption as it affects the market demand curve. Consequently, the monopolist may produce a higher level of total consumption compared to the competitive equilibrium level. The monopolist internalizes the positive externality and can potentially generate higher total consumption.

Therefore, the difference between the optimal level of total consumption and the level of total consumption in equilibrium will depend on the elasticity of the market demand curve, the presence and size of the positive externality, and the monopolist's ability to capture additional profit from internalizing the externality.

To determine the specific difference in this case, you would need to analyze the shape and elasticity of the demand curve, the monopolist's pricing strategy, and the specifics of the externality and utility function.