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^(1/2))+m +(G^(1/2)) 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.
QUESTION: What is the difference between the optimal level of total consumption minus the amount of total consumption generated by the market?
To find the difference between the optimal level of total consumption and the amount of total consumption generated by the market, we first need to determine the optimal level of total consumption.
In this scenario, consumption of the good generates a positive externality. This means that the social benefit from consuming the good exceeds the private benefit enjoyed by the consumers in the market. To incorporate this externality, we must find the level of consumption that maximizes the social welfare.
Given the utility function of consumers and the constant marginal cost of production, we can find the optimal level of consumption by equating the marginal social benefit (MSB) to the marginal cost (MC).
The marginal social benefit is the sum of the marginal private benefit (MPB) and the marginal external benefit (MEB). The MPB is the derivative of the consumer's utility function with respect to consumption (q). Differentiating the given utility function, we have:
d(MPB)/dq = d((1/2)*(q^(1/2))+m +(G^(1/2)))/dq
= (1/4)*(q^(-1/2))
The MEB is the derivative of the total utility with respect to the total consumption in the market (G). Differentiating the total utility function, we have:
d(MEB)/dG = d((1/2)*(q^(1/2))+m +(G^(1/2)))/dG
= (1/2)*(G^(-1/2))
The marginal cost (MC) is constant at 1.
Setting the MSB equal to the MC and solving for q, we have:
(1/4)*(q^(-1/2)) + (1/2)*(G^(-1/2)) = 1
Simplifying the equation, we get:
(q^(-1/2)) + 2*(G^(-1/2)) = 4
Now, to find the optimal level of total consumption (G*), we can plug in the value of q into the equation to eliminate it:
(4 - 2*(G^(-1/2)))^2 = 4
Solving this equation will yield the optimal level of total consumption in the market, denoted as G*.
Once you have obtained the optimal level of total consumption (G*), you can subtract the actual amount of total consumption generated by the market (G_market) to find the difference:
Difference = G* - G_market
Please note that to determine the actual amount of total consumption generated by the market, you might need additional information, such as the demand curve or quantity supplied by the competitive firms.