Suppose that preferences over private consumption C and public goods G are such that these two goods are perfect substitutes, that is, the marginal rate of substitution of public goods for private goods is a constant b > 0. Determine the optimal quantity of public goods that the government should provide, and interpret your results. Make sure you show all of the relevant cases. What happens when b changes, or when q changes?

Repeat part (a), except with perfect complements preferences, that is, for the case where the representative consumer always wishes to consume private consumption goods and public goods in fixed proportions, or C = aG, with a > 0.

Please see explanation below.

Explanation:

Public goods are goods consumed collectively, they are provided for all members of a community,

no one can be excluded from their consumption. The consumption by one person does not decrease the consumption possibilities for others. Public goods are available for everybody without paying, and these goods cannot be rationed: they are either provided for the whole community, or for no one. Examples of public goods include the public lighting system, public roads, radio broadcasts, national defence, lighthouses, town pavements, etc.

Private goods, on the other hand, are goods consumed individually, and if a unit has been consumed by

someone, then no one else can also consume the same unit. Private goods are scarcely available, and consuming a unit will decrease the amount available for further consumption. Therefore consumers compete for private goods, i.e. private goods are rival in consumption. Consumers can consume them if they pay the price, non-payers are excluded from consumption.

In the first scenario, given that both the private good and public good are perfect substitutes, the optimum quantity produced by the government is at the point where marginal social cost is equal to the marginal social benefit. This optimum output is lower than that of the private firm because the price of public good is higher than price of private good (since marginal social cost > marginal private cost).

If b increases, that means consumers are willing to give up more units of public goods for one unit of the private good. Therefore, the quantity produced by the government will reduce.

For the second part of the question: C = aG, where a > 0.

This implies that equal or more units of the private good is consumed with a particular units of public good. The optimum output still remain at the point where marginal social cost is equal to marginal social benefit but this output level is lower than if the two goods were to be perfect substitutes.

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