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?
(b) 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.
Consider the model of public goods in the last section of this chapter.
(a) 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?
(b) 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.
The correct answer and explanation is:
Optimal Quantity of Public Goods Provision by the Government
(a) Perfect Substitutes Case
When private consumption CC and public goods GG are perfect substitutes, the consumer’s utility function implies a constant marginal rate of substitution (MRS) between the two goods: MRS=bMRS = b
where b>0b > 0 represents the rate at which the consumer is willing to trade public goods for private consumption.
The government’s budget constraint is: C+qG=YC + qG = Y
where qq is the marginal cost of providing one unit of GG and YY is total income.
Since the consumer views CC and GG as perfect substitutes, they will only consume public goods if: b≥qb \geq q
In this case, the government should allocate all resources to GG, providing G=YqG = \frac{Y}{q} and C=0C = 0.
If b<qb < q, the consumer values private consumption more, so the government should provide no public goods (G=0G = 0) and allocate all resources to C=YC = Y.
Comparative Statics
- When bb increases: The consumer places a higher value on public goods, so the government will allocate more resources to GG.
- When qq increases: The cost of public goods rises, so if q>bq > b, the government provides fewer public goods.
(b) Perfect Complements Case
If consumers require private goods and public goods in a fixed proportion: C=aG,a>0C = aG, \quad a > 0
the government’s budget constraint is: C+qG=YC + qG = Y
Substituting C=aGC = aG: aG+qG=YaG + qG = Y G(a+q)=YG(a + q) = Y G∗=Ya+q,C∗=aG∗G^* = \frac{Y}{a + q}, \quad C^* = aG^*
Comparative Statics
- When aa increases: Consumers demand more private goods relative to public goods, reducing GG.
- When qq increases: The cost of public goods rises, reducing the optimal GG.
Now, I’ll generate an image illustrating these cases.
Here is the conceptual economic diagram illustrating the optimal provision of public goods under both perfect substitutes and perfect complements cases. Let me know if you need any modifications!
