Answer to Question #231076 in Macroeconomics for BENSON

Question

(a) consider three consumers who care about the consumption of a private good and their consumption of a public good. their utility function are given by

Ui=XiG i = 1, 2, 3

where Xi is consumer i's consumption of the private good and G is the amount of the public good consumed by all. the unit cost of the private good is $1 and the unit cost of the public good is $10. individual wealth levels are w1=30, w2= 50 and w3 = 20.

(i) compute the marginal rate of substitution between G and X for each of the three consumers.

(ii) derive the samuelson condition for this model

(iii) derive the aggregate resource constraint and compute the optimal level of public good consumption.

(b) brief discuss the relevance of the samuelson condition in consideration to provide public goods in your country

Answer 1

Solution:

a.). i.). The marginal rate of substitution between G and X for each of the three consumers:

MRS_GX = $\frac{MU_{Gi} } {MU_{Xi}}$

Ui=XiG i = 1, 2, 3

MU_Gi = $\frac{\partial U} {\partial G} = Xi$

MU_Xi = $\frac{\partial U} {\partial X} = Gi$

MRS_GX = $\frac{Xi}{Gi}$

MRS_GX = $\frac{P_{G} }{P_{X}}$

$\frac{Xi}{Gi} = \frac{1}{10}$

Gi = 10Xi

Budget constraint for consumer 1:

I = PxX + PgG

30 = X + 10G

30 = X + 10(10Xi)

30 = X + 100X

30 = 101X

X = 3.4

G = 10X = 10 $\times$ 3.4 = 34

MRS_GX for consumer 1= $\frac{34}{10} = 3.4$

Budget constraint for consumer 2:

I = PxX + PgG

50 = X + 10G

50 = X + 10(10Xi)

50 = X + 100X

50 = 101X

X = 2.02

G = 10X = 10 $\times$2.02 = 20.2

MRS_GX for consumer 2 = $\frac{20.2}{10} = 2.02$

Budget constraint for consumer 3:

I = PxX + PgG

20 = X + 10G

20 = X + 10(10Xi)

20 = X + 100X

20 = 101X

X = 5.1

G = 10X = 10 $\times$5.1 = 51

MRS_GX for consumer 1= $\frac{51}{10} = 5.1$

ii). Samuelsen condition:

∑MRS_1,2,3 = MRT

 

b.). The Samuelson condition is a condition for the efficient provision of public goods. Any optimal allocation is such that the sum of the quantity of private goods consumers would be willing to give up for an additional unit of public good must be equal to the quantity of private good that is actually required to produce the additional unit of a public good. When satisfied, the Samuelson condition implies that further substituting public for private goods or vice versa would result in a decrease of social utility.