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Homework answers / question archive / Draw the indifference curves for the utility function U(x1, x2) = x1 + 3x2
Draw the indifference curves for the utility function U(x1, x2) = x1 + 3x2. 2.2 What is the marginal rate of substitution evaluated at an arbitrary consumption bundle (x1, x2)? 2.3 Suppose that p1 = 5, P2 = 2, and m = 10. Find the utility-maximizing consump- tion bundle (among those that satisfy the budge constraint) for this agent. You should be able to do this without using any calculus: it should be clear from your indifference curves. 2.4 Given any pair of prices P1, P2 and income m, what is (are) the utility-maximizing consumption bundle(s)? (Assume that p? > 0, p2 > 0, m > 0.) Consider the following three cases separately: 21 > P2/3, P1
Given, utility function;
U(x1, x2) = x1 + 3x2
This utility function is for perfect substitute goods, therefore the indifference curves will not be curves but straight lines with slope 1/3;
2) Marginal rate of substitution (MRS)
MRS is the rate at which consumer is willing to give up some units of one good in order to gain one more unit of another good. It is the slope of indifference curve. It is calculated as;
MRS = MU1 / MU2
= d/d1(x1 + 3x2) / d/d2(x1 + 3x2)
= 1+0 / 0+3
MRS = 1/3
3) P1 = 5
P2 = 2
Income, M = 10
The budget constraint will be;
P1 x1 + P2 x2 = M
5x1 + 2x2 = 10
The slope of budget line is the ratio of prices of the two goods;
Slope of BL = P1 / P2
Slope of BL = 5/2
As we can see, the slope of budget line = 5/2 is greater than slop of indifference curve = 1/3;
MRS < P1 / P2
1/3 < 5/2
Therefore, the consumer will only consume x2 and not x1.
So, the utility maximising bundle will be; (0,x2)
5x1 + 2x2 = 10
5(0) + 2x2 = 10
2x2 = 10
x2 = 5
The utility maximising bundle is (0,5)
4) Given any P1, P2 and M to find utility maximising bundle there will be 3 cases;
CASE 1: P1 > P2/3
P1 > P2/3
P1 / P2 > 1/3
This means when ;
P1 / P2 > MRS
When slope of the budget line is greater than slope of indifference curve, this means that the budget line will be steeper than IC, in this case the consumer will consume only of good x2 and 0 of x1. This is because both the goods are perfect substitutes and price of good 1 is more than good 2, he/she will consume only of 2nd good and none of 1st.
CASE 2: P1 < P2/3
P1 < P2/3
P1 / P2 < 1/3
This means when ;
P1 / P2 < MRS
When slope of the budget line is less than slope of indifference curve, this means that the budget line will be flatter than IC, in this case the consumer will consume only of good x1 and 0 of x2. This is because both the goods are perfect substitutes and when price of good 2 is more than good 1, he/she will consume only of 1st good and none of 2nd.
CASE 3: P1 = P2/3
P1 = P2/3
P1 / P2 = 1/3
This means when ;
P1 / P2 = MRS
When slope of the budget line is equal to the slope of indifference curve, this means that the budget line will be as same slope as the IC, in this case the consumer will consume both the goods equally. This is because both the goods are perfect substitutes and when price of good 2 is equal to good 1, he/she will consume both of the goods in equal proportion.
5) Now,
U(x1, x2) = 1 + 2ln(x1 + 3x2)
The marginal rate of substitution, i.e the slope of the indifference curve will be:
MRS = MU1 / MU2
= d/d1(1 + 2ln(x1 + 3x2)) / d/d2(1 + 2ln(x1 + 3x2))
=
MRS = 1/3
As we can see that with new utility function also, the slope of the indifference curve, i.e the MRS is 1/3.
Therefore, all the parts (1-4) will remain same for this new utility function also.
please see the attached file.
