1. An individual has a utility function defined over goods a and b that is given
ID: 1136679 • Letter: 1
Question
1. An individual has a utility function defined over goods a and b that is given by U(a, b12a 2b (a) Compute the Marginal Rate of Substitution (MRS) between a and b evaluated at the consumption bundle (a = 10, b = 4). Would this MRS be the same if it was evaluated for any consumption bundle (a A, b B), A > 0, B > 0? Why or why not? If not, compute the MRS at the consumption bundles (4, 10) and (5,5). Whether the MRS is the same or not, graph the Indifference Curves that pass through the three consumption bundles (10,4), (4,10), and (5,5) (b) Characterize the Indifference Curve that passes through the point (a 10, b- 4). What is the value of a associated with b- 5 on this IC? 2. Same as Question 1, but with 3. Same as Question 1, but with U(a, b) min(2a, b) 4. Same as Question 1, but with U(a, b2lna b 5. A consumer faces fixed market prices for a and b of pa and pb. Her income is given by I. If her utility function is that given in Equation (1), find her demand functions for a and b. If Pa-4, pb = 2, and 1 = 40, how much a and b would she (optimally) consume? 6. Same as Question 5, but where her utility function is given by Equation (2) 7. Same as Question 5, but where her utility function is given by Equation (3) 8. Same as Question 5, but where her utility function is given by Equation (4)Explanation / Answer
Given Utility function is U=min[2a,b]. This is a fixed proportions utility function which means that the consumer always consumes good a and b in a fixed proportion. The proportion is given as 2a=b. So the consumser will use all its income to bu both a and b such that all his income is used and he consumes a and b in a proportion such that 2a=b.
Given, Pa=4 and Pb=2 and I=40.
The consumer's budget constraint is 4a+2b=40. Substitution the fixed proportion into this budhet constraint we get the consumer's optimal bundle as
4a+2*2a=40. Solving for a we get a=40/12 and thus b=2a=2*(40/12)=40/6
Thus consumer's optimal bundle is (a.b)=(40/12,40/6)
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