Bruce has the quasi log-linear utility function, U (x_1, x_2) = x_1 + 2 In(x_2)

Question

Bruce has the quasi log-linear utility function, U (x_1, x_2) = x_1 + 2 In(x_2) Bruce has an income of $100 and faces prices p_1 = p_2 = 20. (a) What is the marginal rate of substitution for this utility function? (b) Solve for Bruce's optimal bundle. (c) Suppose Bruce's income falls to $20. What will happen to his optimal bundle? Is the MRS = MRT at the new optimal bundle? Dilbert spends his entire income of $-50 a month on hours of internet access (x_1) and other goods (x_2). Draw his budget constraint for each of the following situations. Please use a separate graph for each part and label all axes and intercepts. (a) The price of all other goods is $1 and the price of internet access is $5 per hour. (b) The prices are the same as in part (a), but Dilbert's grandmother sends him an extra $20 per month. (c) Dilbert's income is once again $50.00. However, the internet service provider charges the following rates: First four hours: $5.00 per hour Next six hours: $2.50 per hour All hours thereafter: $1.00 per hour

Explanation / Answer

3.

a.

U = x1 + 2lnx2 -------------------------(1)

Differentiation of eq.1 w.r.t. x1

MUx1 = dU/dx1 = 1+0 = 1

Differentiation of eq.1 w.r.t. x2

MUx2 =dU/dX2 = 2/x2

MRSx1x2 = MUx1/MUx2 = 1/(2/x2) = x2/2

MRSx1x2 = x2/2

b.

MUx1/MUx2 = Px1/Px2

1/(2/x2) = 20/20 = 1

X2/2 = 1

X2 = 2

Budget equation is :

100 = Px1*x1 + P2*x2

100 = 20*x1 + 20*2

X1 = (100-40)/20 = 3

Thus, consumption bundle is x1 = 3 and x2 = 2

c.

If income falls to $20

Then,

20 = Px1*x1 + Px2*x2

20 = 20x1 + 40

X1 = -1

Thus, at $20 income only 1 unit of x1 / x2 will be fetched and his optimal bundle will not be achieved.

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