Question

5. John Paul derives utility from the consumption of cigars and brandy on friday evenings. His utility from c cigars and b glasses of brandy is given by: (a) First find Uc. Interpret you answer: What does your answer mean to J.P? b) Suppose cost or money is not a problem for J.P. How many cigars and glasses of brandy will he consume on a friday evening? Verify that your answer indeed maximizes U Hint: Hessian (c) Continue to assume cost is not a problem. However, J.P's doctor advises that he should limit the sum of glasses of brandy and cigars consumed to 5; i.e., c+b-5. What is the optimal consumption given this advice?

Answer 1

Solution:

U(c, b) = 20c - c² + 18b - 3b²

(a) Ucb is the mixed partial derivative, with differentiation of U with respect to c, and then the whole differentiation with respect to b. So, Ucb = $\frac{\partial }{\partial b}(\frac{\partial U}{\partial c})$

$\frac{\partial U}{\partial c}$ = 20 - 2c

So, $\frac{\partial }{\partial b}(\frac{\partial U}{\partial c})$ = 0

Or Ucb = 0. This implies that change in utility due to change in consumption of glasses of brandy, is not influenced by how J.P. changes his consumption of cigars. In same context, the change in rate of how utility changes due to change in consumption of cigars is also not influenced by how consumption of brandy changes. In simple terms, it basically gives an idea of independence between consumption of two goods, influencing the utility attained.

Denoting optimal consumption bundle by (c*, b*)

b) Since, money is not a problem, no budget constraint (unconstrained optimization):

We calculate the optimal consumption using the first order conditions (FOCs): $\frac{\partial U}{\partial b}$ = 0 and $\frac{\partial U}{\partial c}$ = 0

$\frac{\partial U}{\partial c}$ = 20 - 2c

So, from FOC, we have 20 - 2c* = 0, giving us c* = 20/2 = 10 cigars

And, $\frac{\partial U}{\partial b}$ = 18 - 2*3b

So, from FOC, we have 18 - 6b* = 0, giving us b* = 18/6 = 3 glasses of brandy

So, on a Friday evening, J.P. will consume 10 cigars and 3 glasses of brandy.

Verifying that the obtained consumption bundle indeed maximizes the utility: (using Hessian matrix methodology)

A combination maximizes utility if $(\frac{\partial ^2U}{\partial b^2})(\frac{\partial ^2 U}{\partial c^2}) - (\frac{\partial ^2 U}{\partial b \partial c})$ > 0 and $\frac{\partial ^2U}{\partial b^2}$ < 0, $\frac{\partial ^2U}{\partial c^2}$ < 0

$\frac{\partial ^2 U}{\partial b \partial c}$ is another way of writing $\frac{\partial }{\partial b}(\frac{\partial U}{\partial c})$ , which we have already seen, equals 0.

$\frac{\partial ^2U}{\partial b^2}$ = $\frac{\partial }{\partial b}(\frac{\partial U}{\partial b})$ = -6 < 0 and, $\frac{\partial ^2U}{\partial c^2}$ = $\frac{\partial }{\partial c}(\frac{\partial U}{\partial c})$ = -2 < 0, which is negative.

$(\frac{\partial ^2U}{\partial b^2})(\frac{\partial ^2 U}{\partial c^2}) - (\frac{\partial ^2 U}{\partial b \partial c})$ = (-6)*(-2) - (0)² = 12 > 0

Thus, we have verified that this gives the maximum utility.

c) Now we have one constraint of c + b = 5 (so, constrained optimization)

Then, c = 5 - b, substituting this in the utility function, and then doing the maximization problem.

U(b) = 20(5-b) - (5-b)² + 18b - 3b²

U(b) = 100 - 20b - (25 + b² - 10b) + 18b -3b²

U(b) = 75 + 8b - 4b²

Then, maximizing by differentiating U(b) with respect to b (finding the first order condition):

$\frac{\partial U(b)}{\partial b}$ = 0 + 8 - 2*4b

$\frac{\partial U(b)}{\partial b}$ = 0 then gives us 8 - 8b* = 0 or b* = 1

Putting this in the constraint above, we have c* = 5 - b* = 5 - 1 = 4

Thus, given the advice of constraint, optimal consumption for J.P. is 4 cigars and 1 glass of brandy.

Answer 2

Question5: Paul video derives utility only from CDs and DVDs. His utility function is ? = √? ∗ ? a. Plot Paul’s indifference curves for: U=5 U=10 U=20 (C refers on CDs and put it on horizontal axes, D refers to DVD and put it on vertical axes) b. Suppose Paul has $200 to spend and that CDs costs $5 and DVD costs $20, draw Paul’s budget constraint on the same graph as his indifference curves. c. Suppose Paul’s spends all his income on DVDs, how many can he buy and what is his utility? d. If Paul buys 5 DVDs , how many CDs can he buy ? what is his utility?

Answer 3

Shireen has a monthly income of $20 that she allocates among two goods: Meat (M) and Chickens (C). a. Suppose meat cost $2 per pound and chickens cost $4 per pound. Derive her budget line equation and draw her budget constraint putting (M) on the vertical axes (y-axes). b. Find her marginal rate of substitution. c. Suppose also that her utility function is given by the equation U (M,C) = 2M + C, what combination of meat and chickens should she buy to maximize her utility? What is maximum utility will Shireen realized?

Answer 4

Shireen has a monthly income of $20 that she allocates among two goods: Meat (M) and Chickens (C). a. Suppose meat cost $2 per pound and chickens cost $4 per pound. Derive her budget line equation and draw her budget constraint putting (M) on the vertical axes (y-axes). b. Find her marginal rate of substitution. c. Suppose also that her utility function is given by the equation U (M,C) = 2M + C, what combination of meat and chickens should she buy to maximize her utility? What is maximum utility will Shireen realized?