Fall Eco11, 2008
Simon Board
Economics 11: Homework 1
30 September, 2008 Due date: Tuesday 14th October. Instructions: You are required to write up your solution separately and independently, although you are encouraged to discuss and work in groups. Please write your name, student ID number, and the name of your TA on the front page of the assignment that you hand in. Also, please put boxes around your nal answer to each part.
1. Utility Functions
Let X = {x |0 x 4 }. Suppose an agent receives utility u(x) = x =4x from consuming x slices of pizza. Which of the following utility functions represent the same preferences as above: (a) v (x) = 2x if 0 x 2, and v (x) = 4 x if 2 < x 4. (b) v (x) = 2x if 0 x 2, and v (x) = 8 2x if 2 < x 4. (c) v (x) = 4 (2 x)2 if 0 x 4. (d) v (x) = 4 (2 x)3 if 0 x 4. (e) v (x) = 4 (2 x)4 if 0 x 4. (f) v (x) = (2 x)2 if 0 x 4. if 0 < x 2 if 2 < x 4
2. Demography
This exercise considers the eect of changes in survival rates on the rate of population growth. 1
Eco11, Fall 2008
Simon Board
An agent chooses to divide her income M between general expenditure, x, and the number of births b. A birth has survival rate s [0, 1]. The cost of a birth is c. The cost of raising a child is k . The agents utility is u(b, x) = 2 sb + x
and her budget constraint is M x + cb ksb + (a) Write down the Lagrangian for the agents problem. Assuming there is an internal optimum, whereby x 0 and b 0, solve for the optimal number of births. (b) Show that if M is suciently high, there is indeed an internal optimum. (c) Suppose c = 0. How does the number of births depend on s? Provide an intuition. (d) Suppose k = 0. How does the number of births depend on s? Provide an intuition. (e) Suppose k = 5 and c = 1. For which value of s is the number of births maximised?
3. Consumption with Three Goods
Suppose and agent consumes three goods, {x1 , x2 , x3 }. Her utility is x1 + x2 + x3 The agent has income M = 1. (a) Suppose the prices are p1 = 1, p2 = 1 and p3 = 1. Solve for the agents optimal consumption. (b) Suppose the prices are p1 = 3, p2 = 1 and p3 = 1. Solve for the agents optimal consumption.
4. Consumer choice problem
An agent chooses a consumption bundle x to maximise her utility. Her utility function is u(x1 , x2 ) = 1 x1 x2 21 2 1 + x2 x2 22
Eco11, Fall 2008
Simon Board
where we assume 0 x1 1 and 0 x2 1. The agent has income M = 1. (a) Suppose the prices are p1 = 2 and p2 = 2. Solve for the agents optimal consumption. (b) Suppose the prices are p1 = 3 and p2 = 2. Solve for the agents optimal consumption. (c) Suppose the prices are p1 = 6 and p2 = 2. Solve for the agents optimal consumption.
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