The purpose of this problem set is to motivate you to learn/review material related to equality constrained optimization for simple economic problems. Please read Appendix A of the course lecture notes/textbook and try the following questions. Pay particular attention to the Method of Lagrange and the section on concavity properties of functions. These two problems will be practice using the Method of Lagrange. Additionally, the last step in each problem asks you to interpret the mathematical outcomes in words. This is important as we are trying to illustrate economic trade-o↵s using mathematics. The math is just there to discipline our stories so we should be able to tell our stories using words that are consistent with the math that we used.

It might be daunting the first time you attempt such problems but the answers will be given in tutorial in Week #2 and you will be graded on the perceived e↵ort of your attempt. In other words, try by putting pencil to paper (or e-pencil to computer/tablet). Unless you receive special permission, you are to submit a hardcopy of your work at the beginning of tutorial if asked to do so by your tutor. Remember, you are allowed to work in groups but the best way to learn is to do the questions yourself, bash your head against the wall and try again, and only after that, talk to your study buddies to learn from each other.

1    Question #1

Consider the classic consumer’s choice problem of an individual who is allocating Y dollars of wealth amongst two goods. Let c1  denote the amount of good 1 that the individual would like to consume at a price of P1  per unit and c2  denote the amount of good 2 that the individual would like to consume at a price of P2  per unit. The individual’s utility is defined over the consumption of these two goods (only). Suppose we allow the individual’s happiness to be measured by a utility function u(c1 ,c2 ) which is increasing and strictly concave in both goods while also satisfying the Inada condition, limc1!0   = limc2!0   = 1.

The Inada conditions simply say that the slope of the utility function becomes vertical in the direction of the good that has its consumption level go to zero. By assuming increasing and strictly concave utility in both directions, we are assuming that the shape of the utility function is such that, holding constant the amount of one good, as the individual consumes more of the other good, they are happier but each additional unit of consumption yields less extra happiness than the previous unit (diminishing marginal utility of consumption). These assumptions imply that the first-order partial derivative of the utility function with respect to good one, u1 (c1 ,c2 ) = , and good two, u2 (c1 ,c2 ) =  , exist and are positive. Moreover, holding c2  constant, if c1  increases, u1 (c1 ,c2 ) remains positive but gets smaller. Similarly, holding c1  constant, if c2 increases, u2 (c1 ,c2 ) remains positive but gets smaller.

1. Let the budget constraint faced by the individual be P1 c1 + P2 c2  = Y . In words, interpret the meaning of the mathematical budget constraint.

2. The individual’s problem is to choose the consumption bundle (c1 ,c2 ) optimally. Specifically,

max{u(c1 ,c2 )}

c1 ,c2

subject to (c1 ,c2 ) satisfying the budget constraint, P1 c1  + P2 c2   = Y . Using the Method of Lagrange, let λ be the Lagrange multiplier on the budget constraint. The the Lagrangean function can be written as

L(c1 ,c2 , λ) = u(c1 ,c2 )+ λ (Y − P1 c1 − P2 c2 ) .

Take the first order-derivative of the Lagrangean function with respect to c1 , c2  and λ . Set each of them equal to zero in order provide the equations that can be used to identify the Lagrangean function’s“critical points”.

3. Take the first-order conditions for c1  and c2  as a system of two equations. Use one of these equations to find an expression for λ . Then use that expression to eliminate λ from the other equation. This will give you a trade-o↵ between purchasing more c1  at the expense of c2  or vice versa.

4. In words, interpret the economic trade-o↵ between good one and good two that you obtained in the previous step.

2    Question #2

This question is very similar to Question #1 but recasts the problem in terms of a single type of good consumed over time instead of two di↵erent types of goods within the same period of time.

Consider the problem of an individual that has Y dollars to spend on consuming over two periods. Let c1  denote the amount of consumption that the individual would like to purchase in period 1 and c2  denote the amount of consumption that the individual would like to consume in period 2. The individual begins period 1 with Y dollars and can purchase c1  units of the consumption good at a price P1  and can save any unspent wealth. Use s1 to denote the amount of savings the individual chooses to hold at the end of period 1.

Any wealth that is saved earns interest at rate r so that the amount of wealth the individual has at his/her disposal to purchase consumption goods in period 2 is (1+r)s1 . This principal and interest on savings is used to finance period 2 consumption. Again, for simplicity, we can assume that it costs P2  dollars to buy a unit of the consumption good in period 2.