international Econ

International Macroeconomics 4: Midterm

1 Instructions

� This exam is worth 100 points total, and has four questions. Each question is worth 25 points.

� In solving the exam, you are allowed to use all of the materials posted on Blackboard as well as the textbook. However, you are not allowed to use any other resources, including consulting the internet, seeking help from others, and whatnot. If there is the slightest suspicion that in solving your exam you have used references that are not allowed, then your exam will be investigated and if proven that you did not abide by the honor code then your grade in the class will automatically be: F. Please note as well that any late exams whose tardiness is not justi�ed by a legitimate and documented excuse, such as a health issue, will automatically earn a grade of zero.

� Please scan and upload your solutions in .pdf format. No format other than .pdf will be accepted.

2 Questions

1. Worth 25 points. Consider an economic agent with instantaneous utility function given by

Ut = ln(ct)+

� (1�ht)

� ,

where: Ut denotes instantaneous utility; ct denotes consumption (a choice variable); ht denotes labor (a choice variable); ;� > 0 are parameters; and t denotes time. (the greek letter �gamma�) is simply a scaling parameter, � is re�ective of the elasticity of intertemporal substitution, and � 2 (0;1) governs the value of the extra unit of leisure. In particular, assume that the agent�s total time endowment is normalized to 1, and therefore 1�ht is the agent�s total time spent on leisure. Let this agent�s budget constraint be given by:

ct � wtht,

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where wt is the exogenously determined wage rate; and the price of consumption has been normalized to 1. The agent�s utility maximization problem can be stated as:

max ct;ht

Ut = ln(ct)+

� (1�ht)

�

such that ct � wtht.

Show in mathematical detail that in this context the agent wants to work the same constant number of hours no matter what. Please show all your work.

2. Worth 25 points. Consider an economic agent with instantaneous utility function given by

Ut = ct � "

1+" (ht)

1+" " ,

which is "quasilinear in c:" Above, Ut denotes instantaneous utility; ct denotes con- sumption (a choice variable); ht denotes labor (a choice variable); ;� > 0 are parameters; and t denotes time. (the greek letter �gamma�) is simply a scaling parameter, � is re�ective of the elasticity of intertemporal substitution, and � 2 (0;1) governs the value of the extra unit of leisure. In particular, assume that the agent�s total time endowment is normalized to 1, and therefore 1�ht is the agent�s total time spent on leisure. Using a detailed mathematical procedure for which all work is shown, characterize the shape of the indi¤erence curve associated with this utility function in (ht,ct) space. "Characterize" in this case means arriving at conclusions regarding the slope and type of curvature. For instance, "decreasing and convex," or "decreasing and concave," etc., would be the type of statement your bottom line notes.

3. Worth 25 points. Consider a Korean investor who has I wons (the won is the Korean currency), and has decided to invest this won in Korea or Japan or both. Two things to note here. First, the "both" case involves investing a portion of the I wons in Korea and a portion of the won in Japan. Second, Korea is the "home" country.

(a) Let !won 2 [0;1] be the fraction of the won that the investor invests in Korea. So, !won = 0 means that the investor only invests in Korea, !won = 1 means that the investor only invests in Japan, and !won 2 (0;1) means that the investor invests a portion of the I wons in Korea and a portion of the I wons in Japan.

(b) The investor invests today and plans on "cashing out" next year. The one-year interest rate on the Korean investment is iwon and the one-year on the Japan investment is iU. Therefore, next year the investor gets

!wonI � (1+ iwon)+(1�!won)I � � 1+

� iU +100 �

Fwon=U �Ewon=U Ewon=U

�� :

To be clear, if !won = 0 then the investor invested only in Japan, and if !won = 1 then the investor invested only in Korea. Indeed, with !won = 0 then the

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expression above becomes

I � � 1+

� iU +100 �

Fwon=U �Ewon=U Ewon=U

�� the expression above reduces to I � (1+ iwon) and if !won = 1 then the earlier expression reduces to I � (1+ iwon).

(c) Assume that the investor�s instantaneous utility function is U = ln(c) and that the investor�s only source of income is the planned investment discussed above. Moreover, suppose that the investor can decide what Fwon=U is by "buying" a contract at the cost � �I �Fwon=U, where � > 0 is a parameter. It follows that the investor�s budget constraint is

c+� � I �Fwon=U � !wonI � (1+ iwon)

+(1�!won)I � � 1+

� iU +100 �

Fwon=U �Ewon=U Ewon=U

�� ;

and the Lagrangian is

L = ln(c)+�

8>< >:

!wonI � (1+ iwon) +(1�!won)I �

h 1+

� iU +100 �

Fwon=U�Ewon=U Ewon=U

�i �c�� � I �Fwon=U

9>= >; :

What are the agent�s optimal choices of !won and Fwon=U?

4. Worth 25 points. Consider an open economy that experiences a positive, one-time temporary positive shock to output. Assume that output is initially $1;200 per year and the world real interest rate is 6%. In year 0, output increases by 20%, returning to its initial level thereafter. In this case, the present value of output is equal to 21;440 and the present value of consumption is equal to 1;213:6. What is the trade balance equal to in year 0?

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