• This is an open book assignment. No generative AI such as ChatGPT permitted.

• This is an individual assignment — group work is not permitted.

• You have until 10.00h AEST on April 6, 2023 to 1) download and 2) work on the assignment and to 3) upload your answers. You are required to upload your answers as a single, legible pdf ile to the relevant TurnitIn folder. E.g., you can handwrite your answers and then convert them to a pdf ile.

• Please keep your pdf ile reasonably small (say  5MB).

• You may email your pdf ile to [email protected] as proof of your time of submission if you experience technical dificulties with uploading it. You then still have to upload the same pdf ile to TurnitIn as soon as possible.

• File formats other than pdf are not permitted. You may not submit multiple iles.

• Where an extension has not been approved, the following penalties apply to late or non-submission: A penalty of 10% of the maximum possible mark of the problem set will be deducted per day for up to 7 calendar days, at which point any submis- sion will not receive any marks unless an extension has been approved. Each 24 hour block is recorded from the time the submission is due.

• By undertaking this assignment you will be deemed to have made the following declaration:“I certify that

– my submission is entirely my own original work, and no part of my answers has been copied from any other source or person except where due acknowl- edgement is made,

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– I am familiar with and understand the implications of UQ’s policies relating to academic integrity and student conduct.”

Instructions: Answer all questions.  Show all your work — you must explain how you arrived at your answer. Partial credit may be awarded if a substantial part of the answer is provided.

Total Questions: 3

Total Marks: 50

Answer all questions.

Show all your work — you must explain how you arrived at your answer.

Partial credit may be awarded if a substantial part of the answer is provided.

[Generally, correctly following through after a mistake earns 50% follow-through marks. Follow-through marks are lower than that if the mistake simpliies the subsequent question/task.]

Question 1 (6 marks)

Suppose net investment income is NII  =  100, the international asset position is A  = 3000, the international liability position is L  = 4000, and the rate of return is 5 percent, r = 0.05.

(a) (3 marks) Economist John Green, a strong advocate of the dark matter hypothesis, believes that A is not accurately recorded. Calculate the amount of dark matter and the“true”international asset position, which we will denote TA, consistent with Green’s view.

Solution: From 100 = NII = r · TNIIP = 0.05 · TNIIP we infer TNIIP =

2000. From the text, we infer that TNIIP = TA — L (in analogy to NIIP = A — L). Thus 2000 = TA — 4000 and TA = 6000 [2 marks]. Note that NIIP = A — L = 3000 — 4000 = — 1000. Hence Dark Matter = TNIIP — NIIP = TA — A = 3000 [1 mark].

(b) (3 marks) Financial analyst Nadia Gonzalez does not believe in the dark matter hy- pothesis. Instead, she believes that A is accurately measured. In her view 5 percent is actually the rate of return on assets rA  = 0.05, and the rate of return on the coun-

try’s international liabilities, rL , is different.  Find the value of rL  consistent with Gonzalez’s view.

Solution:  100  = NII  = rA A — rL L  = 0.05 · 3000 — rL  · 4000 implies rL   =  = 504000  = 0.0125.

Question 2 (14 marks)

In answering this question, assume that there are no valuation changes of assets, that the net international compensation to employees equals zero and that there are no net unilateral transfers.

Consider a three-period economy that at the beginning of period 1 has a net foreign asset position of — 175. In each of the three periods 1, 2 and 3, GDP is 200. The interest rate on bonds held between any two consecutive periods is 6 percent; that is, r0  = r1  = r2 = r = 0.06.

(a) (4 marks) For this part of the question only, assume that in period 1, the economy runs a current account deicit of 5 percent of GDP. Find the trade balance in period 1 (TB1), the current account balance in period 1 (CA1), and the country’s net foreign asset position at the beginning of period 2 (B1).

Solution:

• CA1  = —0.05 · 200 = — 10 [1 mark].

• Substituting above values into the equation CA1 = TB1 + rB0 yields

— 10 = TB1 + 0.06 · ( — 175)

and thus TB1 = 0.5 [2 marks].

• From CA1 = B1 — B0 we infer that B1  = — 185 [1 mark].

(b) (1 mark) State the transversality condition for this economy.

Solution: B3 = 0.

(c) (4 marks) For this part of the question only, assume that in period 1, the economy runs a current account deicit of 5 percent of GDP and that in period 2, the trade balance of the economy is zero, that is, TB2  = 0. Is the economy living beyond its means? To answer this question ind the economy’s current account balance in period 3 and the trade balance in period 3. Is this value for the trade balance feasible? [Hint: Keep in mind that the trade balance cannot exceed GDP.]

Solution:

• CA2  = rB1 + TB2 = rB1 = 0.06 · ( — 185) = — 11.1.

• Therefore, B2  = B1 + CA2  = — 195 + ( — 11.1) = — 196.1.

• CA3  = B3 — B2 and B3  = 0. Therefore, CA3 = 196.1 [2 marks].

• Substituting values into CA3  = TB3 + rB2 yields

196.1 = TB3 + 0.06 · ( — 196.1)

and thus TB3 = 207.866 [1 mark].

• Since GDP = 200, this value of the trade balance is not feasible [1 mark].

(d) (5 marks) Compute as a percentage of GDP the maximal current account deicit in period 1 that is feasible for the economy.

Solution: Rearranging

B0  = —  —  —

yields

TB1  = — (1 + r)B0 —   TB3      —   TB2   

[1 mark].

Question 3 (30 marks)

Consider a two-period model of a small open economy with a single, perishable good. Let preferences of the representative household be described by the utility function

lnC1 + lnC2 ,

where C1  and C2  denote consumption in periods 1 and 2, respectively.  Each period t  =  1, 2, the household receives proits t  from the represenative irm it owns.  The production technologies in periods 1 and 2 are given by

Q1 = 3.5 · I0(0) .75

and

Q2 = 4 · I

where Q1 and Q2 denote output in periods 1 and 2, respectively, I0  = 39.0625 is exoge- nously given and represents the investment from“period 0”and I1 denotes the investment in period 1. Observe that the irm invests in period t — 1 to be able to produce goods in period t.  The household and the irm have access to inancial markets where they can borrow or lend. The irm inances its investments by issuing debt (both in“period 0”and in period 1), as in the lecture. Assume that there exists free international capital mobility and that the world interest rate, r* , is 5% each period (i.e., r0  = r1  = r*  = 0.05, where

rt is the interest rate on assets held between periods t and t + 1). Finally, assume that the household’s initial net asset position is B0(h) = — 10.

(a) (1 mark) Compute the initial net foreign asset position of the economy.

Solution: B0 = B0(h) — D0(f)  = B0(h) — I0 = —49.0625.

(b) (1 mark) Compute the irm’s output Q1 and proit 1 in period 1.

Solution: In period 1, the irm’s output is Q1  = 3.5 · I0(0) .75  = 3.5 · (39.0625)0.75  = 54.6875 and the irm’s proit is  1  = Q1 — (1 + T0)I0    13.6719.

(c) (3 marks) Compute the irm’s optimal level of investment in period 1 and its proit in period 2.

Solution: Using the irm’s optimal investment condition

MPK = d(4I10.75 )dI1  = 0.75 · 4I0.25  = 1.05 = 1 + T1  = MCK

we obtain I1   66.6389 [2 marks] and Q2  93.2944. Hence

 2  = 4I (1 + T1)I1   93.2944 — 1.05 · 66.6389  23.3236 [1 mark].

(d) (5 marks) Derive the optimal levels of consumption in periods 1 and 2.

Solution: From the household’s budget constraints in periods 1 and 2,

C1 + B 1(h) = (1 + T0)B0(h) +  1 ,

C2 + B2(h)   = (1 + T1)B1(h) +  2 ,

尸山、使

=0

we obtain the household’s intertemporal budget constraint

1 + T1                                                    1 + T1 .

Substituting  1    13.6719,  2    23.3236, T0  = T1  = 0.05 and B0(h)  = — 10 into (1)

yields

C1 + C2105  25.3849

[1 mark]. Combining this equation with the household’s utility maximization condi-

tion

1C1  = (1 + T1)1C2

[1 mark] yields

C1    1225.3849  12.6924 [2 marks],

C2  = 1.05 · C1    13.3270 [1 mark].

(e) (3.5 marks) Find the country’s net foreign asset position at the end of period 1 and, for each of the periods 1 and 2, the country’s savings, trade balance and current account balance.

Solution: [0.5 marks for each of the following]

• TB1  = Q1 — C1 — I1   — 24.6438.

• CA1  = TB1 + r0B0    — 27.0969

• S1  = I1 + CA1  39.5420

• B1  = CA1 + B0  = CA1   — 76.1594

• TB2  = 4I C2   79.9674

• CA2  = TB2 + r1B1    76.1594

• S2  = I2 + CA2 = CA2    76.1594

Now suppose that the government at the beginning of period 1 publicly announces an investment subsidy. Speciically, for each unit of investment that the irm makes in period 1, the government promises to pay the irm a subsidy of s2  ∈ (0, 1+r1 ) units of the good in period 2. The government inances the subsidy by charging the household a lump-sum tax T2 in period 2. The government neither has other expenditures nor other revenues. In particular, T1  = 0.

(f) (1 mark) Write down the government’s budget constraint in period 2.

Solution: s2I1 = T2 .

(g) (4 marks) Write down a formula for the irm’s proit in period 2. Derive the optimal investment condition and calculate the optimal investment as a function of s2 . Using a MPK-MCK-graph, illustrate in a igure how the optimal investment and the irm’s period-2 proit 2 change after the subsidy is introduced.

Solution: We have

 2  = 4I (1 + r1 )D1(f) + s2I1  = 4I (1 + r1 — s2 )I1

[1 mark]. The optimal investment condition  2(/)(I2) = 0 is

3I0.25  = 1 + r1 — s2

[0.5 marks]. Solving for I1 yields the optimal investment

I  (s1(*) 2 ) = ( 1.05 — s23)1025   = ( 31.05—s2)4

[1 mark] We can see from the following (schematic) graph that if the subsidy is introduced then optimal investment increases from a level of  66.6389 to I1(*)(s2 ) and period-2 proit  2 , originally the green area, increases by the dark green area [1.5 marks].

MPK, MCK

MPK = 3I0.25

 66.6389 I1(*)(s2 )                                                                 I1

(h) (1.5 marks) Write down the household’s period 1 and period 2 budget constraints. Derive the household’s intertemporal budget constraint.

Solution: From the household’s budget constraints

C1 + B 1(h) = (1 + r0 )B0(h) +  1 ,

C2 + B2(h)   = (1 + r1 )B1(h) +  2 — T2 ,

尸山、使

=0

[1 mark] we obtain the household’s intertemporal budget constraint

C1 +  = (1 + r0 )B0(h) +  1 +

[0.5 marks].

(i) (2 marks) Derive the economy’s resource constraint.  Compare it to resource con- straint that holds without the subsidy. Provide intuition for your comparison.

Solution: Substituting 1  = A1F (I0) — (1+r0 )I0 ,  2  = A2F (I1) — (1+r1 — s2 )I1 and B0(h) — I0 = B0 into (2) yields

C2            1 + r1 — s2                T2                                                                     A2F (I1)

1 + r1             1 + r1                  1 + r1                                                                    1 + r1   .

Using the government budget constraint to simplify this equation, we obtain the econ- omy’s resource constraint

C2                                                                               A2F (I1)

4I10.75

 3.1719 +

[1 mark]. This equals the resource constraint with s2 = 0 [0.5 marks]. The intuition is that feasible investment levels are not affected by the subsidy, and that the subsidy’s has a neutral effect on the household’s budget:  the HH receives the total subsidy payment as part of 2 and has to pay the same total in form of T2 . Thus, the subsidy does not affect the feasibility of triples (C1, C2 , I1 ) [0.5 marks].

(j) (6 marks) Assume that s2  = 0.1. Derive the household’s optimal consumption path and the current account balances CA1  and CA2  in periods 1 and 2, respectively. What effect did the introduction of the subsidy have on the optimal consumption path and on CA1?  Provide a detailed explanation of the effect on C1 , C2  and CA1  of introducing the subsidy and intuition for your results.  Is the household better off after the subsidy was introduced?

Solution:  Given s2   =  0.1, the optimal investment is I1      99.4468.  Therefore, the resource constraint simpliies to C1 + C2105    23.6927. Combining the resource constraint with household’s utility maximization condition

1C1  = (1 + r1 )1C2

yields

C1    1223.6927  11.8463 [0.5 marks],

C2  = 1.05 · C1    12.4386 [0.5 marks].

We can now obtain:

• TB1  = Q1 — C1 — I1   — 56.6056.

• CA1  = TB1 + r0B0  = TB1   — 59.0587 [0.5 marks]

• B1  = CA1 + B0  = CA1   — 108.1212

• TB2  = A2F (I1) — C2   113.5273

• CA2  = TB2 + r1B1    108.1212 [0.5 marks]

• Effects of subsidy:

– Relative to the case s2  = 0, C1 ↓ and C2  ↓ [0.5 marks]. The HH is worse off with the subsidy [0.5 marks].

– Also CA1 ↓, CA2 ↑ [0.5 marks].

• Intuition [2.5 marks]:

– The cost of the subsidy, s2I1   =  0.1 · 99.4468    9.9447, exceeds the increase in proit  2  = [4I (1 + r1 )I1] — 23.3236  31.4915 — 23.3236  8.1679.

– Therefore, ( 2 — T2) < 0 and the introduction of the subsidy acts like a negative“endowment”/income shock in period 2.

– Consequently, the HH is poorer, consumes less, and is worse off with the subsidy.