Solution PS4 - F2022 Econ 102-2

Fall 2022 Econ 102 Problem Set 4 (Practice Final) Prof. Saki Bigio November 21, 2022 READ THESE INSTRUCTIONS before you start writing: There are 4 questions. Your maximal score is 110 points. All questions are mandatory to recieve full credit. DEADLINE: November 30 at 3:30 PM. NOTE THAT POINTS VARY BETWEEN QUESTIONS. Name: UID: Grade: Q1 ................. Q2 Q3 Q4 TOTAL 1

1. Multiple Choice Question (20 Points - 4 points each). 1.A Assume that a Peruvian company, DMB LLC, just reported its earnings this year (year 0). The reported revenue was $9 million and the reported cost was $10 million. Its revenue is expected to grow 6% annually, while its cost is expected to grow 2% annually. Mark ALL the CORRECT statements. For this question, profit = revenue − cost. First, apply the Gordon Formula to calculate the present value of all future revenues and the present value of all future costs separately. Then, calculate the present value of all future profits, which equals the present value of all future revenues minus the present value of all future costs. a) The first year that the profit of the company becomes positive is year 3. b) If the discount rate is 12%, the present value of all future profits of the company is $50 million. c) If the discount rate is 10%, the present value of all future profits of the company is $111 million. d) If the discount rate is 5%, the present value of all future profits of the company is infinite. e) The present value of all future profits of the company is always positive no matter what the discount rate is. SOLUTION: a,c,d. If r > g then PV = d 1 r − g . 1.B Suppose the capital share in Canada is α = 2/5. Mark ALL the CORRECT statements. For this question, use the growth accounting formula given in class. a) If capital increases by 10%, labor hours increase by 5%, and total output increases by 10% relative to last year, then TFP should decrease by 1%. b) If capital increases by 15%, labor hours decrease by 5%, and TFP increases by 5% relative to last year, then total output should increase by 5%. c) If capital increases by 10%, TFP increases by 10%, and total output increases by 10% relative to last year, then labor hours should increase by 10%. d) If labor hours increase by 10%, TFP increases by 5%, and total output increases by 15% relative to last year, then capital should increase by 1%. e) None of the above. SOLUTION: e. Apply the formula of Solow residual. 1.C Suppose the Federal Reserve sells Treasury Bills and Treasury Bonds in the open market. Compared to a scenario in which no such open market operations took place, which of the following answers are CORRECT? a) Banks short of reserves end up with more loans. b) Banks short of reserves end up with the same amount of deposits. 2

2. Ricardian Equivalence VS Government with Money (30 Points). Suppose your friend who works as a consultant for the government in country A knows that you have taken ECON102. He/She has to come up with a plan to stimulate economy. So, he/she decides to consult you about what the government in country A should do. The household in country A lives for only two periods. In each period, they are endowned with income y 1 and y 2 . They could choose to consume goods in period 1 or save to consume in the next period. If they choose to save, they will recieve a nominal interest rate of i . The household in country A also loves to hold cash. In this economy, cash serves as a storage of value but doesn’t pay any interest rate. The household’s utility function is: u ( c 1 , M 1 , c 2 ) = log( c 1 ) + θ log M 1 P 1 + β log( c 2 ) The household’s budget constraint in each period is: P 1 c 1 + P 1 a 1 + M 1 = P 1 y 1 − P 1 τ 1 P 2 c 2 = P 2 y 2 + P 2 τ 2 + M 1 + (1 + i ) P 1 a 1 1. (5 points, easy) Your friend proposes that government in his/her country can print more money to finance their spending. The new government budget constraint is the following: P 1 G 1 + M 1 + (1 + i ) B 1 = P 1 τ 1 + B 2 + M 2 . He/She further proposes that the amount of money print will be exponentially increasing at the rate of µ : M t +1 = (1 + µ ) M t . Re-write the government budget constraint as a function of money demand, noting that money demand (for real balances) is M d ( Y, i ) = M S P . Since the government doesn’t plan to borrow, you can drop B t from the budget constraint. 2. (10 points, moderate) Note that the houshold has a utility from money that takes the form ϕ M t P t = θ log M t P t . We can derive money demand by using the following condition (you don’t need to re-prove this, but it becomes evident from setting up the household’s Lagrangian): ϕ ′ M 1 P 1 = i 1 + i u ′ ( c 1 ) , where u ( c ) = log( c ) . Solve for money demand M 1 P 1 in terms of c 1 . 4

3. (10 points, hard) Now, suppose the household knows that the government will increase the money supply every period at the rate of µ . Substitute the money demand that you derived in (2) into the government budget constraint in (1) and replace i = r + µ . Find the change in consumption with respect to a change in government spending ( dc 1 /dG 1 ) keeping interest rate fixed, where the government finances its spending only by printing money. 4. (5 points, moderate) Explain the result in (c). Can the government stimulate the economy when its spending is financed by printing money alone? What is the implication for government fiscal policy in this model, compared to the Ricardian Equivalence result we’ve seen in this class? SOLUTION: 1. P 1 ( G 1 − τ 1 )= M 2 − M 1 G 1 − τ 1 = M 2 P 1 − M 1 P 1 = M D 1 ( Y, i )(1 + µ ) − M D 1 ( Y, i ) = µM D 1 ( Y, i ) 2. θC 1 = i 1 + i M 1 P 1 M 1 P 1 = θ i + 1 i c 1 = θ 1 + 1 i c 1 3. G 1 − τ 1 = µM D 1 ( Y, r + µ ) = µ θ 1 + 1 i c 1 dc 1 dG = 1 µθ 1 + 1 i 4. 5

( c ) : 1 C = λP ( l ) : θ l = λw ( λ ) : PC = w (1 − l ) Rearranging: 1 PC = θ wl wl Pθ = C Solving for leisure: P wl Pθ = w (1 − l ) l = θ (1 − l ) l = θ 1 + θ Hence: h = 1 − l = 1 1+ θ . Finally, C = w P 1 1+ θ 1. Solving the problem for θ = 1 yield: h = l = 1 2 and c = w P 1 2 2. Solving the problem for θ > 1 yields: h = 1 1+ θ < 1 2 , l = θ 1+ θ > 1 2 and c = w P 1 1+ θ < 1 2 . Notice ∂h ∂ ( w/p ) w/p h = 0 : labor supply does not change with wages. Also, ∂c ∂ ( w/p ) ( w/p ) c = 1 , i.e. consumption increases one-to-one with real wages. 3. The government transfers income to the consumer proportional to labor income. This means that the transfer is TR = τwh . Replacing in the budget constraint we obtain: PC = w (1 − l ) + TR = w (1 − l ) + τw (1 − l ) = (1 + τ ) w (1 − l ) The consumer problem results: max C,l log C + θ log l s.t. PC = (1 + τ ) w (1 − l ) 7

Taking FOCs: ( c ) : 1 C = λP ( l ) : θ l = (1 + τ ) λw ( λ ) : PC = (1 + τ ) w (1 − l ) Rearranging: 1 PC = θ wl (1 + τ ) wl (1 + τ ) Pθ = C Solving for leisure: P wl (1 + τ ) Pθ = (1 + τ ) w (1 − l ) l = θ (1 − l ) l = θ 1 + θ Hence: h = 1 − l = 1 1+ θ . Notice that labor supply did not change with the subsidy. For consumption: C = w P 1+ τ 1+ θ , which is larger after the introduction of the subsidy. 4. Now the government collects taxes from the consumer. The total tax payed by the consumer is T , so the budget constraint results: PC = w (1 − l ) − T . The problem results: max C,l log C + θ log l s.t. PC = w (1 − l ) − T Taking FOCs: 8

5. The benchmark labor supply is h = 1 2 . Using the solution for hours supplied for part d we have an equation for taxes: w + θT w (1 + θ ) = 1 2 w + θT = θ + 1 2 w θT = θ − 1 2 w T = θ − 1 2 θ w To verify this, we check that the benchmark leisure decision of l = 1 2 will also yield the same tax. Using the solution for leisure hours for part d we have an equation for taxes: θ 1 + θ ( w − T ) w = 1 2 w − T = θ + 1 2 θ w − T = 1 − θ 2 θ w T = θ − 1 2 θ w Solving for consumption: C = w P w + θ ( θ − 1 2 θ w ) w (1 + θ ) − θ − 1 2 θ w C = w P 1 + θ − 1 2 1 + θ − θ − 1 2 θ w C = w P 1 + θ 2(1 + θ ) − θ − 1 2 θ w C = w 2 1 P − θ − 1 θ The First Welfare Theorem states that the consumer can not be better off with taxes. We need to compare utility in both scenarios: U ( T = 0) = ln w P 1 1 + θ + θ ln θ 1 + θ U ( T > 0) = ln w 2 1 P − θ − 1 θ + θ ln 1 2 10

Notice that labor supply is larger under the tax. This implies that the utility is lower, at least on the leisure part. For consumption, we can show when consumption is lower under the tax: w P 1 1 + θ > w P 1 2 − θ − 1 2 θ w θ − 1 2 θ w > w P 1 2 − 1 1 + θ θ − 1 θ > 1 P 1 − 2 1 + θ θ − 1 θ > 1 P 1 + θ − 2 1 + θ θ − 1 θ > 1 P θ − 1 1 + θ 1 P < 1 + θ θ P > θ 1 + θ This aligns with our result from part (4). If we assume that P > θ 1+ θ , then the tax will result in both lower consumption and lower leisure. Hence, the tax would make the consumer worse off. 11

SOLUTION 1. In the first period: τ 1 c 1 . In the second period: τ 2 c 2 2. Period 1: (1 + τ 1 ) c 1 + I = w (1 − l ) + Π F 1 Period 2: (1 + τ 2 ) c 2 = (1 + r ) I + Π I + Π F 2 Intertemporal: (1 + τ 1 ) c 1 + (1 + τ 2 ) c 2 1 + r = w (1 − l ) + Π F 1 + Π I + Π F 2 1 + r − T 3. L = max c 1 ,c 2 ,l ln( c 1 − θ (1 − l ) 1+ ϵ 1 + ϵ ) + β ln( c 2 ) + λ ( w (1 − l ) + Π F 1 + Π I + Π F 2 1 + r − (1 + τ 1 ) c 1 − (1 + τ 2 ) c 2 1 + r ) First order condition with respect to c 1 1 c 1 − θ (1 − l ) 1+ ϵ 1+ ϵ = λ (1 + τ 1 ) First order condition with respect to c 2 β c 2 = λ 1 + τ 2 1 + r First order condition with respect to l θ (1 − l ) ϵ c 1 − θ (1 − l ) 1+ ϵ 1+ ϵ = λw The consumption-leisure decision θ (1 − l ) ϵ = w 1 + τ 1 The Euler equation becomes c 2 c 1 − θ (1 − l ) 1+ ϵ 1+ ϵ = β (1 + r ) 1 + τ 1 1 + τ 2 Using wage definition it becomes the following c 2 c 1 − θ w θ (1+ τ 1 ) 1+ ϵ ϵ 1+ ϵ = β 1 + r 1 + τ 2 13

4. Start with good producing firms. For period 1: max L Π F 1 = A 1 L α − wL For period 2: max K Π F 2 = A 2 K 1 − α − r K 2 K Their optimal choices of firms taking prices as given are αA 1 L α − 1 = w = ⇒ L = αA 1 w 1 1 − α (1 − α ) A 2 K − α = r K 2 = ⇒ K = (1 − α ) A 2 r K 2 1 α Investment firm: max I r K 2 I 1 + r − I which implies that r K 2 = 1 + r . 5. Use labor market clearing condition. L =1 − l Next use the consumption leisure and firms optimal choices to obtain L θL ϵ (1 + τ 1 ) = αA 1 L α − 1 which implies L = αA 1 θ (1 + τ 1 ) 1 1+ ϵ − α When A 1 decreases the only policy the government act on is to decrease value added tax on consumption in the first period τ 1 . To determine the instrument τ new 1 we write the following. αA 1 θ (1 + τ 1 ) 1 1+ ϵ − α = α ( A 1 − µ ) θ (1 + τ new 1 ) 1 1+ ϵ − α which implies the following. τ new 1 = A 1 − µ A 1 (1 + τ 1 ) − 1 No, the policy cannot prevent the recession in the first period. To see this write the GDP in the first period. GDP 1 = A 1 L α L stays the same while A 1 decreases to A 1 − µ . Hence GDP in the first period decreases. 14