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This will be a graph shifting question, which asks about the intuition of both the AD-AS figures and the I-NS figures, specifically relating them to long-term growth. Suppose that the economy uses (only) two major inputs in its production: capital (K) and labour (L). Suppose also that we are thinking about long-term growth, specifically through productivity growth. 1. Suppose that there is an increase in the productivity of input in this economy. What would we expect to happen to Y in the long-run? Explain your answer. [2 points] Now suppose that the Aggregate Demand (AD) remains fixed through this period, and that we begin in long-run equilibrium. 2. If AD remains fixed, and the growth in the economy's productivity occurs, then what would happen to (i) the price level (p), (ii) Real GDP (Y), and (iii) the price of inputs (L & K ) in the economy? Explain your answer using the AD-AS model. [2 points] Suppose that the government is concerned about stability in the price level and is concerned about the impact of this long-term growth on the long-run price level. 3. What can the Government do to maintain stability in the price level? Explain your answer using the AD-AS model. [2 points] 4. What would be the impact of your answer to the previous question on the government budget deficit? What would need to be true for this to have a negative effect on the deficit? [1 point] Assume that the budget deficit here increases. 5. What would be the impact of the change in Y and the increase in the budget deficit on this economy's long-term interest rates? Explain your answer using the I-NS model. [ 3 points].

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3.3 Macro Economic Models

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1. This question is on the application of the Binomial option pricing model. PKZ stock is currently trading at 100. Over three-months it will either go up by 6% or down by 5%. Interest rates are zero. a. [25 marks] Using a two period binomial model to construct a delta- hedged portfolio, price a six month European call option on PKZ stock with a strike price of £105. b. [3 Marks] Using your answer from the first part, together with the put-call parity, price a put option on the same stock with same strike and expiry. COMP0041 SEE NEXT PAGE 2 2. This question is on the Binomial method in the limit δt → 0. [40 Marks] The binomial model for pricing options leads to the for- mula V (S,t) = e−rδt [qV (US,t + δt) + (1 − q) V (DS,t + δt)] where U = eσ √ δt, D = e−σ √ δt, q = erδt −D U −D . V (S,t) is the option value, t is the time, S is the spot price, σ is volatil- ity and r is the risk-free rate. By carefully expanding U,D,q as Taylor series in δt or √ δt (as appropriate) and then expanding V (US,t + δt) and V (DS,t + δt) as Taylor series in both their arguments, deduce that to O (δt) , ∂V ∂t + 1 2 σ2S2 ∂2V ∂S2 + rS ∂V ∂S − rV = 0. COMP0041 SEE NEXT PAGE 3 3. This question is on probability and Monte Carlo a. Consider theprobabilitydensity function p (x) fora randomvariable X given by p (x) = { µ exp (−µx) x ≥ 0 0 x < 0 where µ (> 0) is a constant. i. [15 Marks] Show that for this probability density function E [ eθX ] = ( 1 − θ µ )−1 Hint: You may assume µ > θ in obtaining this result. ii. [20 Marks] By expanding ( 1 − θ µ )−1 as a Taylor series, show that E [xn] = n! µn , n = 0, 1, 2, .... iii. [15 Marks] Hence calculate the skew and kurtosis for X. COMP0041 CONTINUED ON NEXT PAGE 4 b. [32 Marks] An Exchange Option gives the holder the right to exchange one asset for another. The discounted payoff for this contract V is V = e−rT max (S1 (T) −S2 (T) , 0) . The option price is then given by θ = E [V ] where Si (t) = Si (0) e (r−12σ 2 i )t+σiφi √ t for i = 1, 2, and φi ∼ N (0, 1) with correlation coeffi cient ρ. Youmayassumethatauniformrandomnumbergenerator isavail- able. Use a Cholesky factorisation method to show( φ1 φ2 ) = ( 1 0 ρ √ 1 −ρ2 )( x1 x2 ) , where ( x1 x2 ) is a vector of independent N (0, 1) variables and has the same distribution as ( φ1 φ2 ) . Give a Monte Carlo simulation algorithm that makes use of anti- thetic variates for the estimation of θ. COMP0041 SEE NEXT PAGE 5 4. This question is on finite differences a. [30 Marks] Consider a forward difference operator, ∆, such that ∆V (S) = V (S + h) −V (S) , (4.1) where h is an infinitessimal. By introducing the operators D ≡ ∂ ∂S ; D2 ≡ ∂2 ∂S2 show that ∆ ≡ ehD −1 (4.2) where 1 is the identity operator. Hint: start by doing a Taylor expansion on V (S + h) . By rearranging (4.2) show that D = 1 h ( ∆ − ∆2 2 + ∆3 3 − ∆4 4 + O ( ∆5 )) . Hence obtain the second order approximation for ∂V ...

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Macro Economics Exam Short Essays: Please select and answer seven ( 7) of the following essay type questions. A. Please explain with specific examples for each category: “Among risks the Fed is taking: that some programs won’t work, that officials won’t be able to unwind them, that politicians will grow accustomed to directing the central bank to fix problems its tools aren’t designed to solve, and that public discontent about the central bank’s choices will erode its authority over time. “ WSJ 04 28 2020 B. What specific Macro-economic phenomenon is being described. Explain how this phenomenon impacts the balance sheet of the Federal reserve and of a commercial bank, “The Fed has a unique power, the ability to create money by crediting banks with funds they can lend. That helps it guide the cost of money, which is the interest rate” WSJ 04 28 2020. C. Please describe four major initiatives that the Federal Reserve has undertaken, in order to address the impact of COVID-19 to the economy. D. What does the “Bagehot’s Dictum,” refer to? Please explain in the context of the effect of the current effect of the COVID-19 to the economy. E. Suppose the economy is suffering in a recessionary period. Firms are facing increasing inventories and individual consumers are increasing their saving to prepare for hard times ahead. What is likely to happen to the economy and can it correct itself and grow toward full employment in the short run? How is this different and how is it similar to the impact of the COVID-19 in the economy? F. Describe the role of business inventory change in determining the equilibrium level of GDP and changes in the level of GDP. G. Differentiate between an induced increase in consumption and an autonomous increase in consumption. How are they represented on a graph? H. The expenditure schedule and the aggregate demand curve show much the same thing, with one crucial difference-the price level. How does the price level affect the two schedules? I. Define the terms recessionary gap and inflationary gap. Why do they occur? Please frame your answer within the context of the COVID-19 impact in the economy. J. Why do booms and recessions tend to be transmitted across national borders? Please frame your answer within the context of the COVID-19 impact in the economy. K. The total expenditure schedule in Macroland begins with these initial levels (in billions of dollars): Income = 1,000; Consumption = 900; Investment = 200; Government = 300; Net Exports = −100. If the MPC = 0.75 and income increases in increments of 200, find the equilibrium level of income. If full employment requires an income level of 2,000, what (if anything) should the government do? Indicate both the direction of the spending change and the size of the spending change. L. What is a multiplier? How does the multiplier effect occur? Please provide examples of at least three types of multipliers in an economy. Please provide some examples.

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This is based on lab activity in week 34. You will explore the DNS protocol, and briefly explain the purpose of DNS protocol and its working mechanism in a general context. You will identify a normal DNS request-response message pair, and explain the protocol operation focusing on the transport layer and DNS fields, and the purpose of that DNS query and how that query is resolved. Further, you will identify a DNS anomaly, and explain the nature and type of.

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