Solution to Discussion Section Week 9 SOME KEY CONCEPTS - IS Curve, Short-run Fluctuation, Multiplier, Consumption Smoothing - The Monetary Policy EXERCISE Question 1 (IS Curve and Short-run Fluctuation) The IS Curve shows the relationship between short-run output (Ỹt ) and real interest rate (Rt ) on horizontal and vertical axes, respectively, expressed in percentage. The relationship is given by Ỹt = a b (Rt r), where a, b and r (marginal product of capital) are parameters. (a) If the output is at the potential (Ỹt = 0) when the real interest rate equals to r, what is the value of a? Solution: Substitute Ỹt = 0 and Rt = r into Ỹt = a b (Rt r), we have a = 0. (b) How sensitive are the economy to the real interest rate? (Hint: Elasticity) Solution: Since Ỹt and Rt are in percentage, the change of Ỹt with respect to Rt , namely, dỸt /dRt , is the elasticity of output gap with respect to real interest rate. Hence, b measures the elasticity of output gap with respect to real interest rate, i.e. if the real interest rate increases by 1%, then output gap decreases by b%. (c) Plot the IS curve. Be careful on the values on the axes. Solution: See Figure 11.2 in p.p. 277 of the textbook. (d) Suppose there is a technological advancement shock so that the potential output increases from Ỹt = 0 to Ỹt = ↵ at the same real interest rate r. What does this represent on the IS curve? Draw the curve. Solution: See Figure 11.4 in p.p. 279 of the textbook. Question 2 (The Multiplier) Recall from the class that if the consumption also depends proportionately on short-run ⇥output, namely, ⇤ Ct /Y t = ac + xỸt . We have the multiplier eﬀect towards the IS curve such that Ỹt = 1 1 x a b (Rt r) . The multiplier is 1 1 x . (a) If you want to draw the IS curve with multiplier, compare the slopes of the IS curves with and without the multiplier, what would be your conclusion? Solution: The IS curve with multiplier is flatter than without multiplier. This follows from the fact that the slope of IS curve with multiplier comes from dỸt /dRt = b/ (1 x), which is greater than b. Thus, on the diagram with short-run output (Ỹt ) and real interest rate (Rt ) on horizontal and vertical axes, respectively, we have that the slope of IS curve being (1 x) /b, instead of 1/b; therefore, (1 x) /b = 1/b + x/b > 1/b. Thus, the IS curve with multiplier is flatter. (You should always be careful with axes when drawing any economic diagram.) What if now, we also have that investment also depends proportionately on short-run output as well, namely, It /Y t = ai b (Rt r) + z Ỹt .. (b) What would happen to the size of multiplier? Solution: This question is covered in the lecture on Thursday before the spring break. Recall previously on how we derive the IS curve, we consider the following from the following equations: Consumption Ct = ac + xỸt Yt 1 Investment Government Export Import Output It = ai b (Rt r) + z Ỹt Yt Gt = ag Yt EXt = aex Yt IMt = aim Yt Yt Ct It Gt EXt = + + + Yt Yt Yt Yt Yt Combining all equations to the output, letting ac + ai + ag + aex Yt Yt (1 Yt 1 Yt x z) Ỹt Ỹt ac + ai + ag + aex aim b (Rt = ac + ai + ag + aex aim 1 = a = 1 b (Rt r) ⇥ 1 a b (Rt x z r) 1 = a, to receive that: aim = IMt Yt r) + z Ỹt + xỸt b (Rt r) + z Ỹt + xỸt ⇤ The final equation is the IS curve with multiplier eﬀect from both consumption and investment. Hence, the multiplier increases from 1/ (1 x) to 1/ (1 x z). In other words, from the book, with multiplier eﬀect only on consumption, the multiplier is 1/ (1 x), now the multiplier becomes 1/ (1 x z). (c) Is there any restriction on x and z? Solution: Since the multiplier is 1/ (1 x z), we require x + z < 1, else, we will have negative value of multiplier. (d) Facing recession, the government sees that the output drops by $100. The government wishes to restore the output level, what should be the size of tax cut to achieve this? Solution: The government can simply cut the tax by the amount $100 ⇥ (1 x z) and let the multiplier eﬀect work through to achieve the $100 output. Question 3 (Consumption Smoothing) Why should you take something from the future and consume today? The marginal utility from consuming T-bone steak is 100, 80, 60, 40, and 20 for the 1st, 2nd, 3rd, 4th, and 5th piece, respectively. Suppose you live for 2 periods. In Period 1, you have suﬃcient income for just 1 piece of steak. In Period 2, you have suﬃcient income for 5 pieces of steak. Assuming that you can shift the consumption across time without any cost, what should you do? Explain. Solution: With initial income scheme, his total utility would be 400 utils. (Consuming 1 piece of steak today gives you 100 utils and 5 pieces of steak tomorrow gives you 300 utils.) However, if you can shift 2 pieces of steak from tomorrow to today, the utility received today is 240 utils, while tomorrow you consume 3 pieces of steak to receive 240 utils. This gives you 480 utils in total, which is higher than initially. Hence, you should smooth your consumption to maximize your utility. Question 4 (The Monetary Policy - MP Curve) The MP curve is represented by the real interest rate equation, Rt = it ⇡t , where it and ⇡t are nominal interest rate and inflation rate, respectively. One key assumption is that, inflation does not adjust quickly, namely, constant. The IS-MP diagram combines IS and MP curves with short-run output (Ỹt ) and real interest rate (Rt ) on horizontal and vertical axes, respectively. 2 (a) Suppose ⇡t = 2% and it = 4%, draw an MP curve. Solution: On the coordinates of IS-MP diagram, the MP curve is a flat line at Rt = 2%. (b) Central banker uses the nominal interest rate as the policy instrument. Seeing the economic boom as the IS curve is shifted rightward, central bank must curb down the economic expansion by increasing the nominal interest rate. Draw two IS curves and two MP curves that exhibit this story. Solution: See the right pane of Figure 12.5 in p.p. 306 of the textbook. However, we start oﬀ with IS’ and MP’, then we have IS and MP curves, namely, reverse the order. 3