Public Affairs 974 Menzie D. Chinn Fall 2009 Social Sciences 7418 University of Wisconsin-Madison Problem Set 1 Answers Due in lecture on Tuesday, 29 September. ""Box in− your answers to the algebraic questions. 1. Policy in an IS-LM model Suppose the real side of the economy is given by: (1) YAD= Output equals aggregate demand – an equilibrium condition (2) GICAD ++≡ Definition of aggregate demand (3) )( FTYcOCC +−+= Consumption function, c is the marginal propensity to consume (4) TTAtY=+ Tax function; TAis lump sum taxes, t is tax rate. (5) FFT= Transfers function; FTis lump sum transfers. (6) IINbi=− Investment function (7) GGO= Government spending on goods and services and the monetary sector is given by: Eq.No. Equation Description (8) MPMPds= Equilibrium condition (9) MPMPs= Money supply (10) MPkY hid=+ −μ Money demand 1.1 Graph the IS and LM curves on a single graph. Show the vertical intercepts, the slopes, and the intersection. Also show what each curve depends on.1.2 Show what happens if government spending on goods and services are increased by ΔGO. Where )1(11tc −−≡αv and hbktc /)1(11ˆ+−−≡α 1.3 Using the same graph as in 1.2, compare what happens if government spending on transfers is increased by ΔFT, where ΔFT is numerically equivalent to ΔGO. LM|μ,, PMvv IS| Av i0 Y0 Y bA/v⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−PMhh1μ i IS| GOA Δ+v Y1 GOΔαv GOΔαˆ LM|μ,, PMvv IS| Av i0 Y0 Y Slope = k/h ⎟⎠⎞⎜⎝⎛−−−=btcslope)1(1bA/v⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−PMhh1μ iNotice the output increase is smaller now, by a factor c (the marginal propensity to consume). 1.4 Show what happens if the only fiscal policy is an increase in government spending by amount ΔGO, and the Fed increases the money supply to keep the interest rate constant. What is the multiplier in this case? LM|μ,, PMvv IS| Av i0 Y0 Y bA /v⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−PMhh1μ i IS|GOA Δ+v Y1 GOΔαv GOΔα Y3 LM|μ,, PMMvvΔ+ LM|μ,, PMvv IS| Av i0 Y0 Y bA /v⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−PMhh1μ i IS| FTAΔ+v Y2 FTcΔαv FTcΔαˆ1.5 Suppose the economy is given by equations (1)-(10), but equation (3) is given by: (3’) WORTHFTYcOCCγ++−+= )( Where WORTH is net private sector household wealth (assets minus liabilities), including housing and equity. What happens when net worth falls by ΔWORTH? As worth falls, then consumption falls, which is then multiplied by the multiplier. 1.6 Compare what happens if a lump sum tax cut of $100 billion occurs if consumers either spend or “rebuild their balance sheets” (i.e., save). For simplicity, set c = 0.7, t=0, and γ=0.05. If consumers behave in the Keynesian fashion, then the standard multiplier is applied: TAcY Δ−=Δαˆ Assume perfectly accommodating monetary policy: 3.233)100()7.0()7.01(1=−××−−=Δ−=Δ−TAcYα If the entire tax cut is saved on the first round, but the resulting higher income is consumed in a normal fashion, then (assuming accommodating monetary policy again): 7.11)100()05.0()7.01(1=−××=−=Δ−=Δ−TAYγα but if there is no MPC for subsequent rounds of spending/income, then 5)100()05.0(=−×−=Δ−=Δ TAYγ 2. Liquidity Trap Consider the following diagram. LM|μ,, PMvv IS| WORTHA,v i0 Y0 Y ⎟⎠⎞⎜⎝⎛+−−−=bmtcslope)1(1 bA /v ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−PMhh1μ IS| WORTHWORTHA Δ+,v WORTHΔγαv Y42.1 Using the above graph, show what happens if expansionary monetary policy is used (label the curve shift assuming Δ(M/P) ). LM|μ,, PMIS| A i0 Y0 Y bA / YZ PMhh1−μ LM|μ,, PMM Δ+⎟⎟⎠⎞⎜⎜⎝⎛Δ+−PMPMhhv1μLM|μ,, PM IS| A i0 Y0 Y bA / YZ PMhh1−μ iIn this case, the increase in the money supply cannot increase output in excess of where the IS curve intersects the horizontal axis at YZ. That’s because of the liquidity trap, wherein at a certain point, monetary policy cannot affect interest rates and hence investment. 2.2 What is the maximal impact on income that can be obtained using monetary policy? See above. 2.3 Show what happens if expansionary fiscal policy is used (label the curve shifts assuming ΔGO). 3. Given an economy defined by equations (1)-(7) in problem 1, setting t=0, and: Eq.No. Equation Description (8) MPMPds= Equilibrium condition (9) MPMPs= Money supply (10) ⎟⎟⎠⎞⎜⎜⎝⎛++−+=PBMjhikYPMdμ Money demand (11) BuD = G - T 3.1 Analyze the implications of running a budget deficit for one period induced by increasing government spending, assuming the initial budget deficit is zero. Use an IS/LM diagram, clearly indicating what you think happens (curve shifts, etc.). LM|μ,, PM IS| AAΔ+ i0 Y0 Y bA / YZ PMhh1−μ Y5Where )/( PBGO Δ=Δ 3.2 Assume the initial budget deficit is very large. Show what an increase in government spending does. Now )/( PBGO Δ<Δ The LM curve shifts up so much that the resulting level of output is lower than initially. LM| PBM ,, IS|A’ i0 i Y0Y Y”0 i’0 IS|A $αΔGO αΔGO Y’0 LM|PBBM,,Δ+i1 Y1 LM| PBM ,, IS|A’ i0 i Y0 Y Y”0 i’0 IS|A $αΔGO αΔGO Y’0 LM|PBBM,,Δ+i1 Y1 GOhbjΔ⎟⎠⎞⎜⎝⎛−1ˆα3.3. Can monetary policy change the result obtained in 3.2? If so, show how, using an IS-LM diagram. Expansionary monetary policy could keep the interest rate constant. In that case, one would obtain the result from Keynesian cross. 4. Leverage, liquidity, and bank balance sheets 4.1 Consider two banks, H (high bank capital) and L (low bank capital). High Bank Capital Low Bank Capital Assets Liabilities Assets Liabilities Reserves $9M Deposits $90M Reserves $10M Deposits $96M Loans $71M Bank Capital $10M Loans $70M Bank Capital $4M ABS $20M ABS $20M Bank capital is the equity of the owners (shareholders) of the bank. ABS stands for asset backed securities. Calculate the return on equity (ROE) for each bank, if the rate of return on loans is 5%, and 10% on ABS, and the interest rate on deposits is 2%. For H, [(71 × 0.05 + 20 × 0.10) – (90 × 0.02)]/10 = 0.375 For L, [(70 × 0.05 + 20 × 0.10) – (96 × 0.02)]/4 = 0.895 LM| PBM ,, IS|A’ i0 i Y0Y Y”0 i’0 IS|A $αΔGO αΔGO Y’0 LM|PBBM,,Δ+i1 Y1 LM|PBBMM,,Δ+Δ+4.2 Show what happens to each of the bank balance sheets when the asset backed securities lose 25% of their value. (20 × .75) = 15, so liabilities