ECN6520 Macroeconomic Analysis 2020/21
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ECN6520
Macroeconomic Analysis
Autumn Semester 2020/21
1 An infinitely-lived household’s lifetime utility can be expressed as follows:
U0 = t
β t o 
− γ 
、
where ct is consumption and nt are hours worked . σ and η are parameters deter- mining the curvature of the utility function . The parameter determining the dis- utility of labour is γ . The representative household produces goods, yt , using a Cobb-Douglas technology in hours worked and the capital stock:
yt = at kt(a)n t(1)_a
The parameter α denotes the share of capital and lies between 0 and 1 . Total factor productivity (TFP), at evolves as follows: ln at = ρ ln at _ 1 + et where et is an iid shock with a zero mean . The capital stock, kt is pre-determined in period t and evolves as follows:
kt+1 = yt − ct + (1 − δ)kt
The firm belongs to the household, which chooses consumption, hours as well as next period’s capital stock .
(a) Set up the Lagrangian function and derive the first-order conditions for ct , nt and kt+1 . (20 marks)
(b) Derive the so called Euler equation and show how the value of σ determines the volatility of consumption . (20 marks)
(c) Use the first-order conditions for ct and nt to analyse the effect of an exogenous increase in consumption (assume that household wealth has risen) on the supply of labour . (20 marks)
(d) In your own words, explain how TFP shocks (shocks to at ) can lead to business cycles . (40 marks)
2 A firm, existing for two periods, produces output using capital and labour . The firm’s production at any time i can be described by a simple production function zi ki(a)n 
The firm faces the following profit function expressed in real terms:
z1 k1(a)n1(1)_a +k1 − k2 +(s1 +d1 )a0 − s1 a1 − w1 n1 + 
ai , si and di denote the quantity and price of shares as well as dividend payments of shares held by the firm in period i in real terms . n and k denote labour input and capital stock, respectively. The firm has to borrow to invest in new capital stock . The amount it can borrow is constrained by the value of its stock holdings:
(k2 − k1 ) = Rs1 a1
where R < 1 is a parameter that limits investment spending to a fraction of the firm’s stock holdings .
(a) Set up the constrained optimisation problem and derive the first-order con- ditions for n1 , n2 , k2 and a1 . (20 marks)
(b) Use the first-order conditions to derive the firm’s demand for capital sched- ule . (30 marks)
(c) Carefully explain the effects of an unexpected increase in R on the capital stock, the share price and on output . (50 marks)
3 A country’s consolidated government budget constraint in period t is defined as: Pt gt + Bt _ 1 = Tt + Ptb Bt + Mt − Mt _ 1
Where g is government spending, B denotes bonds held by the private sector, Pb is the price of bonds, T denotes taxation and M the money supply. The consumer price index is denoted by P .
(a) Use the consolidated government budget constraint to derive an intertem- poral government budget constraint . (20 marks)
(b) In your own words, explain why a non-Ricardian fiscal expansion will always end in inflation . (40 marks)
(c) In your own words, explain the mechanism behind the fiscal theory of the price level . (40 marks)
4 Assume an economy where households receive utility from consumption of goods and disutility from hours worked . Households maximise expected utility:
U0 = t
β t ln oct − 
、
where the parameter ω > 1 . Households maximise expected utility subject to an infinite sequence of flow budget constraints:
o o
β t (Pt wt nt + (St + Dt )at _ 1 + Mt _ 1 + (1 + it _ 1 )Bt _ 1 ) = β t (Pt ct + St at + Mt + Bt )
t=0 t=0
In addition, households face the following cash-in-advance constraint in every period:
o o
β t Mt = β t Pt ct
t=0 t=0
Pt ct is nominal consumption, (1 + it ) is the nominal interest rate on bonds bonds, Bt is the level of bond holdings, St is the share price, at the stock of shares purchased by the household in period t , Dt is a dividend payment and Mt is the money stock held by agents at the end of period t . wt nt denotes the household’s labour income .
(a) Set up the intertemporal Lagrangian and derive the first-order conditions with respect to ct , nt , at , Bt and Mt . (20 marks)
(b) Show how the presence of a cash-in-advance constraint distorts the consumption-labour decision . Explain why a nominal variable can affect the choice between two real variables . (40 marks)
(c) What happens to consumption and price of bonds if the cash-in-advance constraint becomes non-binding [when the multiplier on the CIA constraint becomes zero] . (40 marks)
2022-08-11