Monetary Economics: Solutions Problem Set 1

Transcription

1 Monetary Economics: Solutions Problem Set 1 December 14, 2006 Exercise 1 A Households Households maximise their intertemporal utility function by optimally choosing consumption, savings, and the mix of their asset portfolio ( how much to hold in real money rather than in claims to capital ) in each period taking prices ( π t, w t, r t ) and the government transfer τ t as given. In doing so they have to satisfy their per-period budget constraint as well as a No Ponzi-Game Condition. The No Ponzi-Game condition restricts their borrowing. The net present value of households expenditure is not allowed to exceed households net present value of their income stream. The household s maximisation problem can be expressed by a Lagrangian L = t=0 { (c γ β t t m 1 γ t ) 1 ε 1 + λ t [(1 + r t )a t + w t c t (r t + π t )m t + τ t a t+1 ] 1 ε The first order conditions are c t = 0 : m t = 0 : c γ 1 t m 1 γ t (c γ t m 1 γ t c γ t m γ t ) ε = λ t (c γ t m 1 γ t ) = (r ε t + π t )λ t = 0 : λ t = β(1 + r t+1 )λ t+1 a t+1 The No Ponzi-Game condition implies T βt 1 λ T a T 0 1 }

2 You can see from the Euler Equation that this is the same as equation (2) in the script. Solving the Euler Equation forward you get λ T +1 = λ 0 β T T t=0 (1 + r t) = λ 0 β T R T So R 1 T a T +1 = (β T λ T +1 a T +1 )/λ 0. But λ 0 is just some constant, so condition (2) of the script is the same as my condition above. Again, as the households have no interest in holding valuable assets at the end of their life ( at T ), the condition will hold with equality T βt λ T +1 a T +1 = 0 The above condition together with the three first order conditions and the budget constraint characterize optimal behaviour of the households. Combining the first order condition for consumption with the first order condition for money holdings, we get the household s demand for money as a function of consumption and the opportunity cost of holding money, the nominal interest rate r t + π t : ( ) ( ) 1 γ ct m t = γ r t + π t To summarise, optimal households behaviour is characterised by the following four equations ( ) ( ) 1 γ ct m t = (1) γ r t + π t λ t = β(1 + r t+1 )λ t+1 (2) B Firms T βt λ T +1 a T +1 = 0 (3) a t+1 = (1 + r t )a t + w t c t (r t + π t )m t + τ t (4) Firms operate in a competitive market. They maximise profits by choosing labour and capital input optimally, taking prices as given. Firms profits are given by 1 π t = L[k α t (r t + δ)k t w t ] 1 Note that this is equivalent to π = K α L 1 α (r + δ)k wl. Rather than choosing K and L, the firm can choose the optimal factor ratio k and then the scale of its operation L. This is so because the production function has constant returns to scale in K and L jointly. 2

3 The optimal factor ratio k t is given by the first order condition r t = αk α 1 t δ (5) which gives the demand for k t as a function of the price of capital r t. To get an expression for wages we can make the following argument: The total stock of labour is fixed ( each household provides a constant amount of labour in each period irrespective of the wage rate w t and there is a fixed number of households). If w t is such that k α t (r t + δ)k t w t > 0 each firm demands L = as this demand maximizes profits. However such a wage rate w t will not clear the market as firm demand will exceed labour supply. If w t is such that k α t (r t + δ)k t w t < 0 firms labour demand will be L = 0 for else they will make losses. Thus the only market clearing wage w t will be such that k α t (r t + δ)k t w t = 0, i.e. C Government w t = k α t (r t + δ)k t (6) The government s monetary policy is such that the nominal stock of money M t grows at a constant rate θ. We define this growth rate as M t+1 M t M t+1 Divide this expression by P t+1 to get = θ M t+1 P t+1 M t P t P t P t+1 = θ M t+1 P t+1 By the definition of π t = (P t+1 P t )/P t+1 this yields the evolution of the real money stock m t as a function of the growth rate of nominal money and the inflation rate ( ) 1 πt m t+1 = m t (7) 1 θ We assume that the government runs a balanced budget in each period. This implies that all its revenue from expanding the nominal stock of money ( this revenue is called seignorage ) is redistributed to the households via the lump sum transfer τ t P t τ t = M t M t 1 τ t = M t P t M t 1 P t 1 P t 1 P t = m t (1 π t 1 )m t 1 3

4 which together with equation (7) implies D Equilibrium τ t = θm t (8) In a competitive equilibrium, for a given sequence of equilibrium prices (r t, w t, 1, π t ) t=0, the equilibrium allocation (k t, h, c t, m t ) t=0 has to clear all markets. There are four markets in this model: the consumption goods market, the labour market, the capital market, and the money market. By Walras Law we know that if three markets clear, so will the fourth market. Therefore I will ignore the labour market in what follows. Consumption goods market clearing implies that whatever is produces has to be either consumed or invested. Thus, the resource constraint imposes consumption goods market clearing k α t + (1 δ)k t = c t + k t+1 Households capital supply is implicitly characterised by the Euler Equation. It describes the optimal savings behaviour for the household and therefore how much capital will be provided by households to the market. Firms capital demand is given by the firm s first order condition with respect to k t. The equilibrium interest rate has to be such that the Euler Equation and the first order condition hold jointly. This is the case if λ t = β(1 + αk α 1 t+1 δ)λ t+1 Money supply is characterized by equation (7), households money demand is given by (1). The equilibrium inflation rate and the equilibrium interest rate have to be such that demand equals supply. This is the case if ( ) [ ( ) ( ) 1 1 γ ct m t+1 = 1 1 θ γ m t αk α 1 t ] δ Notice that as λ t is a function of m t and c t, the above three equations are three first order difference equations in three endogenous variables c t, k t and m t. Together with the transversality condition T βt λ T +1 (m T +1 + k T +1 ) = 0 m t 4

5 and an initial value for the capital stock k 0 they describe the dynamic evolution of the economy. 2 For our purposes it will prove more convenient to work with money supply and demand as well as capital demand and supply separately, so the core equilibrium equations we are interested in are 3 λ t = β(1 + r t+1 )λ t+1 money demand and capital demand c t = kt α + (1 δ)k t k t+1 ( ) 1 πt m t+1 = m t 1 θ ( ) ( ) 1 γ ct m t = γ r t + π t r t = αk α 1 t δ E Steady State In a steady state, the endogenous variables c t, m t, k t and consequently λ t are constant. We will call this steady state (c, m, k, λ ). From the Euler Equation we then get that r = 1 β 1 But this directly pins downs the steady state capital stock using firms capital demand ( ) 1 k αβ 1 α = 1 β + δβ 2 For the technically interested: To fully pin down the evolution of k, m, and c we are still missing one initial or terminal condition in addition to k 0 and the transversality condition. We will know the initial value of the nominal money stock M 0, but we won t know m 0, which is determined by the initial price level P 0. But P 0 is not predetermined. The equilibrium of the above dynamic system will change with P 0. This is the source of multiple equilibria in this model as we will see in exercise 2. 3 You might have noticed that I have not used the expression for wages w t,the balanced budget condition for the government, and the household s resource constraint yet. These three equations are redundant by Walras Law. To check this yourself, plug them into the household s per period budget constraint. Using the money supply equation you will see that you end up with the economy s resource constraint, a condition we have already stated. 5

6 Once we know the steady state capital stock, we know steady state consumption from the resource constraint c = (k ) α δk From money supply, steady state inflation is given by π = θ Therefore steady state real money balances can be derived from money demand as ( ) ( ) 1 γ c m = γ r + θ Notice that neither k nor c depend on the growth rate of money θ or any other monetary variable. In steady state, the marginal product of capital is uniquely determined by the subjective discount rate of households and depreciation. As the marginal product of capital only depends on k, this condition pins down the capital stock. But this via the resource constraint directly determines steady state consumption. We can conclude that superneutrality holds in this model. Increases in θ only drive up the steady state rate of inflation. As r does not change with θ this means that the opportunity cost of holding money goes up. c being constant means that households will decrease steady state real money balances. But this will decrease steady state utility. Exercise 2 If the marginal utility of money only depends on money itself and is linear in m, this implies that the utility function is additively separable in consumption and money and quadratic in money, i.e. u(c, m) = g(c) + Am Bm 2 g( ) is some unspecified concave function. We assume that the real side of the economy is in its steady state, that is c t = c and k t = k. Remember from exercise 1 that the standard Sidrauski Brock model is superneutral. This means that the steady states of c and k can be determined independently of the equilibrium in the money market. Therefore in this model we can actually consider a situation where c and k are in steady state ( and thus constant ), and analyze the equilibrium evolution 6

7 of m t separately. Money supply is given by m t+1 = ( ) 1 πt m t 1 θ Money demand is u m (c t, m t ) u c (c t, m t ) = r t + π t Given the above utility function and the assumption that the real side is in steady state this yields A Bm t g (c ) = 1 β 1 + π t In a competitive equilibrium the inflation rate has to be such that money demand equals money supply. We can solve the above money demand for π t and substitute this into supply in order to impose equilibrium. Then we have m t+1 = 1 1 θ ( 1 β A Bm t g (c ) ) m t To simplify notation, we can rewrite the above equation as where ν = m t+1 = νm 2 t γm t ( ) ( ) [ 1 B θ g (c ) and γ = 1 θ β A ] g (c ) γ is positive because in a steady state for the money market 1 β A Bm g (c ) = 0 Bm = A g (c ) β > 0 1 β < A g (c ) We can now write our equilibrium condition for the money market as m t+1 m t = νm 2 t (1 + γ)m t (9) This will yield a graph like the one depicted in the lecture notes ( page 5 ). The change in real money is constant at two points, m = (1 + γ)/ν, the steady state, and m t+1 = m t = 0. But this tells you that there can 7

8 be inflationary bubbles in this economy ( provided that the transversality condition is satisfied. We ll check this in the end. ). For any initial value of m 0 such that m 0 < m, we know from (9) that the real money stock will decrease in every period and will eventually converge to zero. Let s see what happens to the price level. The nominal stock of money will still grow at rate θ, however the real stock of money will monotonously go to zero. But this means that the inflation rate has to be higher than the growth rate of nominal money. Therefore π t > θ > 0 for all t and the price level P t will eventually converge to infinity. This means that money becomes worthless, a unit of the consumption good will be worth infinitely many units of nominal money. Lastly, to make sure that this indeed is an equilibrium, we need to make sure that the transversality condition is satisfied. It says T βt λ T +1 a T +1 = 0 From the first order condition for consumption we know that if the real side is in steady state u c (c, m T +1 ) = g (c ) = λ T +1, so λ T +1 is constant. Furthermore in steady state a T +1 = k + m T +1 and m T So the transversality condition is indeed satisfied as β < 1. Exercise 3 A Households T βt g (c )(k + m T +1 ) = 0 As in exercise 1, only that now households also have to decide how many hours to work. The Lagrangian for this problem is L = β t {V (φ(c) + ψ(h, m))+ t=0 λ t [(1 + r t )a t + w t h t c t (r t + π t )m t + τ t a t+1 ]} 8

9 We have four first order conditions = 0 : φ (c t )V ( ) = λ t c t = 0 : ψ m (h t, m t )V ( ) = (r t + π t )λ t m t = 0 : ψ h (h t, m t )V ( ) = w t λ t h t = 0 : λ t = β(1 + r t+1 )λ t+1 a t+1 The transversality condition is as in exercise 1 T βt λ T +1 a T +1 = 0 Combining the first two first order conditions we get an expression for money demand ψ m (h t, m t ) = (r t + π t )φ (c t ) (10) and combining the first and the third first order condition we get the labour supply for households ψ h (h t, m t ) = w t φ (c t ) (11) To summarise: Optimal household behaviour is characterised by the following five equations ψ m (h t, m t ) = (r t + π t )φ (c t ) (Money Demand) ψ h (h t, m t ) = w t φ (c t ) (Labour Supply) λ t = β(1 + r t+1 )λ t+1 (Euler Equation) a t+1 = (1 + r t )a t + w t h t c t (r t + π t )m t + τ t B Firms T βt λ T +1 a T +1 = (Budget Constraint) (Transversality Condition) Suppose there a L households in the economy. Suppose furthermore that output is produced with a constant returns to scale production function using labour and capital F (K, Lh). Let us first transform this production function into intensive form, i.e. output per hour worked 1 F (K, Lh) = F Lh ( K Lh, 1 9 ) f ( ) k h

10 So profits for a firm that employs h t hours of labour and has a capital-labour ratio k t are ( ) kt π t = h t f w t h t (r t + k t )k t h t Optimal factor demands as a function of prices are given by the first order conditions ( ) r t = f kt δ (12) h t and C Government w t = f ( kt h t ) ( ) f kt kt (13) h t h t As in exercise 1: m t+1 = ( ) 1 πt m t (Money Supply) (14) 1 θ τ t = θm t (Balanced Budget) (15) D Steady State In steady state all the endogenous variables c t, h t, k t, m t and consequently λ t are constant. Then by the Euler Equation r = 1 β 1 From the capital demand equation we then get ( ) k f = 1 h β 1 + δ You see that unlike in exercise 1, this condition does not directly pin down the steady state capital stock, but only the steady state ratio k /h. Divide the resource constraint by h to see that the same is true for steady state consumption ( ) c k h = f δ k h h 10

11 Only the ratio c /h is pinned down as a function of subjective discount rate and depreciation rate. Form the money supply equation we once again have π = θ Now let us consider money demand and labour supply in steady state. Under the functional form ψ(h, m) = [(H h) γ + m γ ] 1/γ money demand is [( H h m ) γ + 1] 1 γ γ = φ (c )(r + θ) and labour supply is [( m H h Divide these two equations to get ) γ + 1] 1 γ γ = φ (c )w ( ) m 1 w H h = 1 γ r + θ Both w and r are functions of k /h only, which does not change with changes in θ. Therefore the ratio m /(H h ) decreases with increases in the growth rate of nominal money θ. Now consider labour supply [( m H h ) γ + 1] 1 γ γ = φ (c )w The left hand side of this equation is a decreasing function in θ. The right hand side decreases in c. Thus ifθ increases, c has to go up. This establishes that increases in the growth rate of nominal money increase steady state consumption. Now what about k and h? Consider the resource constraint c h = f ( ) k δ k h We know that if θ increases, so does c. But the right hand side of the resource constraint is uniquely determined by β and δ, so it can t change with 11 h

12 θ. Therefore h has to increase with θ. But this also means that k increase with θ as the ratio k /h has to stay constant. Finally let us determine the behaviour of m. Notice that we can express m as m H h = (H h )/m We know that the numerator decrease with θ, and the denominator increases with θ. It follows that m decreases with θ. Intuitively what happens is that higher money growth drives down the marginal disutility of working. Hence households are willing to supply more labour as the marginal utility of consumption stays unaffected ( this is because φ and ψ are additively separable ). So it must be the case that even though the leisure to real money holdings ratio increases in θ, the absolute amount of leisure decreases in θ. 12