Equilibrium Analysis of a Two-Period Economy with Uncertainty, Study notes of Introduction to Macroeconomics

Download Equilibrium Analysis of a Two-Period Economy with Uncertainty and more Study notes Introduction to Macroeconomics in PDF only on Docsity! 213 O VERLAPPING G ENERATIONS B USINESS C YCLE M ODEL • To discuss equilibrium business cycle models we need to introduce a labor supply decision by households and we will compensate by simplifying the consumption side of the model. Let pref- erences be U(c 0 ,n) + V(c 1 ). Throughout this discussion we shall consider only stationary solu- tions to the relevant first order conditions although the above discussion of non-stationary solutions can be readily extended to the models discussed in this section of the notes. With labor supply in the first period, the demand for cash at the end of the first period is = p 0 (n-c 0 ) and this is spent on goods next period = p 1 c 1 . 1. Example 1 - Constant Money Supply • As above, assume money balances are constant per capita at m. The stationary solution will be the constant price one. We get equilibrium c 0 , c 1 , n as solutions to (1) subject to c 1 = n - c 0 (2) Again, p will be given by m = p c 1 . 2. Example 2 - Transfers to the Old Generation • Now assume the quantity of m is changed via transfers to the old generation. Then balances next period are given by x. Let the transfer x be selected randomly from some known distribution. Each period looks like any other except the quantity of money has changed. The state of the sys- tem is described completely by the quantity of money. The strategy is to seek solutions p(m), c 0 (m), c 1 (m), n(m). m̃ m̃ U c0 n,( ) V c1( )+{ } c0 c1 n, , max m̃ 214 • The current price is assumed to be known by traders when they trade. Use p' for tomorrow's price. Traders do not know x and p' but are assumed to know their probability distributions. As- sume traders maximize expected utility (3) subject to = p(n - c 0 ) (4) • Since each person is getting a transfer proportional to his holdings there is no inflation tax (the extra cash received by each individual matches the decrease in the value of money) so intuition suggests c 1 will be unchanged and p will fluctuate with m. Thus we conjecture that and (5) where c 1 is the solution in example 1. • Assume rational expectations so that if these are the solutions for p and p' then we can also use them to model consumer expectations about the determinants of p and p'. In particular, the dis- tribution of p' can be obtained from (6) • Then the problem becomes (7) Ex p', n c0 m̃, , max U c0 n,( ) V m̃x p' -------   +       m̃ p m c1 ----= p' mx c1 -------= Pr p' p≤( ) Pr mx c1 ------- p≤   = E n c0 m̃, , max U c0 n,( ) V m̃x mx -------c1   +       217 • Solve the conditional maximization problem for c = c( ) with c ′ (.) > 0 (15) n = n( ) with n ′ (.) < 0 (16) • Now eliminate c and n from the problem to get a function of . Define f( ) ≡ U[c( ), n( )] (17) The young individual thus chooses to (18) Differentiate (18) to get FONC for a max c n Assume neither leisure nor consumption are inferior goods mz m̃– p ----------------- mz m̃– p ----------------- m̃ mz m̃– p ----------------- mz m̃– p ----------------- mz m̃– p ----------------- m̃ f mz m̃– p -----------------    E V m̃ p' ----   +       m̃ max 218 (19) • Now impose the equilibrium conditions. In equilibrium , money demand = money supply so m(1+z) = (20) Substitute the market clearing condition (20) into (19) to get (21) Equation (21) is then a market equilibrium condition giving current price as a function of expect- ed future price. • The state variables are m and z so we guess a solution p = p(m,z). We further know that m is the known historical quantity of money. If this is doubled we'd expect all future prices to double. Hence we guess p(m,z) = m φ (z). Substitute our guess into the equilibrium condition (22) where we have assumed expectations are rational so that if p(m,z) = m φ (z) is the equilibrium pricing function then p' = m' φ (z') = m(1+z) φ (z') will model individual expectations. • Since m is known to the young, m can be taken through the expectations operator and we can rewrite (22) (23) Also 1+z is known to the young this period and therefore also is a constant with respect to the 0 f′ mz m̃– p -----------------    1 p --–    E V′ m̃ p' ----    1 p' ---+= m̃ 1 p --f′ m p ----–    E V′ m 1 z+( ) p' ---------------------    1 p' ---= 1 mφ z( ) ---------------f′ 1 φ z( ) ----------–    E V′ m 1 z+( ) m 1 z+( )φ z'( ) ---------------------------------    1 m 1 z+( )φ z'( ) ---------------------------------= 1 φ z( ) ----------f′ 1 φ z( ) ----------–    E V′ 1φ z'( ) -----------    1 1 z+( )φ z'( ) ----------------------------= 219 distribution of next period prices. Hence, it too can be taken through the expectations operator in (23) to give , (24) for some constant A. Solve the equation consisting of the left-most and right-most parts of (24) for φ = φ(z,A).2 Then insert φ (z,A) into the expression for A on the right hand side of (24) to get a non-linear equation to be solved for A using the functions φ and V and the distribution for z': (25) Then substitute A into φ = φ(z,A) to get φ. So our guess was correct and we indeed have a solution of the form p = mφ(z). • Now look at the behavior of the solution with respect to shocks z. Differentiate (24) with respect to z: (26) Simplify (26) to get: (27) from which we can conclude 2.Note that this solution will depend on the known functional form for f() (which in turn depends on the orig- inal utility function). 1 z+ φ z( ) -----------f′ 1 φ z( ) ----------–    E V′ 1φ z'( ) -----------    1 φ z'( ) ----------- A≡= A E V′ 1φ z' A,( ) ------------------    1 φ z' A,( ) ------------------= 1 φ -- f′ 1 φ --–    1 z+ φ2 -----------φ′f′ 1 φ --–   – 1 z+φ -----------f″ 1 φ --–    1 φ2 -----φ'+ 0= 0 f′ 1 z+φ ----------- f′ f″ φ ----– φ′–= 222 • Solve the constrained maximization (38) and (39) for c = c( /p) and n = n( /p)) and define f( /p)) ≡ U[c( /p)), n( /p))]. Individuals then choose to (40) The FONC for a max is: (41) • In equilibrium, money demand = money supply so or (42) • Define F(x',p'|m,p) as the conditional distribution of (x',p') given m and p. Also define the func- tion h ≡ −f′. Then from (41) and (42) we get (43) • The state variables here are m, x and θ. Individuals know m but don't know x or θ. Current price movements will give some noisy information on x and θ. • Solution strategy: Guess a solution p = p(m,x,θ) and solve for it. Lucas shows that any solution to (43) is in fact monotonic in x/θ. Then as in example 3 he guesses p(m,x,θ) = mφ(x/θ). He proves there is a unique continuous solution φ(z) to(43) with z/φ(z) bounded, φ strictly positive and continuously differentiable. • Lucas then examines two polar cases: (i) θ = 1 with probability one so trading takes place in a single market and no non-monetary dis- m̃ m̃ m̃ m̃ m̃ m̃ f m̃ p ----    E V m̃x' p' --------   +       m̃ max 0 f′ m̃ p ----    1 p -- E V′ m̃x' p' --------    x' p' ---+= θm̃ mx= m̃ mx θ -------= h mx θp -------    1 p -- V′ mxx'θp' -----------    x' p' --- F x' p' m p,,( )d∫= 223 turbances are present. (Lucas has two markets with fraction θ/2, θ ∈ (0,2), of the young allocated to one market and (1-θ/2) of the young allocated to the other market.) Then p = mx/y* where y* is the unique solution to h(y) = V′(y). This is the classical neutrality of money result. (ii) x = 1 with probability one so the only disturbance is exclusively real. The money supply is fixed. The disturbances have real consequences. Those of the young who find themselves in a market with few of their cohorts (a low θ) obtain, in effect, a lower price of future consumption. They attempt to distribute this gain to the future by holding higher cash balances. This attempt is partially frustrated by a rise in the current price level (raising current expenditure). The solu- tion here is p = mφ(1/θ) where φ is continuously differentiable with an elasticity between zero and one. • Returning to the general case, Lucas notes that the current price informs agents only of x/θ. Lu- cas did not require lnθt ~ N(0,τ2) and lnxt ~ N(0,σ2) as I have written above (and will use short- ly). He did, however, impose restrictions on the distributions of x and θ. In particular he required3 Pr {θ ≤  f(x,θ) = z} is an increasing function of z Pr {x ≤  f(x,θ) = z} is an decreasing function of z With these restrictions on the distributions of x and θ, Lucas shows that the equilibrium price is p(m,x,θ) = mφ(x/θ) where φ is continuously differentiable with an elasticity between zero and one. The monetary changes x have real consequences because agents cannot discriminate perfectly between real and monetary demand shifts. 3.Also, he assumed θ has a continuous symmetric density on (0,2) and, as we noted above, in Lucas' paper there are two markets with θ/2 of young allocated to one market and 1- θ/2 allocated to the other. θ x 224 5. A parametric example • Take lnθt ~ N(0,τ2), ln xt ~ N(0,σ2) and assume U(c, n) = c1+β − kn1+β (44) with − 1 < β < 0; and k1/(1+β) > c/n over the relevant range of c and n so that U is concave and (45) with -β < α < 1. For the U function (44), the solution to (46) subject to n - c = ; n > 0, c > 0 (47) is and (48) so that we may write (with H = k(1+β)(1−k1/β)−β) h(y) = Hyβ (49) for a constant H and - 1 < β < 0. • Guess a solution p(m,x,θ) = mA(x/θ)B for constants A and B. If this guess is correct, the equi- librium condition (43) becomes V y( ) y 1 α– 1 α– ------------= U c n,( ) c n, max m̃ p ---- c k1 β⁄ 1 k1 β⁄– ------------------- m̃ p ----= n 1 1 k1 β⁄– ------------------- m̃ p ----= 227 (61) • Now use the fact4 that if ln y ~ N(µ,σ2) then E y = γeµ where γ = . Then (62) where recall the parameter a is given by the variance ratio (60). Substitute (62) into (57) to con- clude that, for the constant B given by , (63) the equilibrium price is indeed of the postulated form: (64) • Now consider some special cases: (i) Suppose τ2 = 0 and price movements result only from monetary disturbances. Then, from (60) and (63), a = 1 and B = 1. Prices increase in proportion to the monetary shock x. (ii) Suppose σ2 = 0 so there is a smooth monetary policy but real shocks. Then a = 0 and B = (α+β)/(1+β) ∈ (0,1). Price responds to the real shock but less than proportionately because of substitution effects. A low θ is a good event and that causes you to work harder today and consume more tomorrow. (iii) If neither σ2 = 0 nor τ2 = 0 individuals respond in part as if there is a real shock and in part 4.Check a probability or statistics text! E ξα 1–( ) m p,ln[ ] α 1–( ) σ 2 σ2 τ2+ ----------------- x θ --   ln= e 1 2 --σ2 E ξα 1– m p,[ ] K2 x θ --    α 1–( )a= B α β+ 1 β+ ------------ 1 α– 1 β+ ------------a+= p K1 mx θ -------    α β+ 1 β+ ------------ mA( ) 1 α– 1 β+ ------------ K3 x θ --    α 1–( )a 1 β+ --------------------– K4m α β+ 1 β+ ------------ 1 α– 1 β+ ------------+    x θ --    α β+ 1 β+ ------------ 1 α– 1 β+ ------------a+    K4m x θ --    B = = = 228 as if there is a monetary shock. 6. A Phillips Curve • For the special utility function considered above we have (65) where π0 is given by − ln(1 − k 1/β). Substitute the solution (64) for pt (66) into (65) to find . (67) • Also, from (66) (68) • Now average (67) and (68) over markets and use E (ln θ) = 0 to obtain expressions for aggregate employment and average nominal prices given by ln Nt = K5 + (1-B) ln xt (69) ln Pt = Κ6 + 1n mt + B ln xt (70) The aggregate price level follows a random walk as does mt. Aggregate employment fluctuates around a constant level and prices and employment are perfectly correlated. nt θ x,( )( )ln π0 mtxt θtpt ----------   ln+= pt K4mt xt θt ---    B= nt θ x,( )( )ln K5 1 B–( ) xt θt ---   ln+= pt θ x,( )( )ln K6 mt B xt θt ---   ln+ln+= 229 • If the distribution of x is changed to N(0,6σ2) without people knowing that, a big x will be mis- taken for a low θ and employment will be increased. However, people will catch on. They know mt with a one-period lag and hence have a history on xt. There is no long-run trade-off between employment and inflation even though (69) would indicate that such a trade-off exists. • It has been observed that in the model Lucas discussed in “Expectations and the Neutrality of Money” consumption moves counter-cyclically. When a large positive money shock is mistaken for a low θ, current output by the young is higher, but current prices are also higher and current real consumption lower. This resulted from the special structure of preferences Lucas assumed to make the model easier to solve. If we use U(c0, c1, n0, n1) for preferences we can obtain a pos- itive correlation between c0, n0 and the first period price p0. To do so we want first period con- sumption to be more responsive to expected discounted (permanent) income than to anticipated real interest rate movements.