Intertemporal choices
minus 1. This has already been demonstrated in the early, in the long-run equilib
rium, all variables lie on the 45-degree line after the effects of shocks disappear in the intertemporal decision diagrams. The economy deviates from this long-run equi
librium path after hitting by shocks. Next, adding the aggregate supply curve (AS) with a positive slope. At the point where AS is across LRE, there is no intertem
poral substitution. Above the equilibrium, the real interest rate is greater than the time discount factor which means the cost of continuing leisure and consumption are more expensive now, so rational households increase their supply of labour today in exchange for working less tomorrow. Below the equilibrium, the result is converse.
Then, inserting the aggregate demand curve (AD) into the diagram. AD is simply the horizontal sum of demand for consumption (CD) and that for investment (NID).
The slope of AD is negative. In the long-run equilibrium, the intersection point of AD and AS sits accurately on the LRE (Figure 2.11).
2 .1 .5 T h e r e s p o n s e t o sh o c k s
In this part, we are going to show how this basic RBC model reacts to a one
time shock. Assume there is an unanticipated positive technology shock that raises production permanently.
T h e p e r i o d o f t h e s h o c k . In Figure 2.12, this positive technology shock increases output so aggregate supply ( A S \ ) shifts horizontally to the right to A S 2 . Given the same level of the real interest rate, capital and labour are more productive than before. The new long-run equilibrium of AS still lies on the LRE line at point B with an increased output level from Vj to Y 2 . For consumption, it always equals
The Theory and Evidence on Business Cycles 33
Figure 2.11: The long-run equilibrium
Y
income in the long-run equilibrium, since income has increased and if this point is the long-run equilibrium, then the CD line shifts the same distance as AS to the right and is parallel to the original CD curve (i.e. from C D X to C D 2 ) . C D 2 and A S 2
cross at point B on the LRE line. For investment, we are looking at the NID line which reveals the expectation of MPK in the next period. While firms observe that the shock has permanent effects on output, they predict that marginal productivity will be higher in the future. Therefore, the MPK shifts outward, in turn, this results a rise in desire for capital. Subsequently, to increase capital stocks, firms need to invest first, so the NID line moves from N I D x to N I D 2 . Now summing the CD and NID lines levelly to get the AD curve. It can be seen from the diagram that the new AD curve intersects the A S 2 curve at point C which is way above the long-run equilibrium. At the long-run equilibrium, the supply of output fails to meet the demand of output. Firms plan to produce output at Y 2 but demand for labour and investment requires output at Y 3 . In order to reduce the gap between the short
Figure 2.12: The period of the shock
supply and the excess demand, a rise in the real interest rate solves the problem by means of the intertemporal substitution mechanism. An increase in the interest rate results in the expansion of work time, this further gives rise to a rise in labour supply and output. Meanwhile, this rising interest rate reduces the incentive of consumption today by saving more so as to enjoy more consumption tomorrow. Thus, moving along the C D 2 line up from point B. Moreover, firms’ demand for investment also falls since the higher interest rate makes investment more expensive. Hence, the combination effects of intertemporal substitution of labour and consumption drive down aggregate demand along the AD curve to point C where AD equals AS at the interest rate r2, and this produces output Y4.
T h e p e r i o d a f t e r t h e s h o c k . Although, the shock has only one-period effect and there are no new shocks in the next period, the point C is not the long-run equilib
rium, the movements of all these variables are still in progress. The driving force behind this is the assumption of ‘time-to-build’, which is introduced by Kydland and Prescott (1982). Due to lags in the investment progress, those goods that are invested in last period become effective capital, therefore, capital stocks raise with a time lag. This in turn raises production regardless technology shocks, and aggregate
The Theory and Evidence on Business Cycles 35
Figure 2.13: The period after the shock
supply moves further to the right ( A S 3 ) . Consumption demand also shifts to posi
tion C D 3 where it is across A S 3 at point D. This is due to same reason discussed above. The continuing increasing capital accompanied with decreasing in the MPK.
The N I D 2 line shifts backward slightly to N I D 3 . Adding consumption and net
investment together, the AD curve shifts to A D 3 and also produces a new equilib
rium at point E. At this point, income still increases (I5 > Y f ) but the interest rate falls to 73. Furthermore, since this point is still not the long-run equilibrium, the intertemporal substitution mechanism still works to reduce the amplitude of the deviation slowly. The above progress is demonstrated in Figure 2.13.
The progress carries on further ahead as capital stocks continue to increase. It follows that income still raises and the interest rate still falls, at a reduced amount, until all the deviant effects disappear and the economy finally arrives at a new long- run equilibrium at point F (Figure 2.14). At this point, the discount factor equals the reciprocal of the time discount factor (3 minus one once again; the net investment demand line shifts back to the original place ( N I D i ) where net investment is zero;
consumption is equal to income ( C D f ) \ and AD and AS intersect at point F which gives income Ve(> b ) .
Figure 2.14: The new long-run equilibrium
Figure 2.14 The new long-run equilibrium
The Theory and Evidence on Business Cycles 3 7
2 .1 .6 G e n e r a l s o lu tio n o f t h e b a s ic R B C m o d e l
Up to this point I have confined myself to discussing the intuition of the RBC theory using graphs. I now turn to a mathematical treatment in order to derive general solutions of the basic RBC model.
E f f i c i e n c y . To start with, I will derive the efficiency conditions for the maximi
sation problem. In each period £, the economic state is expressed by the technology
A t and capital K t, and households make decisions on consumption C t and leisure time (1 — L t ) (or labour supply Lt), once decisions of those variables are made the paths of other variables are determined. Thus, we need to find the optimal paths for { C T } j l T and {Lr}£2r that maximise utility. Additionally, the capital stock in period £ + 1 is determined jointly by C t and Lt, as stated in equations (2.2) and (2.5). So the household’s maximisation problem is
where the operator E ( - ) indicates the rational expectation of the argument condi
tioned upon information up to time £.
Nonetheless, the productivity shock A t is random and unpredictable, the house
hold’s optimisation problem faces uncertainty. Fortunately, this problem can be solved by using dynamic programming techniques1. This technique leads to the framework of the principle of optimality2 which declares that the choice of { CT, L r } f L t
are optimal only if the remaining choices of {CT, L r } £ L t , (primes denotes next period values, i.e. £' > £) also maximise expected utility over the remaining time horizon (£',£' + 1, ■ • •). Otherwise, it will be benefit to switch to other optimal paths for
1 Dynamic programming decomposes the optimisation problem which includes n variables into n stages where each stage comprises a subproblem with a single variable, in order to derive the optimal solution.
2 The principle of optimality states that future decisions for the remaining stages will constitute an optimal policy regardless of the policy adopted in previous stages.
O O
s . t . K t+1 = { l - S ) K t + A t F { K t , L t ) (2.9)
{CV, L r } f L t, which maximise utility over (£', t ' + 1, • • •). In short, the optimal paths have to be efficient.
To solve the problem, it calls for the Bellman equation which restates the above maximisation problem as the following equation:
A t ) ~ m a x { u ( C t) I — L t ) + f 3 E t [ V ( K t+ i : A t+ X ) ] } (2.10)
{Ct ,Lt }
subject to the same constraint as (2.9). Here, the function V denotes the value func
tion and represents the utility level. The level of intertemporal utility that can attain over the remaining period relies on the availability of capital stocks and the state of technology. Equation (2.10) describes that the optimal choice of (C t , L t) today maximises the sum of current utility u ( C t , 1 - L t ) and the discounted expectation of all future utilities / 3 E t [ V ( K t+ X , A t+ X ) } , where the latter is itself maximised.
Taking the first order condition (FOC) of the Bellman equation with respect to consumption C t and leisure 1 —Lt, and setting equal to zero, this yields the necessary optimality conditions
d V ( K u A t ) 8 u _ rd V ( K t+1 ) 9 K t+1
d C t d C t + 1 t l 9 K t+ 1 ' d C t 1
= f 3 E t { V ' ( K t + 1 , A t+
i.e.
^ ■ = / 3 E t l V ' ( K t + u A t+ 1 ) \ (2.11)
And
d V j K u A t ) m q x ,+1> A t + i ) _ d K t+ i
9 ( 1 - L t ) 9 ( 1 — L t ) 9 K t+1
The Theory and Evidence on Business Cycles 39
and second, the change in the expected value of intertemporal utility caused by the additional change in capital,
E t [ V ' ( K t+ i , A t + i ) \ .However, the value function
V ( K t , A t )does not have prior knowledge of its ex
plicit functional form, moreover, solutions of consumption, labour supply and capital to the maximisation problem are required to satisfy (2.9)-(2.13), concurrently, this makes solving the problem more difficult in general. The existence of a closed form solution to this problem is applicable only under specified restrictions on
u , Fand J. We will explain this later. Before that, it is essential to show that the market equilibrium satisfies the efficiency conditions, despite of the functional forms.
M a r k e t e q u i l i b r i u m .