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foundations of microeconomics

Questions and Answers of Foundations Of Microeconomics

Also, Young's Theorem ensures that FNK FKN. Armed with these useful properties equations (4.29) and (4.30) can be derived. First, totally differentiate FN (N, K):dFN FNN dN + FNK dK.But (P3) ensures
Some production theory. If Y = F(N, K) is a linear homogeneous production function, it possesses several very useful properties (see e.g. Ferguson, 1969, pp. 94-96):(P1) FNN FKK = Y (Euler's
Since the economy is operating under perfect competition, the production function is linear homogeneous (constant returns to scale), and FN and FK depend only on KIN. The expressions for FN and FK
As a final application of the model, we now consider the general case where the model is interpreted at a macroeconomic level, and equations (4.13)-(4.15) are appended with a labour supply equation
Using all this information, the adjustment path is easily seen to consist of a jump from E0 to A at time tA , gradual adjustment from A to B between tA and tE , followed by gradual adjustment from B
The temporary subsidy is announced and introduced in tA = ti and simultaneously announced to be abolished again at some fixed time in the future tE (> tA of course). Our heuristic solution principle
Suppose that the policy maker wishes to stimulate the economy and has decided to do so by creating investment incentives in the form of an investment subsidy. If the policy maker desires the maximum
Temporary or permanent investment subsidy?
The striking (though intuitive) conclusion is that investment goes up initially!Firms in this economy rush to put in their investment orders in order to be able to get the subsidy while it still
E0 to A directly above it (recall that K is fixed in the short run). Between tA and tj the economy moves in a north-easterly direction towards point B, where it arrives at tj. After that, there is
-middy is one wa)K1 94 Chapter 4: Anticipation Effects and Economic Policy
idy we z all this to A at t.TN -adual adi■, -,..en in the Lou Eo to A- on k..Qmpar, ilme able of q fa,
su!
our simple me' tt •lp S of course). Our het _ t maiming must he q
After that, thc le striking '30) tiAlS Ott subsidy while it rq-,ure 4.7. The c-e 7v KFigure 4.6. Investment with full employment in the labour market Figure 4.7. An anticipated abolition of the
The ultimate effect of the abolition of the subsidy is to increase the relative price of investment goods and to shift the K 0 line up and to the left. In the long run the economy ends up at point E
As a first policy measure, consider an anticipated abolition of the investment subsidy, as was for example the case in the Netherlands in the late 1980s. Using the solution principle introduced
In Figure 4.6, the saddle path is derived graphically. The dynamic forces are much more complicated in this case. This is because the steady-state level of q and the q-dynamics itself are now both
Intuitively, steady-state q is downward sloping in K because the more capital is used, the lower is its marginal product. As a result, the discounted stream of marginal products (which is q) falls.
Assume furthermore that the economy is characterized by full employment in the labour market. By normalizing employment to unity (N = 1), the model consists of:K = I (q, si) – SK, (4.25)4= (r + 8)q
The effects of the investment subsidy with full employment in the labour market Up to now we have interpreted the model given in (4.13)-(4.14) as applying to a single firm facing a constant real
Figure 4.5. An anticipated permanent increase in the rate of interest requiring that all jumps occur when something truly unexpected occurs (which is at time tA). Obviously, at tA there is an
As a final example of how the model works, consider the case where the firm hears at time tA that interest rates will rise permanently at some future date tI. This is an example of a so-called
immediate financial correction, the adjustment proceeds smoothly along the saddle path towards the ultimate steady-state equilibrium point E1 .
Figure 4.4. An unanticipated permanent increase in the rate of interest the future) immediately leads to a revaluation of this stream of returns. After the
Clearly, the new equilibrium is at point E1 and the only path to this point is the saddle path going through it. Since K is fixed in the short run, the only stable adjustment path is the one with a
As a second "finger exercise" with the model, consider an unanticipated permanent increase in the exogenous rate of interest r as illustrated in Figure 4.4. Equation(4.22) shows that this shifts the
Figure 4.3. An unanticipated permanent F1160 increase in the investment subsidy very uneven investment plans very expensive. The adjustment over time has also been illustrated in Figure 4.3.
For future reference, the implementation date is denoted by t1 . Hence, an unanticipated shock is a shock for which announcement and implementation dates coincide, i.e. tA = tI. The effects of the
The particular type of stability that is exemplified by the model is called saddlepoint stability: there is exactly one stable adjustment path (called the saddle path)that re-establishes equilibrium
In conclusion, for each given initial level of the capital stock, there is exactly one path towards the steady-state equilibrium. And this is very fortunate indeed, because one would have an
Figure 4.2. Derivation of the saddle path
c -ANON- For It d s'i.e. = 1, .the aid of Fi 0 _ t 0 t r . _ the acilustinenz sadd L * *tin the vital stoci(4.21)88 K* K Chapter 4: Anticipation Effects and Economic Policy qq*
-,-s _ ,.mot ec no-...y be Amor ultimately, ai so 7 '...non. The rtatnail!mak- -.-e Wr It;he 1 means tha
The graphical interpretation is as follows. At point B the value of q is consistent with an equilibrium investment plan. Now take a slightly higher value of q, say the one associated with point B',
For points off the q = 0 line, the dynamic behaviour of q is also provided by (4.20):
This is intuitive: since both the rate of interest and the marginal product of capital are constant (and hence independent of K) , q itself is also constant and independent of K in the steady state.
The Foundation of Modern Macroeconomics where we have used the fact that the marginal product of capital is constant. From(4.20) it is clear that the q = 0 line is horizontal:= 0. ( Kg ) =o
The q = 0 line represents all points for which the firm's investment plans are in equilibrium. By differentiating (4.14) we obtain:(4.19)al wages!ct competition in the goods- 4; on function) renders
The graphical interpretation is as follows. In point A the capital stock is in equilibrium.If K is slightly higher (say at A' to the right of point A), (4.19) predicts that depreciation exceeds gross
The after-subsidy cost of investing falls and as a result firms are willing to invest the same amount for a lower value of q.For points off the K = 0 line, the dynamics of the capital stock is also
least three types of labour tl is interpreted at firm level 1 (and constant); the model d (ii) full employment of'ply equation is added to it e cases in turn.Figure 4.1. Investment with constant real
If K*q* = FOr +dq = (r + 8) dq + q dr, (4.20)87 Chapter 4: Anticipation Effects and Economic Policy(4.13)(4.14)(4.15)ally interesting variations ly, in view of (4.15), some'
4=o
A'
B'A" •k=o
In words, a higher capital stock necessitates a higher level of steady-state gross investment. This is only forthcoming if q is also higher.4.home 4.1. it I86
which implies that the slope of the k = 0 line is:aq aK )1 O. (4.17)
Since the production function is homogeneous of degree one (constant returns to scale), the marginal products of labour and capital are homogeneous of degree zero(see the Intermezzo). This implies
The effects of the investment subsidy under constant real wages If the real wage rate is constant, the assumption of perfect competition in the goods market (and the implied homogeneity of the
Despite its simplicity, the model allows several economically interesting variations to be considered within the same framework. Clearly, in view of (4.15), some assumption must be made about the
The model can now be used to investigate the immediate, transitional, and longrun effects of governmental efforts to stimulate investment. Omitting the (now almost superfluous) time index, the model
The nominal stockmarket value of the firm is P(0) V(0) and the nominal replacement value of its capital stock is Pi (0)K(0). As a result, Tobin's average q is P(0) V(0)/(PI (0)K(0)), which equals
where the term in square brackets on the left-hand side vanishes due to the transversality condition. The final expression of (f) shows that Tobin's average q (designated by 4) equals marginal
where we have used the linear homogeneity of F (i.e. F = FNN + FKK), and equation (a). By substituting (e) into (d) and integrating we obtain:T." t)] e-R(t) (d)f (d) can be expanded by ion (c).
In the final step we have used the linear homogeneity of (1). (i.e. (13. (1)Ii FKK), equation (b), and the following result:= FNN FKK wN (1 si)(1)= FKK + N [FN - 141] - (1 - si) (1)= FKK - pi (1 —
-] = K f(r + 8)X FK pi (1 - si)(13 + X [I - 6K] + r XK= + (1 — si)(DKK +_FKK +11(1— — Oin + = (e). and R(t) is a discounting, t rates up to t:
Chapter 4: Anticipation Effects and Eco'nomic Policy p11111111t of profits, using the (time- indexes for now, we obtain:; uation (4.4) is altered to:[-
The term in square brackets on the right-hand side of (d) can be expanded by substituting the capital accumulation identity, and equation (c). Ignoring time dR(t) dt = r(t).== Fx 1( -= FAK we
In order to establish the relationship between the Lagrange multiplier (X(0)), the capital stock (K(0)), and the real stockmarket value of the firm (V(0)), we first derive the definition:X(t)K(t)
The firm is assumed to maximize the present value of profits, using the (timevarying)real interest rate r(t) as the discount factor. Equation (4.4) is altered to:V(0) f {F (N (t), K(t)) w(t)N(t)- pi"
The Foundation of Modern Macroeconomics
TTi obin's q- theory of investment. In this intermezzo we show that Tobin's averntermezzo age and marginal q coincide under certain conditions. The proof is adapted from Hayashi (1982). Suppose that
We have developed Tobin's marginal q-theory of investment in this section.It is shown in an intermezzo to this chapter that, provided some more specific assumptions are made about the adjustment cost
Equation (4.12) allows for a very intuitive interpretation. The shadow return on the possession and use of physical capital is the sum of the shadow capital gain(4(t)) and the marginal product of
price of capital, which is spending money today on d value is measured by the the form "marginal cost 1 the problem is solved by ii. The first-order condition(4.9)0) (see the Intermezzo on Chapter 4:
nothing: The investment!gory, after its inventor James interpretation of the n to equate the marginal
al..c.--iuty COLA 4,ides 4' YO it VII K(0).And this is exar stoc, and LAI'S by tl and c CC .te the panicular firm (see termezzo Tobin's q-theor\J r`.d by in, ud ki) .1=Z t kA.77; kt)relative price of
J 11,—, 1,,e optimal path for ‘;dL P is shown in an- 10r-
(4.10)This condition can be written in several ways, two of which are:q(t) = (r + 8)q(t) - FK(N(t), K(t)), 82 a&1 0: + F40..Of).Tuation (4.12) alk nIllta.shadow price (to ma pies tne rate of phy- -
Equations (4.6)-(4.7) are in essence static conditions of the form "marginal cost equals marginal benefit". The truly intertemporal part of the problem is solved by choosing an optimal path for the
The first expression in equation (4.8) allows a very simple interpretation of the optimality condition for investment. It instructs the firm to equate the marginal cost of investment (equal to (1 -
The parallel with the expression derived in Chapter 2 (i.e. equation (2.36)) should be noted. Note that we have not used the symbol q for nothing: The investment theory developed here is formally
The interpretation of (4.6) is the usual one: the firm must choose the amount of labour such that the marginal product of labour equals the real wage rate. Note that(4.7) implies a very simple
The firm can freely choose employment and the rate of investment at each instant, so that the following first-order conditions (for t E [0, opo]) should be intuitive:aN(t) =e_rt [FN (N(t), K(o) -
The Foundation of Modern Macroeconomics eLion kir a ad, stock were increased slightly (dK(0)), i.e. q(0)::-=. dV (0)/ dK(0) (see the Intermezzo on Tobin's q below).
1 Note that the method sketched here is a generalization of the Lagrange multiplier method used in Chapter 2. An explanation of the Maximum Principle based mainly on pure economic intuition can be
Formally, q(t) plays the role of the Lagrange multiplier for the capital accumulation restriction. The economic interpretation of q(t) is straightforward. It can be shown that q(0) represents the
Principle the solution to this problem can be found quite easily.' The current-value Hamiltonian can be written as:7-1(t) e-rt [F (N (0, K(t)) - w(t)N(t) - - Mt)] 'S. (' (0)+ q(t) [I (t) - 6K(t)1].
The firm maximizes (4.4) under the restriction (4.3). With the aid of the Maximum
To the extent that shares of this company are traded in the stock exchange, and share prices are based on fundamentals and not on the speculative whims and fancies of irrational money sharks, its
The firm must choose a path for its output such that the present value of its profits is maximized. Since real profits are defined in (4.1), the appropriate discount rate is the real rate of interest
The capital accumulation identity is given by:K(t) = /(t) - BK(t), 8 > 0. (4.3)
(4.1)Chapter 4: Anticipation Effects and Economic Policy where n- (t) is real profit in period t, F(., .) is the constant returns to scale production r. . nction, w(t) is the real wage rate
Assume that the real profit of the representative firm is given by what is left of revenue after the production factor labour and investment outlays have been paid:4,--es are based on L.v , ik las or
4.1 Dynamic Investment Theory In Chapter 2 we sketched a theory of investment by firms that is based on forwardlooking behaviour and adjustment costs of investment. For reasons of intuitive clarity,
3. Embed the investment theory in an IS-LM framework. How do anticipation effects influence the outcome of traditional budgetary policies?
2. Use the investment theory to determine how the government can use tax incentives(such as an investment subsidy) to stimulate capital accumulation. This is an example of fiscal policy where the
1. To complete our discussion of the dynamic theory of investment by firms that was commenced in Chapter 2,
As was acknowledged by Lucas himself, an early statement of the Lucas critique is found in Marschak (1953). For an early application of the rational expectations hypothesis to finance, see Samuelson
The classic articles setting out rational expectations are Lucas (1972, 1973), Sargent (1973), Sargent and Wallace (1975, 1976), and Barro (1976). Papers stressing the stickiness of wages or prices
It is almost universally agreed that the PIP cannot be taken seriously, except perhaps as an extreme position taken to promote a discussion. Furthermore, due to the fact that Fischer and others
As was quickly pointed out by proponents of the new Keynesian school, the REH does not necessarily imply the validity of the PIP. Stanley Fischer pointed out that if nominal wage contracts are set
neck.Alimikaa wage emu 11=Eh at the 111}1.4a, AAA yields i(eyn£S A— LA:: tine p uict that fic her&Agile is vaLM tor t't", A it:asoi,- of hoot .-:net Readi r,tassar articles 3Ct- 4.e F .lorirv;y,
Muth's idea was introduced into the macroeconomic literature in the early 1970s by a number of prominent new classical economists. They argued that under the REH, monetary policy is ineffective (at
To most economists, one of the unsatisfactory aspects of the adaptive expectations hypothesis (AEH) is that it implies that agents make systematic mistakes along the entire adjustment path from the
Intuitively, the asymptotic variance of output is a multiple of the variance of the error term due to the persistence effect via lagged output. If A is close to unity, there is a lot of persistence
The second term on the right-hand side is the variance of the error term (0 -(2 ), and the third term is zero because the error term is independent of lagged output. The term on the left-hand side is
The asymptotic variance of output implied by the process in (a) is calculated as follows. First, we write Et-_yt = AEt-coYt-i xt and work out the square:The Foundation of Modern Macroeconomics Taking
How would a Martian judge the degree of fluctuations in output, not knowing any realizations of output and the error term, but in full knowledge of equation(a) and the stochastic process of the error

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