Phase Diagrams: construction and comparative statics

Transcription

1 1 / 11 Phase Diarams: construction and comparative statics November 13, 215 Alecos Papadopoulos PhD Candidate Department of Economics, Athens University of Economics and Business papadopalex@aueb.r, I detail the steps for the construction of a phase diaram. Then I present different tactics to do comparative statics, dependin on how much alebraically complicated the model is. This is not a comprehensive uide. It focuses on the most popular educational instance of a phase diaram in economics, the representative-household and the overlappin-enerations deterministic, continuous time, saddle-path stable models of rowth. 1. Constructin the phase diaram We will concern ourselves with the phase diaram of the overlappinenerations model of Blanchard and Weil with overnment, since it essentially nests the Ramsey representative household model, as reards the phase diaram. Our oal is to picture in two dimensions the lon run state of an economy, study its stability properties and et a taste of the path towards it. The "lonrun" state was initially called "steady-state", but in order to accommodate the fact that the actual levels of the areate variables continue to row indefinitely (when there is exoenous rowth like that of population and productivity), the profession has started to use the term "balanced rowth path". Sounds nice, the only problem is that in the phase diaram this "path" is just a point, while there is another important path, the "saddle-path towards the fixed point". To avoid confusion, we will keep the "steady-state" terminoloy. The Blanchard-Weil overlappin enerations model with overnment is (eventually) described by the two differential equations c( t) r( t) c( t) n k( t) d ( t) [1] k( t) y( t) c( t) c ( t) n k( t) [2]

2 2 / 11 the production function in intensive form y( t) f k( t) Ak( t) [3] and the market-clearin and profit maximizin condition 1 r( t) f k( t) Ak( t) [4] where y is total output(income), c is private consumption, k is physical capital, d is public debt (held as wealth by the individual), and c is overnment consumption. These manitudes are measured per labor efficiency unit. is the rate of pure time preference, r is the real interest rate, n is the birth rate of new households that also equals the rowth rate of population, is the rowth rate of labor efficiency, and is the rate of capital depreciation. A denotes total factor productivity and is the capital share in production. Preferences have been assumed loarithmic, so the intertemporal elasticity of substitution in consumption is equal to unity. We maintain the (not necessary but crucial, and not unrealistic) assumption n. Note 1: the bar above the two overnment variables indicates that are treated as exoenous to the model, functionin essentially as parameters. So the phase diaram is for iven values of these two variables. Note 2: Consumption and capital here are averae manitudes per efficiency unit, since in this model new households are born every instant of time (formin a eneration), and the representative household of each eneration differs from the representative households of all other enerations as reards the level of capital it owns (they are identical in every other respect). The phase diaram here is a raph in ( ck, ) space where we put consumption on the vertical axis. It becomes informative if we draw in it a) The zero-chane loci for capital and consumption, namely the combinations of consumption and capital that satisfy ct ( ) and kt ( ). The crossins of these loci represent fixed-points of the system, where both differential equations are zero. b) Arrows indicatin how consumption and capital tend to chane (increasedecrease), outside these zero-chane loci, which provides qualitative information on the dynamic tendencies of the system when we are not on the fixed points. These also help establish the stability properties of the latter (a "fixed point" is not necessarily stable).

3 3 / Zero-chane loci and their shape For the zero-chane loci, it is convenient to write them both as "consumption bein a function of capital", since capital is placed on the horizontal axis and it is customary even in economics to have the "arument of a function" on the horizontal axis (except for price-quantity diarams!). So n k d n k d c( t) c r f ( k) [5] k( t) c y c n k f ( k) c n k [6] where I have dropped the time-index to stress that these are static relationships. In order to draw (schematically) the raphs of these functions, we need to do old-fashioned analysis of a function with respect to its arument: calculate derivatives with respect to capital to see whether the function is monotonic or not, determine if and where it crosses the axes. We have c n f k f k n k d k ct ( ) f k ( ) ( ) ( ) 2 [7] This is always positive because consumption cannot be neative (so we examine the space where k : f ( k) ) and because we have assumed that f( k). Therefore this zero-chane loci is monotonically increasin in capital, and moreover it has a vertical asymptote at k ˆ : f ( k ˆ ). For k eq. [5] crosses the consumption axis at zero, because by the Inada conditions, lim f( k) (so the quotient is not an indeterminate form). k Movin to the other zero-chane loci, we have c k k( t) f ( k) n [8] where also This is not monotonic. It has a critical point at k : f ( k) n

4 4 / 11 2 c 2 k k ( t), kk f( k) so it is a maximum. The locus [6] takes the value c when k and also it crosses the axis of capital at another point, since its second derivative with respect to capital is always neative after the maximum point. Finally we note that the vertical asymptote of the consumption locus lies to the left of the maximum of the capital locus, since we have made the assumption n and so f ( kˆ) n f ( k) kˆ k With all this information athered, we can draw representative curves for the two zero-chane loci, without the need to specify numerical values for the parameters: We observe that the system possess two fixed-points, the second and low one due to the existence of overnment consumption (if c the capital locus crosses at the beinnin of the axis and the low fixed point disappears). We will concentrate on the other fixed point, and leave this one as a small application/exercise, after we have finished with the phase diaram.

5 5 / Dynamics off the zero-chane loci We want to know how consumption and capital move when not on the zerochane loci. Namely, we want c( t) c n k d f ( k) [9] k( t) c f ( k) c n k [1] Inequality [9] tells us that for all points in the phase diaram "above" the consumption zero-chane locus, consumption will tend to increase (and it will tend to decrease when below). Inequality [1] tells us that for all points in the phase diaram below the capital zero-chane locus, capital will tend to increase. We represent these dynamic tendencies by drawin arrows. The arrows that are parallel to an axis represent the dynamics of the variable measured on that axis. The two zero-chane loci have split the space in five sub-spaces. We draw arrows in four of them (see the exercise at the end for the fifth), and in each we join the arrows at their beinnins to provide a hint on the joint dynamics that characterize the system:

6 6 / 11 This is a classic picture of a "saddle-path stable" phase diaram: in two of the reions the arrows representin the dynamic tendencies jointly seem to push the system to extreme/corner solutions. In the other two, they seem to push the system towards the fixed-point... but this does not mean that if we start wherever in these two reions, we will end up on the fixed point. There is a unique path that lives in these reions and leads to the steady-state: the saddlepath. Points off the saddle-path, even if inside the two reions that host it, will lead to the other two, clearly unstable reions and then to extreme/corner solutions. The saddle path represents combinations of consumption and capital that will be realized as time passes (i.e. it is a curve qualitatively different than the zerochane loci). On it, variables chane in manitude. The exact shape of the saddle path depends on the alebraic nature of the equations involved. But the certain thin is that it passes from the fixed point, and that it lies wholly inside the two reions. We can add this information to the phase diaram, completin its construction:

7 7 / A special case: the consumption zero-chane locus in the Ramsey model. In the representative-household Ramsey model the differential equation for consumption is c( t) r( t) c( t) [11] and the correspondin zero chane loci is c( t) f ( k) f ( k) [12] (we inore the uninterestin case where c ). This cannot be seen as "consumption as a function of capital", but it is just an equation determinin capital, and it will be a vertical line in the consumption-capital space. But we can still apply the standard methodoloy to determine the dynamic tendencies, combined with assumptions on the production function. Specifically, we want c( t) f ( k) [13] Now, by the assumption f( k), we know that as capital increases its marinal product tends to fall. So in order for the marinal product to be hiher than the value determined by equation [12], the capital level must be lower. So we can conclude that "to the left" of the vertical locus, consumption will tend to increase, while to its riht it will tend to decrease. Exercise 1 Draw the dynamic arrows on the fifth reion of the phase diaram, and verify that the low fixed point is unstable. Exercise 2 Consider the case where n, (and nullify overnment for clarity). Verify that the phase diaram of the Blanchard-Weil model becomes equivalent to a representative-household Ramsey model, with the followin interpretational qualifications: if we assume that at some point in time population becomes fixed, after havin rown in overlappin-enerations fashion for some time interval, then the Blanchard-Weil model still contains heteroeneous households. This means that the consumption and capital variables are still proper averaes here, not the manitudes that characterize identical households. Also, in the Ramsey model we may have rowth of the size of each

8 8 / 11 household (this is what n would represent there), while in the Blanchard-Weil model, n is the birth rate of new households, while the size of each (old or new) household is identical and fixed. Still, the phase diarams of the two for n in the Blanchard-Weil model become mathematically equivalent, as reards the eneral properties of the economy, its fixed point, but also the comparative statics......to which we now turn. 2. Comparative statics methods By "comparative statics" we mean studyin qualitatively how the steady-state of the economy is affected by a chane in some parameter or exoenous variable of the model. The phase diaram permits as also to say somethin about the short-term response of consumption and capital, as well as the properties of their travel towards the new steady state, especially if we assume that we start by bein on the current steady state, which we will do. Moreover, we may want to say thins not only about consumption and capital, but also about output, the interest rate, or the wae level and the savins rate. In eneral, there are three methods to approach the matter: 1) Direct method. Solve and express consumption and/or capital as an explicit function of the exoenous parameters, and then differentiate these expressions with respect to each parameter in turn to see what happens. This method is rarely available -for example, in the Blanchard-Weil model it is not, as we will see in a while, but for the Ramsey model, it is, at least for the capital variable (which is the central variable here). 2) Implicit-function method. Express in a sinle equation the condition characterizin the steady-state, and apply the implicit function theorem. This in principle can always be done, but it does not uarantee an unambiuous conclusion. 3) Phase-diaram method: Turn to the phase diaram, and study further the two zero-chane loci, but now as functions of the parameters, for any iven level of capital. If we can determine how each curve shifts as the parameter under examination chanes, then we are justified in picturin that shift on the phase diaram, and conclude eometrically what will happen to the steady state. This approach in many cases proves to be the easiest one, and more over, it is the only one amon the three that can ive us directly what will the short run response will be, and how are we oin to approach the new steady-state.

9 9 / 11 In the Blanchard-Weil model, combinin the two relations [5] and [6] that toether characterize the steady-state we have, eliminatin consumption n k d c( t), k( t) f ( k) c n k f ( k) G f ( k) c n k f ( k) n k d [14] As lon as the production function is non-linear in capital (as is our case), the above equation determines only implicitly the level of steady-state capital (while in the Ramsey model one can obtain from [12] a direct expression for capital). So the Direct method cannot be applied here. Let's attempt the Implicit-function method, say, as reards a chane in total factor productivity A which is present in the production function f k Ak() t. By the implicit function theorem we have dk G A da G k 1 k f ( k) f ( k) c n k ak f ( k) n f ( k) f ( k) f ( k) c n k n Assume that we start at the current steady state c, k. Then we have that f ( k ) because as an equation, it defines the vertical asymptote to the riht of the zero-chane locus for consumption. Also, f ( k) c n k c. f ( k ) n (you should For the denominator we also have that remember why). So we can compact the expression a bit into k f ( k ) c ak 1 dk kk d ( ) ( ) ( ) A f k n f k f k c n We know that the numerator is positive, so the sin of the derivative will be the opposite of the sin of the denominator. But what is the sin of the denominator? The first term is positive while the second and third terms are neative since f( k ). At a first lance, it is ambiuous. Sometimes, other assumptions of

10 1 / 11 the model may help determine the sin after all, but this is not uaranteed. Moreover, note that intuition does not help here: it is not clear that a hiher total factor productivity will lead to a hiher level of steady-state capital: capital has just become more productive, so maybe we can et away only with hiher consumption while steady state capital will decrease? (After all, we care about capital only because we care about consumption). Note that the sin ambiuity is not related to G A but to G k, so it will be present to all such calculations we may attempt, with respect to any other parameter of the model. So let's try the Phase-diaram method. We are lookin at the two zero-chane loci n k d c, c f ( k) c n k f ( k) How each of these shifts as A chanes? Note that, as in the Implicit-function method, the variables themselves are not differentiated with respect to the parameter we examine (unlike the Direct method) because here they act as an arument of a function, they do not represent a specific value (the steady-state). We have c A 2 c( t) k ( t) 1 k n k d c, k f ( k) A These results provide clear uidance as to how the two zero-chane loci will shift if we increase A : The first tells us that for any iven level of capital the correspondin level of consumption must be lower on the consumption locus. The second tells us that for any iven level of capital, the correspondin consumption level must be hiher on the capital locus. Schematically then, the phase diaram will chane as follows (eliminatin overnment for clarity):

11 11 / 11 C S 2 kt ( ) S 1 kt ( ) ct ( ) ct ( ) K The economy is initially in steady state S 1, characterized by the red loci. The chane in total factor productivity A shifts the two loci which now become blue (the black dashed arrows indicate this shift, they are not dynamic arrows). The new steady state will be at S 2. We et a clear lon-run result: both consumption and capital will be hiher at the new steady-state. We can also conclude that in the mid-term we will see both consumption and capital risin to attain their new steady state values. But the short-term reaction of the decision variable consumption remains ambiuous: whether the new saddle-path will pass "above" or "below" the old steady state (the two straiht blue lines represent possible saddle paths) depends further on the values of the various parameters. But this is what determines whether we will see consumption jump up or down immediately after the increase in A. So we don't et a definite result on that. Still, by this third method we were able to finally obtain some answers to our questions. Exercise 1 Practice on other comparative statics results on consumption and capital, usin the Phase-diaram method. Try also to answer questions about the effects on output, the savins rate, the real interest rate and the real wae. Exercise 2 Turn to the Ramsey model, where the Direct method is also applicable, and try to apply also the Implicit-function method as well as the Phase-diaram method for some of the parameters of the model. Which one is more easy and tractable? Which one provides more definite results amon the three? --