004: Macroeconomic Theory

Transcription

1 004: Macroeconomic Theory Micro Foundations of Various Macroeconomic Systems Mausumi Das Lecture Notes, DSE Jan 17-Feb7; 2017 Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

2 Micro-foundations of various Macroeconomic Systems: So far we have discussed various macroeconomic systems (Classical, Keynesian and their different extensions). Recall that we have presented each system as a bunch of adhoc equations that are supposed to define the macroeconomy as a whole. We made no attempt to derive the underlying micro-behaviour of optimizing agents that would generate these aggregative equations for the macro-economy. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

3 Micro Foundations: We now turn to the precise micro-foundations of each of these equations. Such micro-foundations are necessary for two reasons: There are many implicit assumptions that are made in formulating the aggregative relationships. A precise discussion of the micro-foundations allows us to highlight these assumptions and also check for their validity; As Robert Lucas pointed out, many of the constants in the aggregative systems are not parameters in the true sense of the term; they capture equilibrium behaviour under certain conditions (e.g. ceratin expectations about the policies, environment etc). As those conditions change, people optimally change their equilibrium behaviour; hence these reduced form terms also change their values. So predictions/forecasts about the economy based on these constants could go wrong unless one actually derives these optimal values from the underlying micro-foundations. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

4 Micro Foundations: Classical/Keynesian Production Side Story We start with the micro-foundations of the production relations. Recall that the production side story for the Classical and the Keynesian systems are identical: both assume that firms operate in perfectly competetive market structure with a production technology which exhibits the following properties: Y i = F (N i, K i ) Constant returns to scale (CRS)/Homegenous of degree one: F (λn i, λk i ) = λf (N i, K i ) Positive but diminishing returns in each factor: F N, F K > 0; F NN, F KK < 0 Both inputs are essential: F (0, K i ) = F (N i, 0) = 0 Inada conditions: lim F N (N i, K i ) = ; lim F N (N i, K i ) = 0; N 0 N lim F K (N i, K i ) = ; lim F K (N i, K i ) = 0 K 0 K Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

5 Classical/Keynesian Production Side Story (Contd.) Any production function that satisfies all the above properties is called a Neoclassical production function. Example: Cobb Douglas Technology: Y i = Ni α Ki 1 α ; 0 < α < 1. Sometimes the production function is also written as Y i = AF (N i, K i ) where A is an index of technology that captures the economy-wide productivity level or total factor productivity (TFP). In a static framework, A is typically assumed to be a constant.(notice that A is not firm-specific; it relates to the entire economy). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

6 Classical/Keynesian Production Side Story (Contd.) Suppose there are S firms in the economy, all having access to an identical Neoclassical technology given by: Y i = AF (K i, N i ). Recall that the firms take decisions about: How much final output to produce; How much labour to employ. The firms also decide how much capital to employ. However in this static framework, we shall assume here that this decision is trivial: they employ the total capital stock available in the economy ( K) equally such that K i = K S Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

7 Classical/Keynesian Production Side Story (Contd.) Indeed if the firms are perfectly competitve (which means they are price-takers), then a firm s choice is rather straight-forward. Given W and P, the optimization problem of the i-th firm is defined as: Max.P {N i } AF ( K i, N i ) WN i R K i. This generates the following FONC from the i-th firm: AF N ( K i, N i ) = W P. The above equation implicitely defines the labour demand of the i-th firm as a function of the real wage rate ( W P ), its capital share ( K i ) and the aggregate TFP Index (A): ( ) W N i = ˆf P, K i, A. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

8 Classical/Keynesian Production Side Story (Contd.) Since all firms face the same TFP (A), same real wage rate W P and employ equal share of the aggregate capital stock (i.e., K i = K i for all i and i ), the labour demanded by each firm would be exactly the same. Aggregating over all firms, we get the corresponding labour demand function for the aggregate economy as: ( W N D = S.N i = S ˆf P, K ) ( ) W S, A = f ; f < 0. P Correspondingly aggregate output supplied: Y S : Y = S.Y i = S AF ( K i, N i ) = AF (S K i, SN i ) (by CRS) = AF ( K, N) where the latter describes the aggregate production function for the economy. Notice that the aggregate production technology looks identical to the firms technology. Hence we expect it to obey all the Neoclassical properties. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

9 Classical/Keynesian Production Side Story (Contd.) Notice that the assumption of CRS plays a very important role in generating the equivalence between S.Y i and AF ( K, N). Also notice that aggregation over all firms have been easy because we have assumed that total capital is divided equally across all firms. (Question: will such equivalence hold when the division of capital is unequal across firms?) Now recall that the labour demand function in the Classical (and Keynesian) system was defined in terms of the aggregate production function: N D : AF N ( K, N) = W P On the other hand, the labour demand function that we derived here is a aggregation of the individual firms demend for labour where ( ) W N D = S.N i where N i = ˆf P, K i, A such that AF N ( K i, N i ) = W P How do we know that these two expressions are equivalent? Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

10 Classical/Keynesian Production Side Story (Contd.) Answer to both questions is an emphatic "yes" - although the F N function is not CRS. The clue still lies in the CRS property of the production function F. But we now have to take a more circuitous route. We exploit here another characteristic of the production function which follows from its CRS property: Consider a production function: Y = AF (N, K ). If it is CRS, then Y ( N = AF 1, K ) = Af (k), where k K N N. Thus another way to write the production function is: Y = N.Af (k). From this latter specification, [ Y N = A f (k) + Nf (k) k ] [ = A f (k) + Nf (k) ( KN )] N 2 = A [ f (k) kf (k) ] Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

11 Micro Foundations: Production Side Story (Contd.) In other words, for a CRS production function, the marginal product of labour is a function of the capital-labour ratio employed. (Question: How about the marginal product of capital? Can we derive a similar relationship between Y and k? Try this as a homework. ) K Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

12 Classical/Keynesian Production Side Story (Contd.) Given this property, let us now go back to the two questions asked earlier. Let us take up the second question first: Are N D : AF N (N, K ) = W ( P and N D = SN i : AF N N i, K ) = W S P equivalent? Notice that AF N (N, K ) is the marginal production labour associated with the aggregate production function. Hence applying the above property: A [ f (k) kf (k) ] = W P (i) where the relevant k here is the aggregate capital-labour ratio: K N. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

13 Classical/Keynesian Production Side Story (Contd.) ( On the other hand, AF N N i, K ) is the marginal product of labour S of the i-th firm. Hence applying the above property: A [ f (k i ) k i f (k i ) ] = W P (ii) K i where the relevant k i here is the firm-specific capital-labour ratio:. N i Since equation (i) and equation (ii) are identical equation (though the variable is different) the solutions to the equations must also be the same, i.,e k i = k = C (some constant) Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

14 Classical/Keynesian Production Side Story (Contd.) In other words, even though firms are employing different levels of capital and labour than the aggregate economy, the capital-labour ratio in both cases must be the same, denoted by some constant C. Given this relationship, we can now write the labour demand coming K out of the aggregate profuction function as N D : N = C N = K C. On the other hand, labour demand coming from an individual firm K i can be written as N i : = C N i = K i N i C. Using the above solution, the labour demand coming out of aggregation over all firms is: N D = SN i = S K i C = S K SC = K C. Thus that the two labour demand functions will be exactly the same. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

15 Classical/Keynesian Production Side Story (Contd.) This also should also give us the clue as to what happens when firms differ in terms of their share of capital. For simplicity, let us assume that there are two sets of firms: one set of firms (S 1 in number) is given a capital stock of K 1 while the other set (S 2 in number) is given a capital stock of K 2 < K 1 such that S 1 + S 2 = S, and S 1 K 1 + S 2 K 2 = K. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

16 Classical/Keynesian Production Side Story (Contd.) As we have seen earlier firms belonging to the first group will have a labour demand equation given by: N i : AF N (K 1, N i ) = W P i.e., A [ f (k i ) k i f (k i ) ] = W P ; k i K 1 On the other hand, firms belonging to second group will have a labour demand equation given by: N i (i) N j : AF N (K 2, N j ) = W P i.e., A [ f (k j ) k j f (k j ) ] = W P ; k j K 2 N j Once again, since equations (i) and (ii) are identical, their solutions must also be exactly the same, i.e., k i = k j = C (some constant). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104 (ii)

17 Classical/Keynesian Production Side Story (Contd.) Thus we can write the individual demand for labour functions coming from each set of firms as: N i = K 1 C ; N j = K 2 C. Aggregating over all firms, the aggregate labour demand function will now be given by: N D = S 1 N 1 + S 2 N 2 K 1 = S 1 C + S K 2 2 C = 1 C [S 1K 1 + S 2 K 2 ] = K C Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

18 Classical/Keynesian Production Side Story (Contd.) Notice that this labour demand function (that we have obtained by aggregating over firms with unequal distribution of capital) is exactly identical to the labour demand function that we had obtained earlier (when total capital stock was divided equally across all firms.) Moreover, this also coincides with the demand for labour that is derived by using the aggregate production function. The upshot of this exercise is that when the production function is CRS, the size of the firm does not matter: Different firms may employ different levels of capital and labour, but the capital-labour ratio for every firm is identical As a result when we aggregate over all firms we get identical labour demand function for the economy as a whole (irrespective of the distribution of the capital stock and workforce across firms). This also tells us how crucial the assumption of CRS is in this entire analysis. It is now pertinent to ask: would the aggregate production function necessarily exhibit CRS? Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

19 Aggregate Production Function: Neoclassical or not? Or more generally, is there any reason to believe that the aggregate production technology will necessarily be Neoclassical in nature? An American economist called Marvin Frankel pointed out long time back (AER, 1962) that even when each atomistic firm faces a production function which is Neoclassical in nature, there is no reason why the aggregate production function would also be strictly Neoclassical. Frankel s argument ran as follows: Consider an economy with S identical firms - each having access to an identical firm-specific technology: Y i = Ā t F (K it, N it ) Ā t (K it ) α (N it ) 1 α ; 0 < α < 1. Note that the firm-specific production function exhibits all the neoclassical properties. The term Ā t represents the current state of the technology in the economy, which is treated as exogenous by each firm. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

20 Production Function: Neoclassical or not? (Contd.) Frankel then relates the Ā t term to the aggregate capital labour-ratio in the economy - due to knowledge spillovers and learning by doing : ( ) Kt Ā t = g ; g > 0; where K t = SK it ; N = SN it. N t The idea is as follows: Productivity depends on how quickly workers can adapt themselves to new machines. This is the process of learning by doing. When the aggregate capital stock in the economy rises in relation to its total labour stock, everybody gets greater opportunity to familiarise themselves with the machines; hence productivity improves faster. Moreover, there is knowlege spillovers - workers can learn from one another (without everybody spending time in going through the instructions manual). Both these factors would imply that Ā t would be an increasing function of the aggregate capital-labour ratio. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

21 Production Function: Neoclassical or not? (Contd.) Without any loss of generality, let us assume: ( ) ( ) β Kt Kt g = ; β > 0, N t Corresponding Aggregate Production Function: Y t = Y it = S [Āt (K it ) α (N it ) 1 α] N t = Ā t (SK it ) α (SN it ) 1 α = Ā t (K t ) α (N t ) 1 α. Notice that Ā t is the total factor productivity term - which is given for each firm, but not so for the aggregate economy. Replacing the value of Ā t in the aggregate production technology: Y t = Ā t (K t ) α (N t ) 1 α = (K t ) α+β (N t ) 1 α β. In the special case where α + β < 1, the aggregate production technology is indeed Neoclassical, but not otherwise! Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

22 Production Function: Neoclassical or not? (Contd.) The Classical/Keynesian production story therefore must assume that there is no such externality at the aggregate level. (Is that empirically true? We shall answer that question later when we discuss the empirics of output dynamics). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

23 Micro Foundations: Neo-Keynesian Production Side Story The production side story becomes a little more interesting for the Neo-Keynesian case. Recall that in this case the output supply function is perfectly elastic at some price level P. What kind of micro-founded firm-side story would support this aggreagtive behaviour? It is obvious that we now have to move away from the perfectly competitive set up (i.e., price-taking behaviour by firms) and allow for some form of market imperfection (i.e., price-setting behaviour by firms). An obvious way to motivate this is to assume that the production function exhibits IRS (increasing returns to scale) such that perfect competition is not sustainable in equilibrium and monopoly emerges as the natural outcome. This is the route that we are going to follow now. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

24 Neo-Keynesian Production Side Story: (Contd.) The simplest production function that exhibits IRS is a linear one with a fixed cost. Let the technology be represented by the following production function: { Y = 0 α (N F ) if N F if N > F where α is the constant marginal product of labour employed in the actual production process and F is the fixed cost defined in terms of units of labour. This production function implies that F quantity of labour is required to set up the production unit before actual production can take place. Thereafter every additional unit of labour employed produces α units of output. (We are ignoring the role of capital for the time being, but capital can be easily brought in either as a part of the fixed cost or the variable cost) Question: Why is this production function IRS? Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

25 Neo-Keynesian Production Side Story: (Contd.) Production is now carried out by a monopolist producer who knows the exact demand schedule. Given the demand function, the monopolist producer optimally chooses the price level to maximise his profit: Max. {P } Π = P.Y D (P) WN. Notice that to produce Y D amount of output, the monopolist producer has to employ Y D units of labour in actual production. In α addition, he has to employ F units of labour to set up the production unit. Thus, N = Y D + F. Plugging this in the optimization problem of the α monopolist: [ Y Max. Π = P.Y D (P) D ] (P) W + F. {P } α Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

26 Neo-Keynesian Production Side Story: (Contd.) Corresponding FONC: dπ dp = Y D (P) + [ P W ] α [ ] P W α P [ P W ] dy D α = Y D (P) dy D dp = Y D (P) dy D dp P dp = 0 = 1 ɛ where ɛ dy D P dp is the price elasticity of demand. Y D Rearranging: P = W ( ) ɛ α ɛ 1 If the nominal wage is constant and the demand function exhibits constant price elasticity of demand (which is greater than unity), then the monopolist producer will indeed optimally charge a constant price level P irrespective of the level of demand. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

27 Neo-Keynesian Production Side Story: (Contd.) Notice that in deriving a horizontal AS curve from micro-foundations, we have departed from the standard Neoclassical framework in two different directions: We have assumed a single monopolist producer instead of many producers operating under perfect competition; We have also assumed a different production function that exhibits IRS, since IRS and monopoly go hand in hand. Is the second assumption essential to generate a horizontal AS curve? To check that, let us retain the assumption of a single monopolist producer but let us endow him with a Neoclassical techonology. Let us assume that the relevant production function for the monopolist is: Y = N α K 1 α ; 0 < α < 1. As before let us fix the capital employed by the firm at K and chooses only N. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

28 Neo-Keynesian Production Side Story: (Contd.) As before the monopolist producer knows his demand function Y D (P). To produce Y D amount of output, the monopolist producer now has ( ) 1 Y D α to employ N = K 1 α units of labour. Hence the optimization problem of the monopolist is now given by: Max. {P } Π = P.Y D (P) W [ (Y D ) 1 ] α (P) K 1 α. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

29 Neo-Keynesian Production Side Story: (Contd.) Corresponding FONC: dπ dp = Y D (P) + P dy D dp W α K 1 α α Y D (P) + P dy D Y D (P) (ɛ 1)Y D (P) = W P = W α K 1 α α ɛ ɛ 1 [ ] 1 Y D α 1 dy (P) D dp = 0 dp Y D (P) = W [ Y D (P) ] 1 α Pα K 1 α α α K 1 α α [ Y D (P) [ Y D (P) ] 1 α α ] 1 α 1 P ɛ P dy D Y D (P) dp where as before ɛ dy D P dp is the price elasticity of demand. Y D Notice that now the price charged by the firm depends on the level of Y D even when the price elasticity of demand is constant. Indeed any exogenous increase in demand (Y D ) would now induce the monopolist to charge a higher price (unlike the IRS technology case). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

30 Neo-Keynesian Production Side Story: (Contd.) Let us now go back to the earlier framework where a single monopolist producer setting the price level as a constant mark up over its nominal cost (W ) (under the assumption of constant price elasticity of demand which is greater than unity): P = W ( ) ɛ α ɛ 1 Notice that the level of profit earned by the monopolist is: [ Y Π = P.Y D (P) D ] (P) W + F α ( = P W ) Y D (P) WF α ( ) 1 W = ɛ 1 α Y D (P) WF Hence the monopolist will operate iff the demand is suffi ciently high: Y D α (ɛ 1) F Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

31 Neo-Keynesian Production Side Story: (Contd.) From now on we shall assume that demand is high enough (presumably because G is suffi ciently high); otherwise production process will crash to zero. Under that condition, the monopolist firm always earns a positive profit. But this generated an additional conceptual problem, which is at odds with the way we had specified our macro frameworks earlier. Recall that we had assumed that the entire output produced in the economy eventually goes back to the households who are the owners of all factors of production. That had completed the circular flow of income for the economy, which allowed us to write the consumption demand (coming from the households) as a function of the total income (Y ). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

32 Neo-Keynesian Production Side Story: (Contd.) It also fitted well with a perfectly competetive market structures where the firms in equilibrium were earning zero profits. After paying both the factors their respective returns, there was nothing left with the firms. This also explains why the firms had to borrow when the wanted to invest and hence the borrowing cost (r) was an important determinant of the investment demand. Now, with a monopolist producer which is earning positive profit, the neat chain of reasoning is broken: Even if the households collective own the firm (as shareholders), what proportion of that profit is given back to households as dividend and how much is retained)? If the firm can retain at least part its profit, then in order to invest why must it borrow from the market? How can consumption be a function of aggregate output (Y ) when a part of the output (retained profit) does no go back to the households? How can investment be a function of the borrowing cost (r) alone? Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

33 Neo-Keynesian Production Side Story: (Contd.) If is not easy to answer these questions without changing the basic (ad-hoc) macro structure that we had assumed earlier. And that is exactly what the more recent (DGE/DSGE) macro models do: they build the entire macro framework from first principles. However, a short cut was provided by the New-Keynesians, which allowed them to retain the aggregative macro structure while at the same time generate price rigidity from the micro optimization exercises of firms. The New-Keynesians derive the price rigidity result from a market structure which monopolistically competitive, but not full-fledged monopoly. The advantage of this second approach over the first is that many of the features of perfect competetion (most importantly, the zero-profit condition) can now be retained without completely moving over to a new way of building macro models. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

34 Micro Foundations: New-Keynesian Production Side Story The New-Keynesian structure defines the final consumption good C as a composite good consisting of M different varieties, indexed by j [1, M] which are close but imperfect substitutes of one other. A convenient (and oft-used) definition of C (expounded by Dixit-Stiglitz (1977)) is: [ ] θ C = M η M 1 M (C j ) θ 1 θ 1 θ ; θ > 1; η 1. j=1 where C j denotes the quantities of each variety j consumed and the term (C j ) θ 1 θ captures the importance of this particular variety in the total composite bundle of C. This specific Dixit-Stiglitz formulation requires some explanation. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

35 New-Keynesian Production Side Story: (Contd.) Typically one defines a composite goods bundle as a weighted average of various varieties of goods: C = 1 M M ω j C j j=1 where ω j captures the importance of a particular variety in the bundle. Dixit and Stiglitz simply generalized this idea to argue that the weights (ω j s) themselves could be endogenous. In particular, they could depend on the quantities consumed of a particular variety such that ω j ω j (C j ). They further argued that there is no reason why the composite bundle has to be a simple average of all the varieties (i.e., a linear function of various varieties). It could take a more general non-linear form. As long as the positive relationship between any C j and C is maintained (i.e., in any C j ceteris paribus leads to an in C ), any such non-linear form would suffi ce as an index of the composite good. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

36 New-Keynesian Production Side Story: (Contd.) Moreover, people may care independently about more variety even when the average component is the same. The specification provided by Dixit-Stiglitz is an example that captures all these three ideas: [ ] θ C = M η M 1 M (C j ) θ 1 θ 1 θ ; θ > 1; η 1. j=1 The parameter θ is related to the elasticity of substitution between two different varieties j and j : a higher θ implies greater degree of substitutability between the two. The parameter η captures the preference for diversity ("love for variety"): a higher η implies a greater love for variety. Notice further that when there is only a single variety (i.e., no differentiated goods) such that M = 1, then the above definition collapses to the usual connotation of C - denoting amount of consumption of a single commodity. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

37 New-Keynesian Production Side Story: (Contd.) The production sector is characterized by monopolistic competition: there are many (M) small firms - each producing a particular variety of the differentiated goods and each enjoying (local) market power in its own product market. To put it differently, the producer of each variety j is a local monopolist who sets his own product price P j. At this point we treat the total number of firms/varieties (M) as exogenous. (Later it will be endogenously determined by the free entry condition. More on this later...) Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

38 New-Keynesian Production Side Story: (Contd.) Each monopolist producer of a variety is endowed with a symmetric technology which uses only labour and is IRS: { 0 if N Y j = j F for all j = 1, 2,..., M. α (N j F ) if N j > F As before, the monopolist producer knows his own demand function: Yj D (P j ) Hence profit function of the monopolist producer of the j-th variety is given by: Π j = P j.yj D (P j ) W [ ] Y D j (P j ) + F α Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

39 New-Keynesian Production Side Story: (Contd.) As before, one can show that the monopolist producer of variety j will choose a price P j which will depend on his own-price elasticity of demand ɛ j such that P j = W ( ) ɛj α ɛ j 1 where ɛ j Y j D P j i th variety. P j Y D j Is ɛ j necessarily a constant? is the own-price elasticity of demand for the Moreover, are these own-price demand elasticities for various varieties j = 1, 2,...M necessarily the same? We cannot comment on whether P j is a constant or not without knowing the answers to the above questions. To know the answers to these questions, we need some information about the demand function for each variety. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

40 New-Keynesian Production Side Story: (Contd.) Notice however that consumers do not buy these varieties separately. They buy the composite good as a single unit. This implies that the demand for each variety is a derived demand - which arises out of the total demand for the composite commodity C. So let us give a closer look to the households demand for the composite good. Note that the households demand for the composite good has two distinct component: How many units of the composite good should they buy, given the price of the composite bundle (P)? What should be the desirable composition of each such unit in terms of various varieties? Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

41 New-Keynesian Production Side Story: (Contd.) Suppose households spend Ŷ amount of their income on consumption. This implies that they spend a total amount of Ŷ in buying the composite good. Recall that a consumer buys the composite good bundle as one unit; she does not buy any particular variety separately. Hence she will be paying an uniform unit price P for each such bundle such that PC = Ŷ. If P was somehow known, then we would have also known exactly how many units of the composite good would be bought (which would be given by Ŷ P. ) Alternatively if C was known, then we could have imputed the unit price P as Ŷ C. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

42 New-Keynesian Production Side Story: (Contd.) However, neither C nor P is not known at the moment. Moreover we do not also know the exact composition of this composite good unit in terms of various varieties. And it is this latter question that we ask first. What should be the optimal composition of the composite good such that consumers derive maximum valuation, given their total consumption expenditure of Ŷ? Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

43 New-Keynesian Production Side Story: (Contd.) Or to put it differently, given the total consumption expenditure of Ŷ, what is the optimal way to spread this total expenditure across all the different varieties? The answer to this question lies in the following optimization exercise: subject to [ Max. U(C ) = C M η {C j } M j=1 M P j C j = Ŷ. j=1 1 M ] θ M (C j ) θ 1 θ 1 θ j=1 Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

44 New-Keynesian Production Side Story: (Contd.) A Digression: Who undertakes this optimization exercise? Usually it is assumed that consumers themselves implicitly undertake this exercise since they want to choose their consumption bundle in the most cost-effective way so as to get the maximum valuation for their total expenditure Ŷ. Alternatively, one can think of a retailer who procures these varieties from their respective monopolist producers and sells the composite bundle to the consumers at some unit price P. Operating under potential competition from other retailers, this retailer offers the best possible deal to the consumers (one that maximizes their utility) subject to the constraint that his total cost ( M P j C j ) must be recovered. j=1 Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

45 New-Keynesian Production Side Story: (Contd.) From the FONCs of the above optimization exercise: C / C j C / C j = P j P j ( Cj C j ) 1/θ = P j P j C j = ( Pj P j ) θ C j for all j, j = 1, 2,..., M Question: Recall that in defining the composite good index C, we had related the parameter θ to the elasticity of substitution between two varieties j and j. Now the elasticity) of substitution ) between two varieties j and j is defined as: ( Cj C j ( Pj P j / ) / ( Cj C j ( Pj P j ). Given this definition and given the FONC above, can you find out how the elasticity of substituion here is related to the parameter θ? Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

46 New-Keynesian Production Side Story: (Contd.) Using the above optimal relationship between any two varieties j and j in the definition of the composite good, we can write the maximised value of U(C ) as a function of a particular variety j such that: U (C ) = C = [ M j=1 M η (P j ) θ C j 1 M (P j ) 1 θ Conversely, C j = M( θ 1 θ ) η (P j ) θ C [ ] θ M (P j ) 1 θ θ 1 j=1 ] θ θ 1 for all j = 1, 2,., M. (A) Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

47 New-Keynesian Production Side Story: (Contd.) Recall that the total expenditure made on procuring various varieties that go to the optimal bundle is given by M P j C j = j=1 M P j C j. j=1 Using relationship (A), we can now write the above expression as: [ ] CM ( θ 1 θ M ) η (P j ) 1 θ j=1 = CM [ 1 θ ] θ M (P j ) 1 θ j=1 ( ) [ ] 1 θ+η ηθ M θ 1 (P j ) 1 θ 1 θ j=1 Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

48 New-Keynesian Production Side Story: (Contd.) Since M P j C j has to be identical to the total expenditure on the j=1 composite commodity Ŷ, and the total expenditure, by definition, is the price (notional or actual- charged by the retailer) of the composite good (P) multiplied by its quantity (C), this allows us to write a precise expression of the general price index: P M ( ) [ ] 1 θ+η ηθ M θ 1 (P j ) 1 θ 1 θ j=1 (B) Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

49 New-Keynesian Production Side Story: (Contd.) Using (A) and (B), we can write: ( ) ( ) θ Cj = M (θ+η ηθ) Pj for all j = 1, 2,..., M C P So we have now derived the demand function for each variety j as a function of its own price (P j ), the composite bundle (C), the number of varieties (M) and the aggreagte price index (P): Y D j C j = M (θ+η ηθ) ( Pj P ) θ C for all j = 1, 2,..., M (I) Recall that each producer of a variety j is a local monopolist, which means that he only knows that relationship between the demand for his own product (C j ) in relation to his own price (P j ). In his profit maximization exercise, he treats the aggregate price index P, the number of varieties M and the composition of the commodity bundle C as exogenous. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

50 New-Keynesian Production Side Story: (Contd.) Given the reduced form demand function derived above, what should be the optimal price charged by any monopolist producer of variety j? We have already seem that the optimal price of variety j (P j ) depends on the own-price elasticity of demand ɛ j C j P j P j C j Given the above reduced form demand function for each variety (equation (I)), it is easy to show that ɛ j = θ for all j = 1, 2,..., M Thus each monopolist producer of any variety j will charge the same price P j = W ( ) θ for all j = 1, 2,..., M α θ 1 which is independent of the level of demand (C i or C). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

51 New-Keynesian Production Side Story: (Contd.) A digression: Since in equilibrium all monopolist producers of various varieties face exactly analogous demand functions and therefore charge the same price, the equilibrium in this differentiated goods framework is often called the symmetric" equilibrium. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

52 New-Keynesian Production Side Story: (Contd.) The aggregate price index P (price of a composite unit) in the symmetric equilibrium is given by: P M = M ( ) [ ] 1 θ+η ηθ M θ 1 (P j ) 1 θ 1 θ j=1 ( θ ( ) [ θ+η ηθ { θ 1 W M α = M (1 η) W α ( θ θ 1 ) θ 1 )} 1 θ ] 1 1 θ It seems that we have almost derived a flat AS schedule which is independent of the level of demand, except that we still do not know the value of M!!! Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

53 New-Keynesian Production Side Story: (Contd.) The crucial question therefore is, in this framework of monopolist competition, what determines the equilibrium number of varieties (M)? The underlying assumption here is that anybody can enter the production market and start producing a new variety as long as he/she earns a positive profit out of it. Recall that the profit function of a monopolist producer is given by: [ Π j = P j W ].Yj D (P j ) WF α Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

54 New-Keynesian Production Side Story: (Contd.) We have already solved for Y D j and P j : ( ) θ Yj D C j = M (θ+η ηθ) Pj C P P j = W ( ) θ α θ 1 Thus we can write the profit function of each monopolist as Π j = = [ ( )] W 1 α θ 1 [ ( W 1 α θ 1 = 1 θ M 1 Ŷ WF ( ) θ.m (θ+η ηθ) Pj C WF P )].M (θ+η ηθ) (P j ) θ (P) θ 1 (PC ) WF Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

55 New-Keynesian Production Side Story: (Contd.) Hence for any given level of expenditure Ŷ, the number of firms will adjust (some more firms will join in if 1 θ M 1 Ŷ > WF and some firms will leave if 1 θ M 1 Ŷ < WF ) until in equilibrium: M : 1 θ M 1 Ŷ = WF M = Ŷ θwf Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

56 New-Keynesian Production Side Story: (Contd.) Notice that the zero profit condition would eliminate the profit-related problem that arose in the earlier framework (with a single monopolist). But...is the aggregate price index now independent of the level of demand? The answer is, NO!! Because M itself depends on Ŷ (which in turn is related to the demand for the composite good). Thus in our attempt to provide a micro-foundation to the Neo-Keynesian aggregative macro system we seem to have run into a conundrum: either we fail to get a flat AS schedule wherethe price level does not respond to exogenous demand shocks, or we end up with firm earning positive profits! (Recall that it is not easy to reconcile existence of positive profits with a consumption function which depends only on Y and an independent investment function which depends only on r). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

57 New-Keynesian Production Side Story: (Contd.) Mankiw (Economics Letters, 1988) resolved this problem in the following way: He proposed a framework with where M was exogenously given. Firms earn positive profits but all profits were distributed back to the households. Mankiw futher assumed that there is no investment demand so that the aggregate demand equation was given by: Y D : Y = C + G Startz (QJE, 1989) argued that Mankiw s case represents the short run. In the long run the number of firms will adjust to ensure zero profit. But in the process, it will also generate an aggregate supply schedule which will not be horizontal. (In particular, the optimal price will respond to exogenous changes in demand). We resolve the issue here by simply assuming a suitable parametric configuration such that the M term drops out of the expression for equilibrium price level. The required parametric condition is: η = 1. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

58 New-Keynesian Production Side Story: (Contd.) An important point: Since we now have defined the final good as a commodity bundle rather than a single good, all other components of aggregate demand also now have to be suitably defined in terms of all the varieties. For example, the aggregete government expenditure has to similarly be defined as [ ] θ G = M η M θ 1 1 ( ) θ 1 M Gj θ. j=1 And it is implicitly assumed that given its total expenditure Ḡ, the govenment also optimally chooses to spread it over various varieties. This requires re-specifying the entire system of macro equations in terms of the composite good bundle, which can be easily done. But we leave this exercise for future. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

59 New-Keynesian Production Side Story: Reference Ben Heijdra: Foundations of Modern Macroeconomics, Chapter 13, pages For those who want to explore the issue at a deeper level (thus acquiring knowledge for the sake of knowledge and not for the exams!!), the following two articles are useful: Mankiw, N. G. (1988): Imperfect Competition and the Keynesian Cross, Economics Letters, Vol 26, pp Startz, R. (1989): Monopolistic Competition as a Foundation for Keynesian Macroeconomic Models, Quarterly Journal of Economics, Vol 104, pp Note: All references are meant as supplementary readings; not as substitutes for lecture notes. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

60 Micro Foundations: Keynesian Labour Supply Equation Let us now analyse the micro-foundations of the labour supply schedule of the Keynesian system. Recall that in the Keynesian system the labour supply schedule is perfectly elastic at a given wage rate. The underlying assumption here is that the wage rate is set by the trade union and at that wage rate whatever labour is demanded by the firms is readily supplied. We now try to generate a perfectly elastic labour schedule from the optimization problem of a labour union which controls the entire labour supply. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

61 Keynesian Labour Supply Equation (Contd.): The monopolist trade union cares for both the real wage rate (ω) as well as the level of employment (N). It s preference is captured by the following utility function: V = V (ω, N); V ω, V N > 0; V ωω, V NN < 0. The trade union chooses ω to maximize its utility subject to the labour demand equation, which acts as a constraint on the union s choice of action. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

62 Keynesian Labour Supply Equation (Contd.): The labour union, being the monopolist seller of the labour services, knows its exact demand curve. In particular it knows that competitive profit maximsing firms are going to choose employment so as to equate the marginal product of labour to the real wage rate: N D : F N (N, K ) = W P ω. In particular, let us assume a Cobb-Douglas production function, so that we can precisely calculate the labour demand function as follows: N D : (1 α)an α ( K ) α = ω [ (1 α)a ( K N D ) α = ω ] 1 α. (1) Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

63 Keynesian Labour Supply Equation (Contd.): Formally, the optimization problem of the trade union is given by: MaxV (ω, N) sub. to N = {ω} [ (1 α)a ( K ) α If we knew the precise functional form of V, we could directly replace N is the utility function from the constraint equation and directly solve for the optimal ω. Since we do not know that we are going to use a diagrammatic approach. ω ] 1 α Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

64 Keynesian Labour Supply Equation (Contd.): From the first order condition of the above optimization problem: V ω + V N N ω = 0 V ω = N ω. V N In other words, at the optimal value of ω, the slope of the union s indifference curve should be equal to the slope of the demand for labour,as shown in the diagram below: Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

65 Keynesian Labour Supply Equation (Contd.): Now suppose the labour demand goes up due to some exogenous reason (say, due to an in A or K). Will the union still ask for the same real wage rate? The answer depends on the precise nature of the union s objective function. Often the objective function of the trade union is written more precisely as: ( ) ( V (ω, N) = u(ω) + 1 N N N N ) u(b); u > 0; u < 0. where N is the total labour force in the economy (corresponding to full employment level of labour supply) and u(b) represents the reservation utility of an unemployed worker. (B can be thought of as the unemploymeny benefit or income from some subsistence occupation). The term represents the probability of finding work N N under the unionized set up. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

66 Keynesian Labour Supply Equation (Contd.): Let us further assume that u(ω) satisfies the following properties: u(0) = 0; lim ω 0 u (ω) = ; lim ω u (ω) = 0. With this specific utility function, and noting that from the constraint [ ] function N = (1 α)a( K ) α 1 α ω, we can write the FONC of the labour union as: ( ) u N N N (ω) + [u(ω) u(b)] 1 N ω = 0 1 i.e., N [u(ω) u(b)] 1 α ω 1 α 1 [ (1 α)a ( K ) α] ( ) 1 α = u N N (ω) Simplifying, [u(ω) u(b)] ω = αu (ω) Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

67 Keynesian Labour Supply Equation (Contd.): From the characterization of the u(ω) function, it is easy to show that the LHS is an upward-sloping curve, going from a negative value to ; while the RHS is a down-ward sloping curve going from to 0. (Verify this). Thus there will be a unique solution to the above equation, denoting the optimal wage rate ω. Moreover, this optimal ω is independent of the labour demand parameters (A or K). Any exogenous shift in the labour demand schedule will leave the optimal ω set by the trade union unchanged. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

68 Keynesian Labour Supply Equation (Contd.): Thus under these specific assumptions, we have indeed derived a labour supply schedule which is perfectly elastic at W P = ω; the equilibrium level of employment will be determined by the position of the labour demand schedule: Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

69 Keynesian Labour Supply Equation (Contd.): We have now derived the perfectly elastic Keynesian labour supply equation through micro-founded optimization exercises of agents. There is one problem though: the labour supply function derived by us is horizontal at a given real wage rate, not at a nominal wage rate. In fact as long households/unions care for real values of goods and services, the labour supply decisions will depend on real wage rate (not withstanding its slope) - unless we impose some additional behavioural assumptions on part of the households (e.g., money illusion; constant price expectations etc.), which are not derived from the utility maximization exercise of rational agents. In other words, the Keynesian system - as is specified in the macro equations - cannot be fully retrieved from the optimizing behaviour of perfectly rational agents. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

70 Keynesian Labour Supply Equation: Reference Ben Heijdra: Foundations of Modern Macroeconomics, Chapter 8, pages Note: All references are meant as supplementary readings; not as substitutes for lecture notes. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

71 Micro Foundations: Classical Labour Supply Function and Consumption Function We now turn to the micro-foundations of the Classical labour supply schedule and the consumption function in the Classical/Keynesian system - both of which are based on households optimization problem. Recall that the households take decisions about: their labour supply (which entails an optimal labour-leisure choice) their consumption demand (which entails an optimal consumption-savings choice) The households also decide how much capital to supply, but we shall assume that here the choice is trivial: as long as the rental price of capital is non-negative, they inelastically supply the entire stock of captial that they own. The fact that households care about savings (i.e, current consumption foregone) implies that households live at least for two periods. Hence households optimization problem has to be defined over at least two periods. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

72 Micro Foundations: Household Side Story In what follows, we shall assume that there are H identical households in the economy. Each household consists of a single member who lives exactly for two periods. The agent member of a household works in the first period of his life and is retired in the second period of his life. Thus he earns a wage income only in the first period of his life, although he may earn other forms of non-wage incomes (rent, interest or dividend (if possible)) in both periods of his life. The agent has one unit of time endowment in the first period - which he distibutes between leisure (ˆL h ) and work (1 ˆL h ). Households have an innate preference for current consumption (C 1h ) vis-a-vis future consumption (C 2h ) which translates into differential weightage to the utilities in the two time periods. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

73 Household Side Story: Labour-Leisure Choice & Consumption-Savings Choice Accordingly, we can define the utility maximization problem of agent/household h as: subject to, Max. log(c 1h) + β log(c 2h ) + v(ˆl h ); 0 < β < 1 {C 1h,C 2h,ˆL h} (i) PC 1h + S h = W (1 ˆL h ) + R 1h ; (ii) PC 2h = R 2h (1 + r) S h + Π 2h. Here R 1h and R 2h denote his total non-wage income in period 1 and 2 respectively, while Π 2h denotes his non-wage earnings other than interest income (e.g, rent or dividend) in the 2nd period. Combining the budget constraints for period 1 and 2, we get the lifetime budget constraint of the agent as: PC 1h + PC 2h (1 + r) = W (1 ˆL h ) + R 1h + Π 2h (1 + r). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

74 Household Side Story: (Contd.) We now maximise the agent s utility function with respect to his life-time budget constraint. This yields the following FONCS: (i) C ( ) 2h = β (1 + r) ; C 1h P P 1 (ii) = P C 1h.v (ˆL h ) W. Notice that the term 1 ( ) on the RHS of equation (i) captures the (1 + r) P P relative price of C 2h in terms of C 1h : A rise in r means that for same units of consumption foregone today, one gets more comsumption tomorrow; so the relative price of C 2h in terms of C 1h would be lower; Likewise, a rise in means that by sacrificing the same units of P P consumption expenditure today, one could buy more consumption goods tomorrow; so once again the relative price would be lower. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

75 Household Side Story: (Contd.) Then using FONC (i) in the life-time budget constraint of agent to eliminate C 2h, we get the reduced-form life-time budget constraint in terms of C 1h and ˆL h alone, given as below: ( ) ( ) W W Π (1 + β)c 1h + ˆL h = + R 1h + 2h, P P P (1 + r) }{{} ˆR h ( ) R1h Π where ˆR h + 2h represents the present discounted P (1 + r) P sum of the houshold s non-labour income in real terms (evaluated at current price). The household treats ˆR h as well as W as exogenous (i.e., not being P affected by the household s choice of various household specific variables). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

76 Household Side Story: (Contd.) At the same time, using FONC (i) in the utility function fo the agent, we got the reduced-form utility function in terms of C 1h and ˆL h alone, given as below: [ ( ) ] Ũ(C 1h, ˆL h ) = log C 1h + β log β (1 + r) C 1h + v(ˆl h ) P P = K + (1 + β) log C 1h + v(ˆl h ), [ ( )] where K β log β (1 + r) P P is treated as exogenous by the household. These reduced from utility function and reduced form budget equation allows us to do the analysis in terms of two variables (C 1h and ˆL h ) instead of three (C 1h, C 2h and ˆL h ), although the choice of C 2h is implicit in the choice of C 1h (through FONC (i)). Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104

77 Household Side Story: (Contd.) The advantage of reducing the problem from a three-variable format to a two-variable one is that we can now use an indifference map to charactise the optimal solutions for C 1h and ˆL h (which are the two variables that we are primarily interested in here). We are now going to use the two reduced-form expressions to first derive the optimal labour-leisure choice of the household and then the corresponding optimal consumption-savings choice. Das (Lecture Notes, DSE) Macro Jan 17-Feb7; / 104