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________________________________________________________ Contents (A) The Fisherian Vision The "Fisherian" approach is so named because of the fundamental contributions of Irving Fisher (1907, 1911, 1930), but it might as well be called the "Lausanne" system as it exhibits the same properties and structure of the general equilibrium systems of Léon Walras and Vilfredo Pareto. We must not forget that, as Morishima (1977) reminds us, that perhaps uniquely among all pre-1920 economists, Walras (1874) did attempt to incorporate money, capital and growth into his story in a "wholly Neoclassical" manner. But Walras's efforts were tentative and subsequently ignored. What Fisher did, particularly in his 1930 work, was to return to Walras's task in the context of a simplified general equilibrium model. Fisher treated "macro" as merely "aggregated micro", a jazzed-up "G.E." system (we must recall that when Frisch (1933) invented the term "macroeconomics", he was including the Walrasian economy-wide system of markets into his definition). Consequently, the Fisherian macromodel is a "real" system, i.e. money remains neutral, Say's Law is held to be true, thus there is no possibility of aggregate demand exceeding or falling below aggregate supply. Yet what was the purpose of building a "macromodel" if all it turns out to be is the old Walrasian G.E. model with all the trappings of being "static" and "equilibrium"-based? How is this supposed to explain movements in output as a whole? Well, it will explain these in a very simple way: namely, it asserts that macrofluctuations must arise from elswhere. This may seem like a anti-climatic conclusion, but it is not ducking the issue completely. What the Fisherian approach asserts is that macrofluctuations will never arise from within a perfectly working system. If we see movements in macro variables like aggregate output, the price level, or interest rates, then one of the following must be the cause:
The Fisherian macromodel will be discussed in more detail in this next section, but we already see its essential contours. There is a "real" economy, governed by "real supply and demand" which will determine the aggregate output level, the aggregate (full) employment level and the real (i.e. relative) prices of goods and factors. This is an equilibrium system which, bar any "real" changes in the primitives, remains comfortably stable. Changes in money will not change anything but the absolute price level. Thus, the actual cause of fluctuations in output and employment must arise because of factors which are effectively outside of this system. They arise from money illusions, badly-placed expectations, miscoordination and institutional rigidities, etc. that come into play during what would otherwise be a smooth and natural adjustment process, whether this adjustment is in response to nominal changes (as in (3)) or real changes (as in (4)). The economy will not "generate" these fluctuations by itself. Naturally, the Fisherian model is not wholly or only Irving Fisher's. The contributions of Gustav Cassel (1918), Alfred Marshall (1923), Arthur Pigou (1927, 1933), Ralph G. Hawtrey (1913, 1926), Dennis Robertson (1926) and John Maynard Keynes (1923, 1930), are all really in this Fisherian tradition. They all looked for their explanations for fluctuations in real output in exogenous frictions, rigidities and imperfections in real variables or else confined their explanations of the cycle to fluctuations in the absolute price level. What we (and almost everybody else) calls the Neoclassical macromodel refers inevitably to the Fisherian version of it. It may not seem like a very interesting story, but by staying close to general equilibrium theory, its Neoclassical credentials cannot be doubted. The question is whether this is too high a price to pay. Cannot one create a thoroughly Neoclassical macromodel which says a little bit more about fluctuations than that? One famous but ill-fated attempt to do so was that of Friedrich A. von Hayek (1929, 1931), which we treat elsewhere. The (Fisherian) Neoclassical model of the macroeconomy has what may be regarded as a "supply-determined" equilibrium. Indeed, the first step of Neoclassical theory is to recognize that there are scarce factors that need to be efficiently allocated to satisfy as many wants as possible. Thus, the crux of Neoclassical theory is to take a given endowment of factors and, by a process of successive substitution, determine the most efficient allocation possible. The operator is the price system - or, in a single market, using price-sensitive demand and supply functions to determine the equilibrium price that will clear it. To understand the major relationships, let us the following "catena" or quick summary of the essence of the Neoclassical macromodel:
The essential features of the Neoclassical macromodel are shown diagramatically in Figure 1, with causality running from Quadrant I (upper right) to Quadrant III (bottom left).
In order to work through this diagram, let us then begin with the first line of the catena, the factor markets. Let w/p be the real wage (where w is nominal wage and p is the price level). Let N be labor. Then we are allowed to write out the labor demand function as:
where dNd/d(w/p) < 0 so that labor demand is a negative function of the real wage. This relationship arises from the substitution by profit-maximizing firms between labor and other factors. At high real wages, firms will opt for less labor-intensive techniques and at low wages, they will opt for more labor-intensive techniques. So, if labor is relatively more expensive, the demand for labor declines and thus, the labor demand curve is downward sloping, as shown in Quadrant I of Figure 1. Let us now turn to labor supply. Here we have:
where dNd/d(w/p) > 0 so that labor supply is a positive function of the real wage. The upward-sloping labor supply function in Quadrant I of Figure 1 also arises from substitution - this time between work and leisure on the part of the household. The greater the real wage, the more labor is supplied. The equilibrium in the labor market and is given by:
which gives us the equilibrium real wage (w/p)* and the equilibrium level of employment (N*), a shown in Quadrant I of Figure 1. Given these conditions, then by total differentiation we know that:
i.e. nominal wages (w) are fully flexible and will accompany changes in the price level (p) by the same proportion in order to maintain (w/p)* and, by extension, N*. This flexibility need not be necessarily instantaneous - adjustment can take time - but it is certainly the "long-run" tendency. In sum, in the long run, employment does not change with a different price level since nominal wages will change proportionately. Thus, we have obtained the first line of the catena and the first quadrant of Figure 1. We have also, by extension, obtained the second one as well. Proposing a short-run production function Y = ¦ (N), then given N* from the labor-market clearing we just derived, we obtain, immediately, the level of aggregate supply, Y*. This is shown in Quadrant II of Figure 1. Now, turning to the goods market in Quadrant III of Figure 1, we notice that the aggregate supply curve is horizontal (or vertical, in normal perspective). In other words, aggregate supply, Y*, is derived entirely from factor-market clearing - thus the output of goods in an economy is wholly "supply-determined". The greater the level of employment, N*, the greater the level of output, Y*. Thus, the supply of goods enters the goods market already determined and is invariant to anything that happens within the goods market alone. Now we turn to the third line in the catena. The given output Y* will be factor income which is, in turn, consumed, saved or taxed away, thus:
Aggregate demand, Yd, however, is composed of consumption, investment and government expenditures, thus:
Thus, for there to be an equilibrium in the goods market, for aggregate demand to be equal to aggregate supply of goods (Yd = Y), then it must be that C + I + G = C + S + T or, cancelling out C and flipping G over:
Thus the condition for equilibrium in the goods market is that investment be equal to savings -- where savings here is defined as private sector saving, S , and public sector saving, (the excess of government revenue over expenditures, or T-G). This is equivalent to the loanable funds theory which says that the supply of loanable funds (public plus private savings) be equal to the demand for loanable funds (investment) in equilibrium. What brings this equilibrium about? Movements in the rate of interest. The interest rate adjustes to bring Yd in equality with Y (or, equivalently, I into equality with S). This is in stark contrast to the Keynesian multiplier where it is output itself that adjusts to equate investment and savings. Thus, we shall argue that interest in the Neoclassical model does not affect aggregate supply, but rather affects only aggregate demand. Given that the rate of interest affects aggregate demand, how does it do so? The most straightforward channel is through investment. This is essentially the flow relationship proposed by Irving Fisher (1930) in his "second approximation". The reasoning for the negative relationship between investment and interest rate is given in more detail elsewhere. Thus, investment demand is negatively related to the rate of interest, I = I(r) where I(.) < 0. However, there is a second relationship between r and aggregate demand - namely through consumption and savings. Via Fisher's (1930) "first approximation", we obtain the result that C = C(Y, r), i.e. consumption is a function of income and interest and that CY > 0 and Cr < 0, i.e. increases in income raise consumption whereas increases in interest reduce it. In other words, as interest rate rises, the desirability of saving is higher. Thus, in our general consumption function, C = C(Y, r), we have it that Cr < 0. Thus, we have the main relationships needed for aggregate demand, i.e.
As noted, Cr < 0 and Ir < 0, thus the aggregate demand function is downward sloping with respect to interest. Aggregate supply is invariant with interest. Thus, the third line of the catena is fulfilled by recognizing that Yd = Y will be equated by movements in the rate of interest. All this is shown in Quadrant III of Figure 1. But everybody knows that the interest rate is determined in the market financial assets. How is this related? The adjustment mechanism of the goods market should be expanded upon as it relates to Fisher's (1930) theory of loanable funds. This can be verified by considering the firm's investment decision and the household's consumption-savings decision simultaneously. "Loanable funds" are demanded by firms who need it for investment and loanable funds are supplied by households who need some place to put their savings. If a firm demands loanable funds, it will issue (i.e. supply) bonds; if a household supplies loanable funds, it will seek to buy (i.e. demand) bonds. Thus, I = Bs and S = Bd (note: Bd and Bs must be flow terms, thus they are not the demand and supply of the stock of bonds, but rather the demand and supply of new bonds). The relationship between the three markets is depicted intuitively in Figure 2. If interest clears the loanable funds (e.g. is at r* in Figure 2) market, Bd = Bs, then obviously it is equivalent to saying that I = S and therefore Yd = Y. If interest is too low (e.g. at r1 in Figure 2), so that there is excess demand for goods Yd > Ys or, equivalently, excess investment demand, I > S, which implies in turn that there must be excess bond supply, Bs > Bd. The reverse applies if interest rates are too high (e.g. at r2).
Applying standard market-clearing arguments for the loanable funds market, we can say that if r is too high (e.g. at r2), then the interest will have to fall to equilibriate the loanable funds market (Bd=Bs) and so, by extension, the goods market (Yd=Ys). Step-by-step, the fall in interest leads to an increase in investment demand and an decrease in savings via the effect on intertemporal allocation of consumption. These effects are equivalent, in Neoclassical theory, to a reduction in the demand for loanable funds and an increase in their supply. Note that in this simple model, the change in the rate of interest, r does not affect aggregate supply. We could posit that it does - by some argument that labor supply or labor demand varied with r. But that would complicate our story somewhat -- and our original equations for Nd and Ns do not have r as arguments. Thus, in this simplified model, aggregate demand does all the adjusting. Aggregate supply is "predetermined" in the factor markets. Finally, we should note that we have assumed that household savings are equivalent to the demand for loanable funds without considering that they may desire to place their savings in other assets (such as money or goods themselves). As Keynes (1936) demonstrated, when these other assets are taken into account, there will some quite important modifications to our conclusions. But where is money anyway? The Neoclassicals appended this on top of their existing construction. In the Cambridge cash-balance theory developed by Marshall (1923), Pigou (1917) and Keynes (1923), the interest rate and output will themselves feed into the money market by influencing the demand for real money (Md). i.e.
where Lr < 0 and LY > 0 (the why of the first relationship is hard to reconcile with the loanable funds theory given earlier, but the Neoclassicals thought it logical, although Keynes, who developed this theory, saw it as the first approximation to his liquidity preference theory). If real money demand, Md, is determined by r and Y and r and Y are determined in the real markets for goods and factors, then if these are unchanged, then Ld will be unchanged. We are also given and (exogenous) real supply of money:
where M is the nominal money stock. For there to be money market equilibrium, Md = Ms, or:
However, as money demand is fixed externally and M is controlled by the central bank (an assumption), then all that remains to adjust this market into equilibrium is the general price level, p. The money market is illustrated in Figure 3, which is drawn in nominal terms, thus we have nominal money demand curve pMd = pL(r, Y) and nominal money supply schedule pMs = M where nominal money stock M* is given exogenously and thus only p* is left to be determined by equilibrium condition pL(r, Y) = M.
Having determined the price level, p*, in the money market, do we need to go back and change anything we had before? No. The determination of money market has no impact on the determination of the real variables w/p, N, Y, r and B. Now, we could speculate that a change in the price level might affect real wages (w/p) - the only point p enters anywhere else in the system - but in fact, as we have seen, nominal wages (w) are assumed to be entirely flexible and so will adjust proportionately in the factor markets to maintain the market-clearing real wage (w/p)*, i.e. dw/w = dp/p. This implies that money is neutral, i.e. changes in the supply of money do not affect real variables. Notice that the real money demand function Md = L(r, Y) can be represented via two famous alternative ways. For instance, in the Cambridge cash-balance approach, we can posit the shape Md = kY (where the Cambridge constant, k, varies negatively with r). Alternatively, we can use Irving Fisher's (1911) form and represent money demand function as Md = (1/V)Y (where velocity, V, varies positively with r). Thus, money market equilibrium condition in Figure 3, that Ms = Md or M/p = L(r, Y) can be rewritten as:
which is the "Cambridge equation" or:
which is the Fisherian "equation of exchange". The money market equibrium depicted in Figure 3 is compatible with either of these. Because of the manner in which w/p is completely flexible and given the other relationships we have outlined here, there are three primary features of the Neoclassical model which are preserved here. Firstly, there is a strict dichotomy between the money market and the real market - what happens in the former does not affect the latter. Secondly, and by extension, money remains neutral, i.e. a rise in the supply of nominal money (M) will not affect any other variables other than the price level. A third, related characteristic is that we obtain the conclusion of the Quantity Theory of Money strictly: a change in the money supply will change price level proportionately but not anything else. The summary of the main points of this simplified version of the Neoclassical model are then the following.
We ought to emphasize a fifth feature of this model, namely that the foundations of the model are firmly set in Neoclassical micro-theory: (1) output, factor employment and investment are derived from firms' profit-maximizing decision (2) labor supply, consumption and savings are derived from household utility-maximization and (3) market-clearing conditions are imposed on all markets. Let us assume there are H households and F firms. We will focus on the representative household (the "hth" household) and the representative firm (the "fth" firm), before aggregating. Looking at the micro-level derivation is useful as it forces us to be a little clearer about what is being assumed in the model. Let us turn to the firm's decision, as this brings out the first problem: namely, whither capital? The only factor we referred to in our model was labor. Yet, as we know from typical Neoclassical theory, there must be other factors around which the firm must handle. One of these is capital. Now, since most macroeconomics operates largely in the short period, we usually sweep capital under the rug. However, whatever result we get for equilibrium labor employment and wages, there must be some corresponding equilibrium level of capital and employment and rate of return on capital. However, if the rate of return on capital is equal to the rate of interest (a common assumption in equilibrium, otherwise Wicksellian demons are unleashed), then the rate of interest is determined as the dual in the first line of our earlier catena. Then how does it turn up again in the third line to be determined by aggregate demand and supply? Is the interest rate determined in the first line the same as the interest rate determined in the third line? This is not evident. However, having said that, we must remember that capital and investment are two different things (as emphasized by Fisher (1906)): the first is a stock term, the second is a flow term. Reconciling these has been a perennial headache in economic theory. This difficulty can be disposed of by one of the following three means. The first is by invoking the principle of marginal adjustment costs to capital employment. This is alright, but runs up against a little problem. In particular, if a firm cannot reach its "optimal" amount of capital, then, conversely, it ought not to hire the "optimal" amount of labor corresponding to that capital level but rather a different amount that corresponds to the investment level. We are not guaranteed full employment in this case. The second resolution is to invoke labor supply growth, so that we can choose optimal capital-labor ratios, and all investment is merely what is necessary to accompany labor supply growth. In this case, investment flow is eliminated as a decision separate from the capital stock decision, and so the problem disappears. This may work better, but then this is an issue of growth theory, not "macro" theory in the strictest sense. The third, and simplest, resolution, is to just follow Irving Fisher (1930) and assume all capital is circulating, i.e. there is no K. The firm's decision is solely based on I and N. Does this make sense? Indeed, it can, but then we lose the independence of aggregate supply from the rate of interest (i.e.the aggregate supply curve becomes "upward sloping", rather than vertical). Specifically, following Fisher (1930), let us propose that the representative firm faces a "short-run" production function of the form:
where the output of the fth firm is some function of N, the labor inputs employed by that firm and investment inputs. It is assumed that the¦ N > 0 and ¦ NN < 0, i.e. the marginal product of labor is positive but diminishing. Similarly, it is assumed that ¦ I > 0 and ¦ II < 0, so that we have diminishing marginal product of investment. The profit-maximizing firm will attempt to increase the difference between total revenue (pY, the sale of output) and total costs (i.e. total factor payments, wL + (1+r)I where w is the nominal wage and r is the rate of interest). So, letting p denote the profits of the firm, then its problem is:
This yields us a pair of first order conditions:
Now, let us take each in turn. Consider the first one. The term on the left, p¦ N is merely the marginal value product of labor, so this says that the firm will hire labor until its marginal value product is equal to the nominal wage. There are two other ways of writing this. A simple one is to divide by p, so that:
so now the firm hires labor until the marginal productof labor is equal to the real wage, w/p. Alternatively, we can write it as:
where the curious term on the right is the marginal cost of output, i.e. the cost of hiring labor in order to increase output by a unit. To see this clearly, recall that ¦ N = dY/dN, so we can write w/¦ N = dwN/dY, i.e. the marginal cost of output is the increase in the wage bill (assuming w is constant, so wdN = dwN) from a unit increase in output. Thus, the equality above merely says that a firm hires labor (and thus produces output) until the output price (or marginal revenue) is equal to marginal cost. Whatever way we look at it, we see that given (w/p), we can determine the amount of labor demanded by the firm (Nd) via the condition ¦ N = w/p, i.e. assuming this is invertible, we can write the labor demand function as:
The fact that ¦ N > 0 and ¦ NN < 0 implies that d¦ N-1/d(w/p) = d Nd/d(w/p) < 0, i.e.that the firm's labor demand curve is downward-sloping. The aggregate or economy-wide-level labor demand function is obtained by aggregate firm-level labor demand functions. We must assume (it is not evident), that these are sufficiently well-behaved so that these properties are also true for the economy-wide labor demand function. Let us now turn to investment decision of the firm. We can define p¦ I - 1 in Fisher's language as the "marginal rate of return over cost", or in more Keynesian language, the "marginal efficiency of investment", so MEI = p¦ I - 1. Thus, first order condition p¦ I = 1 + r implies that the firm will employ I (i.e. "invest") until MEI = r, i.e. marginal efficiency of investment is equated with rate of interest. Notice that as ¦ II < 0, then as the rate of interest rises, then, to equate r and MEI, it must be that investment declines - thus the negative relationship between investment and interest rate, succinctly, I = I(r) where Ir = dI/dr < 0. Yet we have been a bit fast and loose with this, as is it makes its seem that the investment decision is independent of wages and the labor demand decision is independent of the interest rate. Formally speaking, this will not be the case in this model (and here we enter into more "Hayekian" ground). To see why, notice what our argument has said: if the wage rises, then demand for labor falls. But why? When wages rose, the profits of the firm declined. There are two possible responses a firm can have in order to restore profits: firstly, by the law the "law of increasing costs", it may have decided to cut output as a whole, and higher output as a whole implies less labor hired, thus the demand for labor declines; secondly, by the "law of technical substitution", a firm may not necessarily change output as a whole, but merely restore profits by changing production methods to a more capital-intensive technique, i.e. moving away from labor and towards investment -- what Hayek (1931) called a "lengthening of the production process". In general, both these effects will probably come into play in order to explain the decline in labor demand when wages rise (details on these effects are discussed in our survey of production theory). In either case, note, the demand for investment goods is affected, so we cannot say that a rise in wages "only" affects labor demand. It will also affect investment demand and thus both the aggregate demand and aggregate supply side of the economy. Similarly, a rise in the rate of interest rate may lead to a reduction in investment demand and a rise in labor demand, and thus employment. So interest rate also affects both the aggregate demand and aggregate supply of the economy. The simplistic macromodel we described earlier does not capture this -- and it is not certain that we are better off now. In principle, then, when we have a Fisherian investment-labor economy, the first order conditions for a maximum imply that at the optimum:
the ratio of marginal products is equal to the ratio of real factor prices (w/(1+r)). The factor price ratio is shown in Figure 4 as the slope of the isocost line. Once (w/(1+r)) is given, the firm automatically knows the amount of labor and investment it will undertake (N* and I*) as the tangency of the highest isoquant and the isocost curve (there is a bit of indeterminacy here, however; see our discussion of the production decision; we would need to use a Paretian general equilibrium model to do this properly).
At any rate, the main lesson is that labor demand Nd is no longer merely a function of the wage, but also of the rate of interest, (1+r). Similarly, investment demand I is not only a function of interest but also a function of the wage rate. The main relationships remain, however: an increase in w/(1+r), which can be due either to a rise in w or a fall in (1+r), will usually lead to a decline in labor demand and/or a rise in investment demand (we qualify this with the term "usually", because there are output effects in this story which may change our conclusions; see our section on production). At any rate, this implies that we still obtain a downward-sloping demand function for labor (as a function of the real wage) and a downward sloping investment function (as a function of the rate of interest). It is just that now we cannot really isolate them from each other. Finally, we should note that the microfoundations of production in the Neoclassical macromodel would get even more complicated if we decided to allow for fixed capital, K, as well. It is for this reason that we are sticking with the Fisherian circulating capital assumption. Let us now turn to the household's utility maximization problem. There are three things we want to obtain here: labor supply, consumption demand and savings. For simplicity, we will do this in separate steps, but in principle, these decisions are all made together at once. The labor supply function is derived from an exercise in utility-maximization on the part of the household. The representative household is assumed to be a rational hedonist - it attempts to maximize what it can consume and minimize the amount of work it has to do. Thus, we can posit a utility function for the household, U = U(C, N), where UY > 0 and UN < 0. UY and UN are the marginal utilities of consuming more goods and supplying more labor, thus our assumption about signs means that consuming goods, C, is utility-increasing while supplying labor, N, is utility-decreasing. The assumption of diminishing marginal utility for nice things implies that the utility function posses the following second derivatives UYY < 0 and UNN > 0. An agent buys goods with income obtained from selling his labor on the factor market (a household can also have endowments of goods or other factors, but we shall ignore that here), thus it faces two parameters in its constraint: the prices of goods and factors (p, w) and the factor it is endowed with (the maximum labor supply). Thus, he faces a budget constraint pC £ wN. In order to recognize that there is a maximum amount of labor supply (call it T - a person cannot work more than twenty-four hours a day), we shall change this constraint by adding the value of maximum labor supply (wT) to both sides so the constraint can be rewritten:
so the agent sells his total labor supply, T, to buy goods, C and "leisure" (T-N). The budget constraint is depicted in consumption-leisure space in Figure 5 by the straight line that emanates from the consumer's endowment, which is at (0, T), with slope -w/p. The agent maximizes his utility subject to this constraint:
s.t.
So, setting up a Lagrangian, and solving for the first order conditions for a maximum:
where l is the Langrangian multiplier. This solution can be rewritten:
i.e. the household will choose between desirable consumption of goods and irksome supply of labor (or, equivalently, desirable leisure) until the ratio of marginal utilities of each is equal to the real wage. This is shown in Figure 5 at point E. Notice, that at the utility-maximizing position E, the household will choose to supply N* amount of labor (and thus consume (T-N)* amount of leisure) and enjoy C* amount of consumption. He achieves utility level U(C*, N*) which is higher than the utility level he would obtain if he was forced to stick to his endowment (U(0, T)).
What will raising the real wage, w/p, imply for the supply of labor, N? The substitution effect argues that the greater the real wage, the more costly leisure becomes relative to consumption foregone and thus the agent will supply more labor. The income effect argues that the greater the wage, the agent can buy the same goods with less work and thus the more appealing leisure becomes (thus the agents supplies less labor). Thus, it is ambiguous. Nonetheless, the common reconciliation is to assume a backward-bending labor supply curve so that at low wages, the substitution effect dominates and at high wages, the income effect dominates. For simplicity, we shall argue here that the substitution effect dominates everywhere so that labor supply increases with the real wage. Let us now turn to the consumption-savings decision, Fisher's (1930) "first approximation". We wish to obtain the result that C = C(Y, r), i.e. consumption is a function of income and interest and that CY > 0 and Cr < 0 or, equivalently, as S = Y - C, then we want a function S = S(Y, r) where SY > 0 and Sr > 0, so savings is a positive function of interest. This relationship is derived via intertemporal optimization of consumption on the part of the household. A two-period illustration of the concept is straightforward. Suppose a household has a lifespan of two periods, receives income in two periods (Y1, Y2) and must make a decision on how much to consume in each period (C1, C2). We can thus posit that it obtains utility from consumption in the form U = U(C1, C2). Although we will not specify a particular functional form is given, we should note that we normally assume that there is positive time preference so that a particular amount of future consumption is worth less in terms of utility than that same consumption amount in the present. If we have perfectly working financial markets, then the household can lend some of its present income to increase future consumption or borrow from its future income to increase present consumption. In the first case, it can curtail present consumption C1 and save an amount S = Y1 - C1 in the first period and receive it back with interest in the second so that second-period income is augmented by (1+r)S, i.e. second period income is Y2 + (1+r)S. If it borrows from the future to the present, all we have to do is consider it "negative" S. Thus, the household faces the following constraints in each period:
so, solving both for savings and equating them, we can collapse this into one "intertemporal" constraint:
which can be interpreted as saying that the present value of the stream of consumption cannot exceed the present value of the stream of income (where, by present value, we see that future consumption and income is discounted by the rate of interest). This is depicted in Figure 6 by the straight line emanating from the intertemporal endowment (Y1, Y2) with slope -(1+r). The household faces the following intertemporal optimization problem:
where, setting up a Lagrangian, we get the following first order conditions:
where U1 and U2 are the marginal utilities of present and future consumption respectively. The solution implies:
i.e. the household will allocate consumption in both periods until the ratio of marginal utilities is equal to the interest rate (or, in other words, until the desirability of present consumption relative to the future is equal to the foregone interest). The extrapolation of Fisher's two-period story to more than two periods is accomplished in the "Life Cycle Hypothesis" (LCH) of Modigliani and Brumberg (1954). Figure 6 illustrates the intertemporal optimization problem. Notice that C* is the chosen intertemporal allocation and Y is intertemporal endowment - thus this individual is a net saver with savings equal to S = Y1 - C1 > 0. Notice that the utility it attains U(C1*, C2*) is greater than U(Y1, Y2), the utility it would receive if it had stuck to its original intertemporal endowment and not saved or borrowed anything at all.
It is not difficult to see that present consumption falls when the rate of interest rises. In other words, as interest rate rises, there is an incentive to save more (i.e. reduce consumption today further in favor of consumption tomorrow). Naturally, there are income effects that need to be accounted for, but we shall presume that this substitution effect dominates. Thus present consumption, C1 is negatively related to the rate of interest, r. Thus, in our general consumption function, C = C(Y, r), we have it that Cr < 0 or equivalently, S = S(Y, r) and Sr > 0, which is exactly what we wanted. It seems that the Neoclassical macromodel stands firm on its own foundations - or does it? We have already hinted at a few problems, let us pursue these for the moment. To see the first difficulty, let us "close" the model by placing all the markets together in one gigantic Walras's Law constraint (we are suppressing prices):
where we have represented four market: money, bonds, goods and labor respectively. They must add up to zero by Walras's Law. We could represent (Yd - Ys) as (I - S) as they are equivalent (assuming government's budget is balanced). The Neoclassical story is then as follows: w/p clears the labor market (so (Nd - Ns) = 0), the price level clears the money market (so (Md - Ms) = 0), leaving the interest rate to clear the goods market (so (Yd - Ys) = 0) and hence, residually by Walras's Law, the interest rate clears the bond market as well (so (Bd - Bs) = 0). All is fine in the Neoclassical kingdom. But this statement allows all markets to affect each other: if there is excess demand in the goods market (Yd > Ys), then somewhere something must be in excess supply. This is simple Neoclassical logic: if something more is demanded and nothing else is supplied, then that demand must come from some deficiency in demand somewhere else. Thus market disequilibria spill over into each other. However, by assumption of dichotomy between money and real sides, we cannot have it that money market disequilibrium affects anything. Thus, we need it that (Md - Ms) = 0 at all times. Thus, the Walras's Law constraint is reduced to:
which is what Oskar Lange (1942) and Don Patinkin (1956) defined as "Say's Law", i.e. the demand for "real" things must equal the supply of "real" things, with no spillovers into the nominal things such as money. Is this a problem? Yes. We have eliminated the manner in which we get the determination of the price level from the money market. Fisher, Pigou, Cassel and company assumed that the price level would adjust by standard Neoclassical market-adjustment arguments and thought no further about it. But the "standard arguments" in Neoclassical theory rely on the concept of substitution between items - i.e. spillovers into other markets. Yet there is no substitution here because the money market is completely isolated from all other markets: money demand is "fixed" from outside and nominal money supply is also fixed. If money demand, for some reason, is different from money supply, there is no straightforward process of "substitution" that leads to the change in price level. The equilibrium condition, L(r, Y) = M/p or Md = Ms is, by the dichotomy assumption, a permanent condition that applies at all times. The problem with this is not only do we not know how the price level is supposed to adjust, but then the price level is actually indeterminate. If money demand always equals money supply, then the price level can be anything -- the money market will still clear. Furthermore, as Patinkin (1956) argues, the mechanisms of the Quantity Theory are violated. How? The Quantity Theory argued that if money supply rises, then prices would rise and the original Fisher (1911) argument was that the excess supply of money led to an excess demand for goods and that would raise prices which would in turn reduce the real supply of money. As excess real money supply fell, excess demand for goods would also fall. The new equilibrium would arise with output unchanged, but price level higher. But if money market is prevented from spilling over into the goods market - as the imposed dichotomy implies - how is this process supposed to happen? This failure was pointed out by Patinkin (1948, 1956) who made the necessary corrections in the Neoclassical model to account for it. His correction was done by violating the dichotomy argument directly. We need nominal variables to somehow affect aggregate demand. He proposed this via a "real balance" or "wealth" effect on consumption. Patinkin's argument was that real money balances, M/p, enter the consumption demand equation so C = C(Y, r, M/p). An increase in the money supply increases real wealth (M/p), which thereby increases C and consequently Yd -- that gives us the necessary excess demand for goods to make the Quantity Theory work: with excess goods demand unmet, the prices for all goods rise, M/p falls back down and so does C and Yd back to Y. Neutrality still holds, but dichotomy is broken. Nevertheless, let us presume now that the money market does clear nicely. A second difficulty nonetheless arises. Consider the possibility of goods demand being so low that it intersects aggregate supply at a negative interest rate. In principle, negative prices are difficult to fathom in general so, in principle, the argument is usually made that the good in question becomes "free". In other words, if the demand and supply for peanuts intersect at a negative price, then we impose the "complementary slackness" condition that price of peanuts is zero, and thus we have, in "equilibrium", an excess supply of peanuts, i.e. peanuts are free. That is alright for a single good, but in the case of goods market, a negative interest rate cannot be rescued by assuming that interest rates are zero and thus "all goods are free". The implication is that if aggregate demand is low enough, the entire economy would be "free". Patinkin (1948) also envisaged that the real balance effect would save Neoclassical macroeconomics from this possibility: the excess supply of goods at zero interest would spillover into an excess demand for money. This would lower the price level and thus lead to an increase in real money supply. By the real balance effect, aggregate demand would rise again - bringing us back into positive interest (and thus an appropriately "non-free" economy). A third problem arises when examining our Walras' Law constraint. Suppose the interest rate clears the goods market, but, concurrently, that we still have an excess demand for bonds. This is possible - but it would imply we had an excess supply of labor. Thus, we can have unemployment even when the goods market clears if we allow spillovers from bond to labor markets. Thus, we can have "too high" interest rates (for the bond market) and "too high" wages simultaneously. Does this make sense? This is not very intuitive - as it implies that firms are somehow substituting labor for bonds. This is not insensible if we think of a firm somehow deciding to use their wage payments to purchase bonds instead (because they offer a "higher return" or something), but the intuition disappears once we ask why firms are reducing their demand for labor when the rate of return on bonds is already "too high"? What does this mean for capital-labor substitution? The intuition is not clear, but it is mathematically allowable in the Walras's Law constraint. Furthermore, if the goods market is in equilibrium but the bond market is not (spilling over into labor), what is the rate of interest which equilibrates both bonds and goods? What are the mechanics implied here? An excess supply of bonds can only be moved back to equilibrium if the interest rate falls, but the interest rate will not fall if the goods market is in equlibrium. What now? If alternatively, we had such a large excess supply of labor so that both bonds and goods markets were in excess demand, then we would be simultaneously saying that the interest rate is expected to rise (from the goods market disequilibrium) and it is expected to fall (from the band market disequlibrium). Is this contradictory? One can suspect that these peculiar predicaments might lie in the fact that the Neoclassical model places together, in the same constraint flows (goods and labor demands) and stocks (bond demands). This could be corrected by simply realizing that they cannot be together as they have different time references. However, many economists - notably John Hicks (1939) and Don Patinkin (1956, 1958) - are adamant about "everything depending on everything else" and permitting stocks and flows to be in the same constraint by just adjusting a little bit. Hicks does make a point to speak of loanable funds entering the constraint in the form of flows (i.e. "lending and borrowing", thus "credit" instead of "bonds") but this does not remove the difficulty of contradictory interest rate movements in the case of an excess supply of labor. Lawrence Klein (1950) however, enters the fray here and surprisingly concludes that actually the bond/credit market has absolutely nothing to do with the rate of interest. The rate of interest, he claims, is determined by investment and savings, period. All the credit market does is determine the price level. This is a bold statement, but if we accept it, then what role remains for the money market? There are two resolutions. The simplest is to claim that we were wrong to differentiate between the credit/bond market and the goods market to begin with. In other words, we placed them separately in the Walras's Law constraint, but that this was illegitimate. Goods demand is bond supply by definition and goods supply is bond demand by definition. There is no "mutual mechanism" between them. The interest rate is determined in one place alone: the goods market. The bond or credit market (stock or flow) is merely another term for "goods market" - they are one and the same thing. Thus, we either include the goods market or the bond market in the constraint, but not both simultaneously. This resolution is not particularly troublesome since we introduced bond demand and supply in precisely this manner. But while it is not counterintuitive, it is not that obvious either: the demand and supply of credit can indeed be reconciled with investment and savings, but are they really the same thing exactly? If so, then we would be arguing in the "real funds" framework of Fisher's (1930) theory, but have precluded the plausible Wicksellian modifications brought in by Robertson (1937, 1940) and Ohlin (1937). A second resolution is to follow Sargent (1979) and remove the simulteneity implied in our Walras's Law constraint and deal with the model in a block recursive form. In short, we impose two Walras's Law constraints, namely:
so that the bond/goods market determines the interest rate and the labor market determines the real wage rate alone. The interest rate, then, is argued to be absolutely determined by the bond market - and so, as the previous analysis insinuates, bond and goods markets are both equilibrated by a single, common interest rate. However, the problem arises that now the labor market must clear at all times (no disequilibrium being implied by the constraint) and thus, analagously with the problem we faced in the money market, the real wage becomes indeterminate! The way out of this would be to include capital - which is what labor substituted for in the first place, was it not? That long-forgotten, out-of-the-system variable must now be included again. In this case, we must rewrite the system as follows:
so the equilibrating mechanism between the factors is (w/p)/r, the ratio of factor returns. In this case, the real wage is still indeterminate, only relative factor returns are determined in equilibrium. But we can now pin down the real wage by recognizing that r falls out of the second bond/goods constraint. Thus, second constraint can be used to pin down a value for r which will, in turn, pin down a value for (w/p). The first equation gives us (w/p)/r, the second r, so using all this, we can determine (w/p) exactly. If, in addition, we can also pin down p in the money market (somehow), then we would have the additional benefit that the nominal wage (w) would also be determinate. The system, then, would become entirely solvable. Or is it? Now we have capital stocks again - and again we must be careful of its relationship with investment. Recall that investment is positive only if capital is not at its optimal level, K* (otherwise there would be no point in investing). Imposing the adjustment cost story does not solve things entirely: in a period where investment is positive, capital would be below optimal capital stock. By the microeconomic theory of the firm, the cost-minimizing choice with capital constrained below K* would be that labor demand would be above optimal labor, L* at the given real wage rate. In fact, a simple isoquant exercise could demonstrate that with constrained capital stock, labor is hired where ¦ L/¦ K < (w/p)/r so that the real wage is actually too high! So much for factor-side equilibrium. As long as investment is positive, things go awry in the factor markets. Attempts at resolutions have been made in investment theory, but they have not been solved to everyone's satisfaction. However, the incorporation of growth theory of the Solowian brand in Neoclassical macromodels has enabled modern macroeconomists to largely circumvent this problem. Note that if labor is growing at some "natural rate", then this "disequilibrium" arising from stock-flow confusion is eliminated almost completely as movements in capital will be accompanied by movements in labor. Thus, we can speak of steady market clearing factor prices and an optimal capital-labor ratio which can be largely reconciled with investment flows. It is a simple maneouvre, but one the old Neoclassicals did not really come up with. The cost of incorporating growth is that the idea of "investment" as an independent behavioral phenomena becomes gradually eclipsed by complete determination by household savings and growth - making rich financial markets and independent firm investment decisions an almost superfluous consideration. In sum, there are particular problems in the formulation of the static Neoclassical model as we have presented it. Of course, one may argue, together with John Hicks (1939: p.154), that these are not real issues: in a general equilibrium system, real wages, interest rates, prices and quantities will be determined somehow, it does not particularly matter where. While this may be true, it does not lend us any insight into the workings of the macroeconomy. What is specifically of interest here is whether interest rates or real wages or whatever equilibrate particular markets. Furthermore, we must note that we are describing a rather simplistic Neoclassical system rather a fully-fledged Walrasian general equilibrium system. Both share the most important features of Neoclassical economics, but there are important points of departure between these as well.
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