Macroeconomics

A SMORGASBORD OF TOPICS IN MACROECONOMIC THEORY - Niloy Kumar Bhattacherjee CONTENTS Part A: Theories of Economic Growth 1.Investment and Capital Formation 2. The Domar Model 3. The Solow Model Part B: The Liquidity Trap 4. The L-M Scehdule and the Concept of the Liquidity Trap Part C: Inflation Theories 5. The Equation of Exchange 6. The Price Equation 7. Demand Pull Inflation 8. Cost Push Inflation 6 6 7 9 5 2 2 3 © Niloy Kumar Bhattacherjee A. Growth Theories – The Models of Domar and Solow §1. Investment and Capital Formation: Capital formation is the process of adding to a given stock of capital. If the process is considered to be continuous, capital stock may be expressed as a function of time, K(t). The rate of capital formation may therefore be denoted by dK/dt. But this rate is identical to the rate of investment flow at time t, denoted by I(t). Hence we can say that dK/dt = I(t). Conversely, therefore, K(t) = ∫ dK = ∫ I(t) dt. We also use the concept of gross investment Ig = I + δK where δ represents the rate of depreciation of capital and δK, the rate of replacement investment. Using the notion of definite integrals, we may express the amount of capital accumulation during the time interval [0, t] for any investment rate I(t) as ∫ 0,t I(t) dt = K(t) – K(0). From this, we see that K(t) = K(0) + ∫ 0,t I(t) dt which gives us the time path of K(t). §2. The Domar Model: Now we set up the context for the Domar model. A change in investment (∆I[t]) will impact aggregate demand and the productive capacity of the economy. The demand effect of ∆I[t] operates instantaneously through the Keynesian multiplier 1/(1-c) or 1/s so that we can write dY/dt = (1/s) . (dI/dt). …1. Also, we define the capacity-capital ratio ρ = κ/K where K is capital stock and κ is capacity (or potential output flow per year). The capacity effect of investment is measured by the change in the level of potential output the economy is capable of producing. Here we see that with a given capital stock K(t) the economy is capable of producing an annual product or equivalently an annual income of κ = ρK rupees. …2. We also see that since κ = ρK, we have dκ/dt = ρdK/dt = ρI. …3. In Domar’s model, equilibrium is defined to be a situation in which productive capacity is fully utilized, that is Y = κ. …4. Given an initial equilibrium position, the condition (4) will reduce to balancing the respective changes in capacity and aggregate demand, that is dY/dt = dκ/dt. …5. To find a solution, let us substitute (1) and (3) into (5). The result is a differential equation: (1/s) . (dI/dt) = ρI => (1/I) . (dI/dt) = ρs …6. From (6), by integrating both sides, we should be able to find an investment path. Hence ρ ρ ∫ dI/I = ∫ ρsdt => ln | I| + c1 = ρst + c2 => ln | I| = ρst + c => | I| = e stec =Ae st …7 Here c1 and c2 are the usual constants of integration, c= c2 – c1 and A = ec. 2 Now setting t = 0 in (7) we get I(0) = A e0 = A, where I(0) is the initial rate of investment, ρ so that from (7) we have I(t) = I(0)e st …8. This result has a disquieting meaning! In order to maintain the balance between capacity and demand over time the rate of investment flow must grow precisely at the exponential rate of ρs, along a path called “the razor’s edge” by economists of that age. So what happens if the actual rate of growth of investment (r) differs from the required rate (ρs)? Let us see! A growth rate of r implies that I(t) = I(0)ert (3) we have dY/dt = (1/s).(dI/dt) = (r/s).I(0) ert. and dκ/dt = ρdK/dt = ρI(t) = ρI(0) ert. => dI/dt = rI(0)e rt. Hence, by (1) and The ratio between these two derivatives, (dY/dt)/ (dκ/dt) = r/ρs tells us something very important about the relative magnitudes of the demand-creating effect and the capacitygenerating effect of investment at any time t, under the actual growth rate of r. If r > ρs, then dY/dt > dκ/dt, so the demand effect will outstrip the capacity effect, causing a shortage of capacity. This is a very curious result indeed! This result states that if investment actually grows at a faster rate than required (r > ρs) the end result will be a shortage rather than a surplus of capacity. Conversely, if investment actually grows slower than required (r < ρs) the end result will be a surplus rather than a shortage of capacity. This simply means, if we give a free hand to individual entrepreneurs to adjust the actual growth rate r they will almost certainly make the wrong adjustment! Investment must be guided extremely carefully along a growth path called the razor’s edge (r = ρs) because any deviation will produce paradoxical consequences. However, upon changing the contextual parameters slightly, more systematic results may be obtained, as seen in the Solow model. §3. The Solow Model: The Solow model has two fundamental constructs, a production function and a capital accumulation equation. For the first, we assume a Cobb-Douglas form: Y = f(K,L) = KαL1-α whence, putting y = Y/L and k = K/L, we get y = kα. Note that dividing by L gives a per unit of labour or per person measure. Recall from your micro-economics course that profit-making firms solve max K,L f(K,L) – wL – rK where Y = wL + rK. We have w = ∂F/∂L = (1-α)Y/L and r = ∂F/∂K = αY/K. For the capital equation we have dK/dt = sY – dK, where dK/dt is the continuous-time version of Kt+1 – Kt, that is, accumulation per period. We assume that s, the gross investment happens because all workers and consumers save a constant fraction s of their income Y = wL + rK and savings and investments are equal because it is a closed 3 economy. The last term on the right states that capital stock depreciates at a constant rate d in every period. We now have a few assumptions to state: (i) Labour force participation rate is constant and equal to n, (ii) Population growth rate is also constant and equal to n. Note that n = (1/L)(dL/dt) since L(t) = L(0).ent. Now given that k=K/L we have (1/k)(dk/dt) = (1/K)(dK/dt) - (1/L)(dL/dt) and y/k = (Y/L)/ (K/L) = Y/K. Based on this, the capital accumulation equation dK/dt = sY – dK can now be restated in per capita or per labour unit terms as (1/k)(dk/dt) = (sY – dK)/K – n = sY/K – d – n = sy/k – d – n dk/dt = sy – (n+d)k. Hence, restated, the two key equations of the Solow model become y = kα … (9) dk/dt = sy – (n+d)k. … (10) Note that when capital per worker increases, it is called capital deepening, whereas when only capital stock grows, but capital per worker does not because of population growth, it is called capital widening. When capital per worker remains constant, it is called the steady-state. That is, by definition, the Solow steady-state is dk/dt = 0. Using this condition with equations (9) and (10) we can solve for the steady-state quantities of capital per worker and output per worker. dk/dt = sy – (n+d)k = skα – (n+d)k = 0. [1/(1- α)] Hence k* = [s/(n+d)] and consequently y* = [s/(n+d)][ α /(1- α)]. This equation provides one answer to the question – why are some countries rich and others poor? The Solow model states in very certain terms that ceteris paribus, countries that have a high savings-investment rate will tend to be richer. Such countries manage to engage capital-deepening processes to their benefit and this results in more output paer capita. Countries with high population growth, however, tend to spend a higher fraction of savings merely to keep the capital-labour ratio constant. This capital-widening requirement makes capital-deepening more difficult and hence these countries continue to be poorer. To explain sustained growth in per capita income, Solow introduces exogenous technological progress in the form Y = f(K, AL) = Kα (AL) 1-α where the technology variable A is labour-augmenting or Harrod-neutral. Technological progress, which is exogenous, occurs when A increases over time. A situation in which capital, output, consumption and population are growing at constant rates is called a balanced growth path. Using this notion and the concept of exogenous technological progress we can show that along the balanced growth path of the Solow model, output per worker and capital per worker both grow at the rate of exogenous technical change. B. The L-M Schedule and Liquidity Trap 4 Recall that the money demand and supply curves are plotted against interest rate r (vertical) – L, M (horizontal) axes respectively. Money supply is usually shown to be vertical or constant. Demand has the usual downward slope to the right, the last segment being relatively flat. If the supply schedule intersects the demand schedule in this horizontal or “flat” part, a further increase in money supply will not cause a reduction in the market interest rate. Individuals will prefer cash to bonds, so the extra money will not enter the bond market, bond prices will not rise and interest rate will not fall. Hence monetary policy will be rendered completely ineffective. This happens in economic deep depression states. Hence, at this stage, the monetary policy is said to be trapped. Correspondingly, the flat part of the money demand schedule is known as the liquidity trap. Liquidity trap sets a floor below which the market rate of interest cannot fall. 5 C. Inflation Theories Let us begin the discussion of inflation from an unusual perspective – the “Quantity Theory of Money”. This will help us understand the said theory as also the neo-classical approach to inflation before we move to more serious platforms. The Quantity Theory of Money is the name given to the neo-classical hypothesis that the general price level is determined by the amount of money made available by the monetary authorities. We will look at two distinct approaches here – the equation of exchange (developed by Newcomb and Fisher) and the price equation or the Cambridge income approach (a simpler version of the Cambridge cash-balance approach developed by Walras, Marshall, Wicksell and Pigou). 1. The Equation of Exchange In the neo-classical world money is used only as a medium of exchange. Hence it is demanded by people for transactions only. Thus in a sense the demand for money is given by PT, the product of the general price level P and the total volume T of transactions carried out over a year or similar period. PT can also be written ∑piti. The supply of money (or the quantity of money) at any one point in time is M. The supply of money over a period of time (e.g. a year) is MV, where V is the “transactions velocity of money” (simply, the average number of times each piece of money changes hands over the year). Hence the equation is MV=PT … (1) Now, in the long run, say 25 years, the volume of trade shows much steadiness in growth inspite of trade cycles. Similarly, the velocity of circulation depends on the monetary habits of people which do not change very fast. Hence we can rewrite equation (1) as …(2), where the bars over variables denote long-run constancy. Hence, apparently, movements in P (e.g. inflation) originate from movements in money supply M. 2. The Price Equation Here the Marshallian demand approach is used so that the demand for money is similar in nature to the demand for any other commodity. Let the real income of a community in any year be Y. The proportion of Y that the community wishes to hold in money is K. Hence the quantity of money is KY and the value of 1 unit of money is KY/M. But this is the opposite of the price level P. Hence the celebrated Cambridge equation is given by P = M (KY)-1 … (3) or M= KPY …(4) Note that P in (3) and (4) is distinct from the P in (1) and (2). The “exchange P” is based on the prices of everything that enters into a money transaction while the “Cambridge P” is based only on the prices of things that form part of the community’s real income. Of course, it may be noted that the “P”s may tend to move together in either direction. From (4), taking logarithms and time rates of change, we can write = + + … (5) or = …(6) 6 the Now, under classical assumption of Y being at full-employment level ( ) and the stability of velocity over time ( , the last equation reduces to = … (7) Thus the Cambridge way has helped establish again that the rate of inflation equals the growth rate of money stock, as stated earlier. 3. The Inflationary Gap of Keynesianism – Demand Pull Inflation The Keynesian theory of inflation can be viewed as a generalization of the Wicksellian dynamics approach. It is based on the axiom that if the aggregate demand exceeds the level consistent with the full employment of productive resources of the society, an inflationary gap is created or the process of inflation is initiated. The same thing can also be demonstrated through a budgetary expansion process. Let us look at the inflationary gap or demand pull model first. In the figure below the horizontal axis measures real ouput Y and the vertical represents consumption C and investment I. The 45o bisector represents potential equilibrium relationships, as usual. Let real consumption be Cf and real investment be If. Also let the full employment level of output be Yf. Then Cf + If is the full employment level of aggregate demand. Now if the level of aggregate demand becomes C0 + I0 which is greater than Cf + If then real output cannot reach Yo as Yo > Yf. Hence at Y=Yf total demand exceeds total supply by AB. The excess demand equivalent to AB in the product market pulls up commodity prices. The gap AB is called the Keynesian inflationary gap. This type of inflation is called demand-pull inflation. The Keynesian inflationary gap can also be illustrated by an aggregate demand-aggregate supply diagram as shown below. Here the horizontal axis measures real ouput Y and the vertical represents the general price level P. Keynesian macro-equlibrium is at point H where aggregate demand ADo intersects aggregate supply ASo. The equilibrium level of output is the full employment level of output Yf. Market clearing price level is Pf. Let aggregate demand shift as shown by arrows from ADo to AD1. With a given aggregate supply curve, the macroeconomic equilibrium should move to H’. The market clearing output level is Yo. But this level of output cannot be achieved because Yo > Yf. Thus a shift in the aggregate demand schedule will only push the general price level to Po. The inflationary gap is given by GH. It is also to be noted that the mere existence of inflation does not automatically eliminate the inflationary gap. But indirect effects of price-hike, which may be manifold, restrain real demand and eventually reduce the inflationary gap. 7 Let us now look at budget inflation. It has already been demonstrated that at full employment level of output a rise in aggregate demand causes inflation. Now let budget expenditures be raised without changing tax rates. Also let aggregate expenditure E consist of consumption expenditure C and government expenditure G only. Also let the budget be initially balanced. Now, to make expenditures in period t we need available wage earnings WAt and this we derive from wages WRt-1 received in the period t-1. Hence WAt = WRt-1 … (8) Wage earnings are reduced by the tax rate r before being used for consumption, so the current consumption is Ct = (1-r)WAt, 0<r<1 … (9) Total government spending G is defined as g% of aggregate expenditure, so Gt = gEt, 0<g<1 … (10) Recall that by definition Et = Ct + Gt … (11) The total expenditure E is received by business and paid out as wages received, WRt so that Et = WRt … (12) The percentage change in general price level Pt or the rate of inflation ∏t is measured as ∏t = = … (13) Substituting equations (8), (9), (10) and (12) into equation (11) we have WRt = (1-r)WRt-1 + gWRt … (14) Initially the economy is assumed to have a balanced budget i.e. g=r. Substituting in (14) we have WRt = WRt-1 - rWRt-1 + rWRt so that WRt = WRt-1 …(15) Pooling (12) and (15) we can write Et = Et-1 …(16) Pooling (13) and (16) we get ∏t = 0 …(17) Thus for a full-employment economy and a balanced budget the rate of inflation is 0. In other words, the general price level is stable. Solving (8) to (11) for Et we find Et = (1-r) Et-1 + g Et = …(18) Now from (13) we have ∏t = = = …(19) Now let us introduce two different fiscal disturbances and check their effects. Disturbance 1: The government decides to raise expenditures to increase its share of output. The tax rate is left unchanged. From (19) we can see that there will be inflation because = = >0 …(20) Disturbance 2: The government decides to keep constant expenditure parameter g and introduce a tax cut. From (19) we can see that there will be inflation because 8 = <0 0<g<1 …(21) Equations (20) and (21) show that parametric rise in g or a cut in r is inflationary. Similarly, a rise in r or a cut in g should be deflationary. This model establishes a link between the rate of inflation and fiscal policy parameters, starting from a situation of full employment. 4. The Cost Push Theory of Inflation The original argument here is that employers are forced by organized trade unions to grant increase in wage rates in excess of the rise in output per head. Thus to maintain profit margins, employers (producers) are forced to charge higher prices. The higher wages enable consumers to buy as much as before in spite of the commodity-price inflation. But the inflation itself induces the labour unions to demand further wage-hikes, and so on. But experts like Samuelson argued that non-union wages also tended to rise, sometimes even in periods of unemployment. Also, the behavior of collective bargaining did not always follow the trajectories stated above. So let us use the aggregate demand-aggregate supply diagram shown overleaf to understand the phenomenon. Here the horizontal axis measures real ouput Y and the vertical represents the general price level P. The economy is at the initial equilibrium point Eo where the aggregate demand ADo intersects aggregate supply ASo. The general price level is Po. The full-employment output is Y*. Clearly the macro-equilibrium at Eo is a Keynesian underemployment equilibrium as Y<Y*. Cost-push occurs when a rise in the cost of production shifts the aggregate supply curve from ASo to AS1, as shown by the bold arrows. The macro-equilibrium shift from Eo to E1 where the new aggregate supply curve AS1 intersects AD0 as a result of the supplyside disturbance. The new general price level is P1 which is greater than the previous level Po. But there are several drawbacks in this model. Cost-push alone, sans demand-pull, cannot explain why inflation can be continuous, what the exact rate of inflation should be and why there are variations in the rate of inflation. Tax policy may be used to ease inflationary pressure from the supply-side. Tax cuts can act as a premium for non-inflationary behavior. This can help check the continuing transmission of cost increases. 9 But at the same time, a progressive income tax structure can accentuate the wage-price spiral. A rise in prices may induce wage-earners to demand constant or higher real post-tax wages, because inflation may push tax-payers into higher brackets. With a proportional rate of taxes the percentage rise in pre-tax income needed to hold posttax income unchanged in real terms may just equal the rise in prices. But if the tax-payer moves into a higher tax-rate bracket, the required percentage increment in pre-tax income exceeds the price-rise. Thus it can be argued that the dynamics of cost-push inflation may generally be magnified by a failure to index the income-tax schedule to inflation. 10