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Lecture III Keynesian Model. Keynes’ General Theory, by all accounts, is difficult to understand

Lecture III Keynesian Model

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- Keynes’ General Theory, by all accounts, is difficult to understand
- For this reason, Keynes’ ideas have come down to us filtered through the eyes of those that either were there when the ideas were being worked out or by later writers that have more or less guessed at what the “great master” had in mind
- In this way, Keynes is very much like Jesus Christ, whose words and meaning have come to us through the “apostles”; although in the case of Keynes, we do have his actual writings to fall back on

- He was trying to work out a new way of looking at the economy that could explain the existence of widespread unemployment when so many people were willing to work for wages well below the market wage
- He referred to this phenomenon as “involuntary” unemployment
- The classical view was that this cannot occur; the wage rate will simply fall to the equilibrium wage and all markets, including the labor market, will be in equilibrium
- Thus general equilibrium at full employment is maintained

- Keynes begins with the notion that aggregate demand for the goods and services in the economy can be decomposed into two parts: a part that depends on the level of output and a second part that is exogenous
- Thus, D = D1(Y) + D2
- We can think of these two as D1(Y) = C and D2 = I, which is exogenous to Y
- Unlike classical theory, Keynes does not really look upon investment demand as depending strongly on the interest rate, although that will not be a major problem to his model

- It is critical to his theory to understand how he felt about the decision to invest or not
- He viewed investment decisions by firms and entrepreneurs as dependent on their “animal spirits”, that is, once businesses are frightened off from the market, perhaps due to continued losses sustained by themselves and other investors, it may be next to impossible to convince them to return
- As a sidebar, much of the discussions we here today about what to do after the financial crisis, concerns the “credibility” of central banks actions; more on this later

- John Hicks in Mr. Keynes and the “Classics” first introduced the IS-LM analysis
- Some of Keynes’ contemporaries argued that Keynes never used such models; however, writings of Keynes discovered after his death did find the he worked on such equations
- The IS curve is the locus of points in i-Y space at which the product market is in equilibrium while the LM curve does the same for the loanable funds market

- Keynes starts with the premise that expenditures E = output Y = C + I in a closed economy with no or limited government
- C depends primarily on Y, but let’s suppose some saving is generated through a rise in I, so C = c(i,Y) which implies that S = s(i,Y) where Si and SY > 0.
- Investment depends only on I and Ii < 0
- Equilibrium requires that S = I or I – S = 0; furthermore, for it to be maintained, the change in I – S must = 0, so d(I – S) = Ii di – (Si di + SY dY) = 0
- Solving for di/dY, we get di/dy = SY dY/(Ii di – SY dY) < 0; thus, the IS curve is down-sloping

- The demand for money depends on both Y and i, where people hold more money to finance transactions at higher levels of Y and economize on money holdings as i rises
- The supply of money is either fixed or rises with i say as financial institutions reduce excess reserves as the return on loans increases
- Again, we need Demand for money L to equal M, the money supply and along the LM curve, d(L – M) = 0
- So, Li di + LY dY – Mi di = 0 and solver for di/dY we get di/dY = LY/(Mi – Li) > 0

- E = A + cY – ai = Y, so (1 – c)Y = A – ai
- Real money demand M/P = mY – bi = Ms/P for monetary equilibrium
- Thus, i = [mY – Ms/P]/b; substituting into first equation we get
- (1 – c)Y = A – a[mY – Ms/P]/b
- Solving for Y we get [1 – c + (a/b)m] Y = A + (a/b)(Ms/P)/b and
- Y = A/[1 – (c – m(a/b))] + Ms/P[1/[m + (b/a)(1-c)]
- Keynes assumed both consumption and investment were relatively insensitive to i (that is, a is small) and demand for money was very sensitive to i when rates are close to 0 (that is, b is large)
- These two imply that changes in the money supply have very little influence on aggregate supply while the Keynesian multiplier is close to 1/(1-c)

- A consol is a perpetuity bond that pays a fixed amount each period, say a year
- S = ∑1/(1+r)n, n = 1, ∞ = 1/(1+r) + 1/(1+r)2 + … + 1/(1+r)n + …
- (1+r)S = 1 + S => (1+r)S – S = rS = 1 and S = 1/r
- So I pay $20 for this console
- If the rate falls to 4%, I make $5, or 20% return for a 1% drop in interest rates; if r goes up to 6%, on the other hand, the price falls to $16.67 and I lose $3.33 or 16.67%
- What if the interest rate is near 0%? The rate can hardly fall, so the next move in interest rates must be upward; that is, I can only lose money (or stay the same) if I buy a consol now
- But I can earn the same return by simply hoarding my cash
- This Keynes referred to as the zero-bound or the liquidity trap

- Let Y = C + I = a + cY + I, a and I are exogenously determined
- Then at equilibrium Ye = [1/(1–c)](a + I), and 1/(1-c) = k, the Keynesian multiplier
- Now suppose Ye < Yp, potential, or full-employment, output
- Keynes argued that government should fill the expenditure gap, which he referred to as the recessionary gap.
- Now Y = a + cY + I + G, and we assume the government expenditures do not affect a, c, or I.
- Then solving for Ye gives Ye = k(a + I + G) which is greater than the previous equilibrium output
- So if kG = (Yp – Ye), so the recessionary gap is g = 1/k(Yp – Ye)

- Y = C + I = 400 + .6Y + 1000 = 1400 + .6Y and solving for Y gives [1/(1-c)] (a +I0) = (1/.4)(1400) = 2.5*1400 = 3500, and k = 2.5

- Now if I declines to 900, AD will decline by k*d(AD) = - 250, so AD = 3250
- This is shown on the Keynesian cross diagram below
- To restore the equilibrium level of AD to the desired 3500 we need add G = 100 and we will be back to the original AD curve

- David Ricardo considered the case of debt financing of government expenditures and conjectured that, if the public perceived the increase in debt as a future tax liability, it might elect to save more, even and equal amount to the debt, as a way to pay the future liability
- Apparently, Ricardo rejected the idea of such foresight of the public, but the notion, called the Ricardian Equivalence Theorem, still bears his name
- But let’s suppose we decide to finance the expenditures with lump-sum tax today
- Then Y = a + c(Y – T) + I + G, T = G is a lump-sum tax

- Now our model is Y = a + c(Y – T) + I0 + G, where T = G
- Then solving for Ye we get Ye = [1/(1-c)](a – cT + I0 + G) = [1/(1-c)] (a + I0 + G – cG) = [1/(1-c)][a + I0 + (1-c)G] = [1/(1-c)] (a + I0) + G, since T = G
- Thus, the “balanced budget multiplier” is 1; if the government spends an amount just equal to Yp – Ye equilibrium is restored at full employment

- Since taxes are collected from individuals and not the country as a whole (since passage of the 16th amendment in 1909), a more realistic model is
- Y = a + c(Y – tY) + I + G, tY = G
- Then Y(1 – c + tc) = Y(1 – c(1 – t)) = a + I + G, so Ye = 1/[1 – c(1-t)] (a + I + G) and k* = 1/(1-c(1-t)) is smaller than before
- Thus the recessionary gap has increased to 1/k*( Yp – Ye)
- In our earlier example, suppose t = .2, then k* = 1/(1 - .6(1 - .2)) = 1/.52 or around 2
- It appears we now need to increase G to 250/2 = 125

- When we substitute the numbers back, they don’t work; Y = 2(a + I + G) = 2(400 + 900 + 125) = 2850
- The reason is we collect too much in taxes; .2*2850 = 570, but we only need 125
- So let’s let the model tell us the optimal tax rate t*; t*Y = G, which we also need to solve for
- The second equation is (1-c(1-t*))Y = a+I+G; so (1-.6(1-t*))3500 = 400+900+G = 1300+G

- We get 3500 = (1/(.4+.6t*))(1300+G); but G = 3500t*
- So we solve for t* in the following equation 3500 = 1/(.4+.6t*) (1300+3500t*)
- Dividing both sides by 3500, we get 1=1/(.4+.6t*)(1300/3500+t*) so
- .4+.6t* = 1300/3500 +t* => .4t* = .4-13/35
- t* = 1-13/14 = 1/14
- G = t*(3500) = 3500/14 = 250!
- The same answer as we got with a lump-sum tax
- Thus, the balanced budget multiplier is again 1; is this just a coincidence? Let’s see.

- Yp = a + c(Yp –t*Yp) + I + G; G = t*Yp
- So Yp = a + c(Yp –t*Yp) + I + t*Yp
- (1-c)Yp = a – ct*Yp + I + t*Yp = (a + I) + t*Yp(1-c)
- Thus, Yp = (a + I)/(1-c) + t*Yp = (a + I)/(1-c) + G
- That is, Y changes 1 for 1 with G; the multiplier on G financed using the optimal tax is still 1
- In general, however, a proportionate tax does reduce the multiplier to 1/(1 – c(1 – t))

- With trade the equation becomes Y = a + c(Y – T) + I + X – M, where X = exports and M = imports
- While exports are generally considered as determined externally to our economy, imports should grow with Y
- In fact, it is often the case that fast growing economies are great exporters; just think about China
- This issue will be covered later when we discuss the monetary approach to the balance of payments

- Y = a + c(Y – t0 – t1Y) + I + G + X – M0 – mY
- Then Y – cY + ct1Y + mY = (1-c(1-t1)+m) = a + I + G + X
- In the literature, the right-hand side variables are called injections; savings, taxes and imports are referred to as leakages
- At equilibrium, injections must equal leakages

- The more open an open is economy the less “bang for the buck”, that is the additional leakage from imports increases the slope so that a monetary change – a movement along the IS curve – will have less affect on Y and more on I
- The addition of export demand, say from greater world output, the further to the right the IS curve will lie

- Exercise 1: Draw a set of IS-LM curves
- What is the effect of an increase in Government spending?
- What happens to equilibrium i and Y?
- Now, how can the Fed reduce crowding out?
- What now happens to equilibrium i and Y?
- What happens to the budget deficit?

- Exercise 2: Draw a set of IS-LM curves
- What is the effect of an increase in central bank credit to lending institutions?
- What happens to equilibrium i and Y?
- What happens to the budget deficit?
- What happens to equilibrium i and Y?

- One issue with deficits financed by bond creation is the public perception of their increased bond holdings
- At one end is the Ricardian Equivalence, which believes that these holdings are viewed as both wealth and as a liability at its limit one for one
- At the other end of the spectrum is the belief that the public sees these bonds only as wealth and therefore will spend even more than the Keynesian model predicts (although this effect was never included in the model; we could show this as C = a + cY + gB, where B is the stock of government bonds held by the public)
- Since some of the expansion in the economy is financed by increased tax revenues, the amount needed to be financed through bonds is reduced and if a portion of the bonds held increases private spending through the wealth effect, the negative impact of the deficits can be reduced, making the Keynesian argument even stronger

- To the extent that future generations may be impacted by the tax liability, the negative Ricardian effect is reduced
- Barro argues, however, that the fact that people bequeath wealth to their heirs indicates that they care about the higher tax liabilities they are leaving them
- But some don’t care; either they have no heirs or they figure the next generation will be so much better off that they can pay the taxes themselves
- Plus the government can borrow more cheaply than the private sector, so the burden is reduced

- Arthur Pigou, a contemporary and (kind of) a friend of Keynes, argued that the price declines will increase the real wealth of the public and thereby increase their expenditures
- Critiques are compelling: effect can be too slow; the fall in prices have negative effect on business optimism; increased bankruptcies reduce investment; postponement of consumption awaiting further price declines; etc.

- Pigou may have won the intellectual debate, showing that the economy, given enough time and flexible wages and prices, would return to full employment on its own
- On the policy side, however, concerns about the speed of adjustment (“In the long run we’re all dead”) led most western economies to adopt Keynesian style fiscal policies
- Even today, this is the most often used model by most politicians and their staffs
- Central banks, on the other hand, have begun to use the alternative DSGE model for their analyses
- This incorporates more of the elements of rational expectations and take into account the Lucas Critique by allowing parameters of their models to adjust and by disaggregating the economy into several sectors
- Even these models, however, are limited as to the amount of disaggregation they employ

- Let’s look at the BP line where BOP = 0
- As Y increases a country wants to import more
- As i increases, more capital flows inward
- Thus as Y increases i must also increase to maintain balance, so BP is positively sloped
- The slope depends on the interest elasticity of capital flows and the income elasticity of imports; the more the interest elasticity of capital flows, the smaller the adjustment in I needed to balance flows, so the flatter the BP curve; the greater the income elasticity of imports the greater the interest rate change needed to balance payments, so the steeper the BP line

- Keynes worked in real values since inflation was not an issue in times of depression; if anything, prices fell
- But Keynesians had to address the issue of what happened as the economy approached full employment
- Let us look at the “stripped down” version of the aggregate supply-aggregate demand diagram

- So the idea of a permanent and stable tradeoff relationship between inflation and unemployment provided additional armor against attacks from those that would worry about the hyper-inflationary danger of permanent deficits or monetary stimulus
- In fact, a thorough reading of Phillips himself shows that he did not intend for the empirical results to be inerpreted by policy makers as an excuse to inflate the economy so as to reduce unemployment

- The events that occurred starting in 1970 demonstrated that the critics were correct after all; continued attempts to stimulate the economy in the face of real supply side shocks took its toll and inflation hit double digits with little effect on the unemployment rate
- This period became known as stagflation, and was only ended with the recession of 1981-83 caused by Paul Volcker
- During this recession unemployment hit 10% for the first time since the Great Depression
- Now we are once again experiencing such high rate of unemployment

- The events in that began at the time of stagflation led most economists to abandon the conventional view and to adopt Friedman view that there is no long-run Phillips Curve
- Friedman viewed the relationship as a short-term fix that would eventually lead to expectations of inflation that would nullify the short-term benefits
- Later he adopted the rational expectations view that even the short run Phillips Curve tradeoff would disappear in favor of a vertical Phillips Curve at the permanent natural rate of unemployment