Relevant and even prescient commentary on news, politics and the economy.

A Warm Wind At the Backs of Some, Generated Off the Backs of Others

This piece offers an understandable comparison between wages and dividend income and neatly summarizes the cost to wage earners. (h/t Mike Kimel)

by Peter S. Meyers

Myers Urbatsch PC

A Warm Wind At the Backs of Some, Generated Off the Backs of Others Yesterday, I learned in this Mother Jones article that workers have increased their contribution to government revenue disproportionately since 1980.  In other words, payroll tax (paid by workers) is a larger portion of government revenue than it used to be.  That’s a macroeconomic analysis, which still doesn’t answer the question of whether rich people are being treated “unfairly” by the current tax system.

So to elaborate a little, let’s take two people who make exactly the same amount:  $100,000 in taxable income (after the standard deduction – let’s not get complicated).  “Worker Taxpayer” earns her money by working (getting compensation by way of a W2) and “Investor Taxpayer” earns her money from dividends in a $4 million stock portfolio she holds (its about 2.5% in yield – about right).  Let’s say they are both unmarried.  Investor taxpayer does not work and has no compensation income.  They are otherwise “equal,” right? (except that investor taxpayer fits the description of those who vituperate about lazy welfare recipients who sit on the couch all day and watch TV, right?)  I’ll keep the rhetoric down, because the facts are outrageous enough to speak for themselves.

Worker taxpayer will pay $7650 in payroll tax, plus $21,617 in income tax (2011 brackets), for a total tax burden of $29,267.

Let’s look at investor taxpayer.  You would think they would be taxed at the same rate as worker, right?  Wrong.  Because investor taxpayer receives all of her income from qualified dividends, they get a “special” tax treatment.  Bear with me, we’re almost done.  Generally, the maximum tax rate for qualified dividends is 15%, BUT HERE it is actually 0% because investor’s other income (remember she doesn’t work) is taxed at the 10% or 15% rate.

To refresh:  worker making $100K pays about $30K in tax.  Investor making $100K in qualified dividends pays $0 – no – tax.  Huh?  Yup. 

What this means is that rich people – who are incented by tax policy to remain on their couches (too much earned income would otherwise trip them into the 15% dividend tax bracket) – are now getting off their couches and going to tea-party rallies to maintain this unfair redistribution of wealth in their favor.  For if they work, they risk having their dividends taxed at 15% (still half of what, say, worker taxpayer paid in taxes, but confiscatory in their view).  Perverse incentive?  Yup.  Does it sound like the rhetoric of the right wingers about unemployed persons and welfare recipients laying on couches and not incented to work?  Hm. . . .

Now let’s say you didn’t work, or you worked very little, and instead you made all of your income from qualified dividends.  The “magic number” (the income threshold you need to stay under to avoid paying any tax on your dividend income) is $69,000 (married), $34,500 (single or married filing separately) or $46,250 (head of household).  Thus, you can actually work a little, and you have all this extra time – to attend rallies, political functions, cook your food, clean your house or do other things that people who actually earn their income from working have to: (a) pay someone else to do (which is not deductible), (b) do in the evenings or on weekends, or (c) simply let it slide.

I will now illustrate how it is almost impossible for someone who is already rich to not get richer, in fact much richer.  Both working taxpayer and investor taxpayer have identical lifestyles and thus spend the exact same amount of money (not likely, given that worker has to pay for commuting expenses – again NOT deductible).  Let’s assume that’s $70,000 per year.  We know that worker taxpayer already paid $30K in tax, so let’s see what they have left to save:  uh, nothing.  Investor taxpayer paid no tax, so what do they have left over to save: $30K.  Exactly the same amount that worker taxpayer paid in taxes.

The rationale for the tax policy you see illustrated above is George W. Bush’s.  In 2003 he said that “double taxation is bad for our economy and falls especially hard on retired people.” He also argued that while “it’s fair to tax a company’s profits, it’s not fair to double-tax by taxing the shareholder on the same profits.”

Its odd to me that the above disparate treatment of otherwise similarly-situated earners is defended on the basis of “fairness.”  Is this 1984?  And I also wonder whether there is a joke in there somewhere – i.e., given that a zero-percent tax bracket would apply to someone who made all of their money from dividends and capital gains, why wouldn’t they retire?  I sure as hell would.  Working too much would bump all of their dividend income into the 15% tax bracket.  Volunteering for the tea-party rally, or perhaps some other Republican cause, would be a far better use of one’s time.

reposted with permission of the author July 23, 2011 post

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Who Cares About Nominal Rigidities?

Tyler Cowen doesn’t much.

I tend to agree with Cowen. Nominal rigidities were quite the thing just before I arrived, so I think they are over rated. However, there are two points one of which is totally twitty and the other of which is a dead horse still being beaten by Paul Krugman.

OK twitty: By definition for there to be unemployment there must be three agents, an employer, an employee and an unemployed person. The unemployed person must be eager to work as the employee does at the employee’s wage. The employer must consider the unemployed person qualified. This means that unemployment can certainly be eliminated if wages fall. At some point, either the employee decides to quit and just live off savings till social security kicks in or the unemployed person decides he or she doesn’t want the job. By definition, wage rigidity is needed to explain unemployment. This is true even if lower wages do not at all cause higher employment. If nothing else super low wages can convince people to leave the labor force eliminating unemployment that way. In this case wage flexibility doesn’t help the unemployed — it makes the alternative of working worse so they consider their horrible predicment the best they can hope for. I said it was twitty.

Second, things are unusual because we are in a liquidity trap. The reason nominal rigidities usually matter is that the real money supply could increase if the nominal money stock staid the same and wages and prices fell. From 1940 through 2008 this meant that wage and price flexibility should have prevented output from fallin. N ow, however, the money supply doesn’t matter since we are in a liquidity trap. In the IS-LM model (M/P) (money divided by the price level) appears. If P is free to adjust, then there can be no problem with insufficient aggregate demand. Therefore in all of the macro literature from 1940 through 2008, nominal rigidities were considered important. The idea here is wages go down so the firms cut prices (to maximize profits they would) so real balances (M/P) goes up so aggregate demand goes up so GDP goes up. There is no need for real wages to fall.

Right now this doesn’t matter as M/P doesn’t matter. But for decades and decades it mattered a lot, so nominal rigidities mattered. In practice, wages and prices are sticky so all reality based macroeconmists (“that’s not enough I need a majority” — Adlai Stevenson) agreed that nominal rigidities mattered. Now not so much. M/P doesn’t matter so P only matters because of debt deflation (lower P makes nominal mortgage debt an ever worse problem) so wage and price flexibility won’t save us so Keynesians don’t talk about it.

As always, don’t confuse “Keynesians” with Keynes. Keynes was not interested in nominal rigidities The General Theory through “The General Theory Restated” included nothing on nominal anything.

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US labor market: wage and salary growth vs. payroll growth

I’ll make this quick, since I’m going to get in trouble for writing on a national holiday. But the pace of annual jobs growth is too slow to generate strong wage and salary income. Much empirical research has been dedicated to the estimation of consumption functions, generally finding that labor income is the primary driver of consumption (here’s a primer at the Federal Reserve Board). However, by extension jobs growth is highly correlated with wage and salary growth, roughly 50% of personal income – this is the relationship I analyze here.

Roughly half of the BEA’s measure of personal income comes in the form of wage and salary, so called labor income and simply referred to as ‘wages’ from here on out. This is highly correlated with nonfarm payroll growth, both in nominal and real terms (92% and 79%, respectively, since 1996). The chart below illustrates the correlation between real wage growth and nonfarm payroll since 1982 (I use real wage so as to account for the effects of inflation).

The annual pace of real wage gains and jobs growth have declined over time (jobs growth is measured using the nonfarm payroll). Simply eyeballing the data, there’s a structural shift roughly around 1996, as listed in the table below.

Using these two time periods, 1982-1995 and 1996-05/2011, I estimate a simple model of real wage growth on nonfarm payroll growth. The chart for the 1996-2011 model is illustrated below; and for reference, the regression results across both time periods are copied at the end of this post.


Note: I do not have time for a full blown econometric analysis. I did, however, perform statistical tests for serial correlation in the errors, unit roots in the transformed data (none), and general modeling tests.

I come to two general conclusions regarding the relationship between real wage growth and jobs growth over time:

(1) Real wage growth has become more persistent over time. In the first period, 1982-1995, just one lag was required to expunge the errors of autocorrelation. Spanning the second period, 1996-2011, three lags were required. The sum of the coefficients on the three lags is 0.87 in the later sample, or current wage growth is highly dependent on previous periods – sticky if you will.

(2) Nonfarm payroll growth has become less significant over time. Spanning the years 1982-1995, the coefficient on annual payroll growth was 0.27 – for each 1pps increase in the annual payroll growth, the trajectory of annual real wage growth increased by 0.27pps. The coefficient dropped to 0.17 in the sample spanning 1996-2011. This is probably a consequence of service sector jobs growing as a share of the labor market. I’d like your ideas in comments as well.

Clearly this is a very simple model but it does highlight that wages are likely stuck in the mud for some time. In May, annual real wages fell 0.6% over the year, having decelerated for 5 of the 7 months since November 2010. Real wages can pick up, but it takes time AND jobs growth faster than the 0.67% annual pace in May 2011.

Ultimately, what this analysis tells me is with wealth effects slowing markedly – the trajectory of the S&P decelerated and house prices continue to fall – it’s going to take a burst of payroll growth to get real wage and salary growth back on track enough to finance US domestic consumption. One caveat to all of this negativity is that oil prices are coming off – this will boost real wage and salary growth directly.

Rebecca Wilder
P.S. I guess this turned out somewhat less ‘quick’ than I had anticipated – not in trouble yet! Gotta go.


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The Unemployment Rate and Compensation Growth

Crossposted at The Street Light.

Last week I took a look at the way that higher labor productivity has not translated into higher worker compensation, particularly during the 1980s and 2000s. This is at odds with classical labor market theory, which suggests that as workers become more productive, their increasing value to firms should cause their wages to be bid higher so that their compensation rises accordingly.

There are a number of possible explanations for the divergence between productivity and compensation, and for how this may play into the broader phenomenon of stagnant wages for average workers. Part of the explanation is that an increasing share of worker compensation takes the form of benefits rather than wages and salaries. As shown in the chart below, fully one-fourth of worker compensation in 2010 took the form of benefits. (Source: BEA personal income data.)

This upward trend has been driven almost entirely by the rise of health care costs in the US, and the corresponding rise in health insurance premiums. Note that the one dip in the series in the late 1990s was due to the widespread implementation of HMOs – but they clearly proved to provide a one-time gain rather than a permanent increase in health insurance efficiency. So part of the reason that workers’ paychecks have not been rising is directly attributable to the rise in health care costs in the US.

But that’s not the whole story, and doesn’t address the question of slowly growing total compensation (as opposed to stagnant wages). There are, I think, reasonable arguments to be made about social and political factors, such as the decline in the power of unions. Along similar lines, Mike Konczal recently wondered to what degree this could be due to the Fed’s consistent and explicit desire to prevent wage increases.

And then there’s plain old supply and demand as a possible explanation. What did the 1980s and 2000s have in common from a macroeconomic point of view? One answer is this: multi-year long periods of high or rising unemployment rates.

The chart below shows, in blue, the seven-year moving average of the portion of increased labor productivity that were paid to workers in the form of higher compensation. During the 1960s and 70s, for example, workers typically received around 80% of gains in labor productivity over any given seven year period. Then during the 1980s that portion fell to about 40%. Meanwhile, the series in red is the seven-year moving average of the unemployment rate.

To make it a little easier to interpret, I’ve color coded the 60 years shown in the chart by shading the periods when workers were losing their share of productivity growth red, while the periods when workers were increasing their share of productivity gains are shaded in green. This helps to make it quite clear that “green” times – i.e. times when workers seem to be enjoying more of the gains in productivity – were periods when unemployment was falling. “Red” times (I guess it actually looks more pink than red in this chart) are clearly associated with periods when the unemployment rate was stagnant or rising.

One implication of this is clear: the high unemployment rate in the US right now, which is expected to decline only slowly over the next several years, is likely to mean that it will be a long time before worker compensation begins to rise as rapidly as worker productivity. Put another way, the overall level of high unemployment right now not only has the obviously enormous personal implications for those who are unemployed — it also is likely to seriously affect the compensation of workers who have never lost their jobs, for years and years to come.

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Growing Productivity, Stagnating Compensation

Crossposted at The Street Light.

Yesterday Ezra Klein had a chart (from a paper by Larry Mishel and Heidi Shierholz at EPI) showing that both private sector and public sector wages have been stagnating for the past several years, and have certainly not kept up with productivity growth. I think it’s useful to look at the relationship between productivity and compensation over a longer time horizon.

The following chart shows labor productivity and real hourly compensation since 1950. (Data from the BLS.) Two things strike me particularly about this graph. The first is how closely the two series track each other between 1950 and 1980. During those 30 years labor productivity in the nonfarm business sector of the US economy rose by 92%; real hourly compensation paid to workers rose by a nearly identical 87%. Classical economic theory says that is exactly what we would expect – as workers become more valuable to firms by producing more output with every hour of labor, firms should compete with each other to employ them, driving up wages by an equal amount.

The second striking feature of this picture is, of course, how much the two series have diverged since the early 1980s. Output per hour of work in 2010 was 87% higher than in 1980, while real hourly compensation was only 38% higher.

The table below shows changes in labor productivity and hourly compensation by decade. Again, let me draw your attention to two features. First, this data confirms that the “great productivity slowdown” of the 1970s and 80s seems to have been vanquished; over the past 15 to 20 years US businesses have been improving productivity at rates as high as during the 1950s and 60s. Yet more evidence that Tyler Cowen’s “Great Stagnation” is not a productivity story.

The second remarkable feature of this table is that the vast majority of the gap between productivity and hourly compensation comes from the 1980s and 2000s, while during the 1990s workers shared in productivity gains nearly as fully as they did in the 1960s. And that, of course, leads us directly to the $64,000 question: what was it about the 1980s and 2000s that made it so difficult for workers to reap the fruits of their more productive labor?

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Are Earnings Rising or Stagnant?

In 2005 Kash Mansori of Angry Bear took a look at Are Earnings Rising or Stagnant? It is an issue worth periodic review:

This question is not as easy to answer as it may first appear. In working on various posts last week I came across an apparent contradiction in the official data on compensation: some series show it rising in real terms, while others show it barely able to keep up with inflation. This discrepancy was also noted by a few readers, who deserve credit for their sharp eyes.

So I thought I’d take a bit of time to sort out these conflicting data series for myself. Here’s what I found. (A warning and apology here: what follows is a relatively econ-geeky post about data details that many may find uninteresting… and I won’t be offended if you stop reading here.)

So what’s my answer to the title question of this post? Personally, if I had to choose just one series to use it would be the P&C series. In addition to being arguably the most complete series, it seems to have done the best job of matching my sense about how the economy has done over the past 20 years. When asked, I think that most people would agree that income growth was indeed much lower during 2002 and 2003 than it had been during the late 1990s; the P&C series bears that out, while the ECI series doesn’t. Meanwhile, the CES series excludes benefits, which I think are a major part of the story today.

But let me reiterate the point that I have made several times now: just because real compensation is rising, that doesn’t mean that people are better off, particularly if nearly all of the gains are just going to paying higher health insurance premiums. This data persuasively illustrates that nearly all of our real compensation gains today (and I do think we’re seeing them) are being eaten up by the monster that we call a health care system in the US. Until we address the profound inadequacies of our health care system, this trend will only get worse.


Spencer took another quick look in 2010, lifted from his email this Saturday:

This is not an easy subject and there is no “right” answer.

A few comments. The average hourly earnings data is for people who earn an hourly wage or punch a time clock as oppose to those earning a salary. As a general rule this is the lower earning segment of the population as college educated professionals and managers are much more likely to be salaried. It reflects what is happening to about 80% of the labor force. Surprisingly, this percent has not changed significantly over the years. But the average hourly earnings data has clearly been changed by the changing composition of employment as more highly paid manufacturing employment has both declined in importance and relative pay.

I always though of the ECI as a measure of what it cost a firm to keep an employee in the same job. It is deliberately designed this way to try to avoid the problem of changing composition of the labor force from distorting the data. There are positive and negatives to this.

The compensation data is the most comprehensive measure of labor payments. The one question I have never resolved to my own satisfaction is how it treats the payment to CEOs and other senior management with stock options. It is a difficult accounting issue at the level of the individual firm, let alone for the aggregate economy. But remember the chart I did a while back of compensation as a share of business cost. From WW II to 1980 this was a very stable ratio with minor cyclical fluctuation around 66%. But since 1980 labors share of the pie has been on a falling trend and is now at around 57% of business cost.
I think this best shows the way the middle class has been squeezed. This measure avoids the issue of how to measure inflation.

The best data for seeing the middle class squeeze is the data published by the Census on family and household income by age,sex, race,etc.,.

All the inflation series have their drawbacks and shortcomings. The PCE deflator does not have the cost of housing and so until the last few years probably significantly understated the true cost of living for the middle class. We are all familiar with the unresolved issue of how the CPI treats housing cost.
The CPI only measures “out-of-pocket” medical expenses and because of this I suspect it significantly
understates and under-weights the impact of rapidly rising medical prices on the middle class. All the Census data uses the CPI-U-RS that recalculated the CPI from 1978 to 2002 using the new methodology for the CPI that incorporates the rental equivalence housing measure and other methodological changes in the way the CPI is calculated.

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One of These Things is Not Like the Others

I try to like the NYTimes Economics Reporting. I really do. Heck, any place that publishes Uwe Reinhardt can’t be all bad.

But David Leonhardt, as he does often enough that I hesitate to read his work, again goes beyond the pale today, and clearly does so deliberately. The offending paragraph:

Twenty-two months after the start of the mid-1970s recession, real weekly pay was down 7 percent. For the early 1980s recession, the decline was 4 percent. Today, thanks to moderate pay growth and scant inflation, pay is 1 percent higher than when the Great Recession began in December 2007.

Let’s (1) remember that wages are sticky and (2) look at this declaration.

Both of the previous recessions are cited as being about 16 months. The current one probably ran 18 for economists’s purposes, and is in its 23rd month for the rest of us. But let’s give him a pass on that.

Note, however, the careful phrasing at the end of the paragraph: “thanks to moderate pay growth and scant inflation.” What does that mean? Well, let’s look at the Annual inflation Rate (CPI) for the actual recessions under discussion:

Gosh; quite a difference! I wonder if Leonhardt is aware of it.

A finger exercise below the fold.

Just for fun, let’s look at the wage changes over those periods. Now, unlike Leonhardt, I’m not going to use real wages. Let’s see if we can figure out what the nominal change in wages is for each of those periods.*

1973-1975 Average Inflation Rate: 10.75. Real wage loss: 7% Wage increase in period: 3.75% (including the residual effects of wage and price controls)

1980-1982: Average Inflation Rate: 7.5% Real wage loss: 4% Wage increase in period: 3.5%

2007-present: Average Inflation Rate: 1.8% Real wage gain: 1% Wage increase in period: 2.8%

I don’t know about anyone else, but I wouldn’t be celebrating the wage “gains” of the current era. (And let’s not even talk about actual wages received, since Barry Ritholz has that territory well-covered and then some).

*If you want to make the case that I should be using real wages, as Leonhardt does, please demonstrate (a) that all wages are renegotiated during a period of inflation, (b) that all parties are able to estimate inflation—even when at relatively unprecedented levels—accurately, and (c) that such negotiations were legally and commercially allowed during the period.

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Labor market rents can cause business cycles

Robert Waldmann

I’m not sure whether (more likely wherE) this has been noted in the literature, but wage differentials not due to differences in workers’ skill are enough to generate a business cycle. A verbal “model” after the jump.

update: additional model with fixed capital added.

The key reference from which this is not quite obvious is Kevin Murphy Andrei Shleifer and Robert Vishny (1989) “Industrialization and the Big Push” Journal of Political Economy vol 97 pp 1003- which contains three models only one of which is The Big Push. Another discusses wage differentials not due to worker characteristics. Since Murphy is one of the authors, they are assumed to be compensating differentials not labor market rents but it doesn’t matter (except for welfare analysis).

OK the problem: Assume that workers all have the same skill yet different wages are paid in different industries. Can this imply multiple Pareto ranked steady states ? Can this imply that there are sunspot equilibria in which GNP changes just because people expect it to change ?

To make the problem hard, assume that everyone is rational, that there is perfect competition (except for the wage differentials) and that all goods are non durable so consumption must equal production and the real interest rate just makes aggregate demand equal aggregate supply.

The last assumption is needed, because a model with two sectors can have a sunspot equilibrium due to the effect of (spontaneous just because people expect it) sectoral shifts on the real interest rate (Boldrin and Rusticchini in Econometrica). I find models which rely on the effect of changes in the real interest rate on the rate of growth of aggregate consumption or labor supply to be unconvincing as there is almost exactly no sign of such effects in the data (that criticism applies to 3 published papers with my name on them).

OK the “model.” Time is continuous. People are free to borrow and save at interest rate r. They have a rate of time preference rho

There is no capital. There are two non-durable goods produced with labor alone. There is nothing odd about good 1. One unit of it is produced with 1 unit of labor.

1) C_1 = L_1

where C_1 is consumption of good 1 and L_1 is employment in the sector which produces good 1.

The price of good 1 is normalized to 1. There is perfect competition so w_1 (the wage paid in sector 1) is equal to 1.

One unit of labor is needed to produce good 2

2) C_2 = L_2

total labor supply is fixed L_1+L_2 = L

However for some reason (it’s ok except for the welfare analysis if its a comepensating differential, that is it is more unpleasant to work in sector 2)

3) w_2= a > 1

there is perfect competition so the price of good 2 P_2 = a.

update 5: I am assuming that each individual worker moonlights working in both sectors and that they all spend the same fraction of their time in each sector. In other words I am making assumptions so that all individuals have exactly the same income. This is silly but makes things simpler.

Now a knife edge very special case for clarity. The goods are perfect substitutes

U = aC_2+C_1

This means that any C_2+C_1 = L can be a steady state equilibrium.

Welfare equals (L + (a-1)C_2)/rho = (aL-(a-1)C_1).

For this knife edge special case, there are a continuum of Pareto ranked steady state equilibria. People are better off if they work in sector 2. Any number can work in sector 2.

Well that’s an extreme example. Now how about this one Good_1 is an inferior good. This means that for fixed relative prices (and relative prices must be fixed) there is an interval of total consumption over which consumption of good 1 declines.

update 3: I think I was totally confused here
update: Assumption immediately below is a correction of the assumption in an earlier version. Sad to say the new assumption is the assumption which I hate.

Consumption/permanent income decreases as the interest rate increases

permanent income is the expected discounted value of the flow of income. Call it pY.

Oh no now I need a graph.

Figure 1

figure 1

Consider possible steady states. Consider income and consumption of good 1. The line shows income as a function of consumption (and production) of good 1 . The curve shows consumption of good 1 as a function of pY.

Note there are two steady states and one is Pareto better than the other (C_2 is higher).

OK now figure 2 shows Y as a function of pY.

figure 2

Define total consumption as C = C_1+aC_2

So there are two steady states one with high output Yg and the other with low output which I will call Yb.

What’s more the economy can jump from one to the other. let’s say it switches according to a poisson alarm clock with the sunspot causing a switch arriving at rate p. Or hell let’s say time is discrete and they switch each period with probability p.

In each steady state consumption equals to income so that the identity C=Y and the equations which give C_1 and C_2 as a function of y (for P_2 fixed and equal to a from the supply side) and Y as a function of C_1 and C_2 both hold.

Furthermore the economy can jump stochastically from one to the other. How ? Well lets say we are in the good state. The national income identity means C=Y. However individuals are free to borrow and save at interest rate r. They decide consumption given permanent income. This means that r can’t equal rho. They know that their income might fall so if y=Yg all agents will try to save if r=rho. They can’t. we must have C=Y because there is no way to invest.

So r must be lower than rho. This means that they are satisfied if the expected marginal utility of consumption increases. It does it jumps up to the higher marginal utility of consumption in the bad steady state with rate p. For any p r can be calculated so that people neither want to borrow nor save if they are currently at the good steady state.

Similarly at the bad steady state. Here r must be greater than rho since income and consumption might jump up.

update 4: Now I add capital to the model. The model above is very strange as there is no fixed capital or trade so consumption must always be equal to production. Also one of the stylized facts about wages, and, in particular, wages which are surprisingly high given worker characteristics is that they are high in industries with a high capital labor ratio. In the model above, both industries have a capital labor ratio of 0.

So now there is a third good K in addition to C_1 and C_2. At any given time total capital equals K = K_1+K_2 where K_i is capital used in sector i. Y_1 and Y_2 are production of each type of good no longer equal to C_1 and C_2. Y = Y_1+Y_2 and C=C_1+C_2

The assumption about wages becomes

5) w_2 = (w_1)a

since now wages depend on the capital labor ratio.

I will assume that the share of capital is constant and the same in each industry. This means that the relative prices of the goods will be constant. For further simplicity I am going to make assumptions so that constant is 1 so P_2 = P_1. This is all for tractability and is not realistic. In fact not only is K/L high in high wage industries but so is the share of capital (capital income)/wL which will be just rK/wL here. But I assume that rK/wL = alpha in both sectors cause it makes things easier. I equations I assume

5) y_1 = A(L_1)^(1-alpha)(K_1)^alpha


6) Y_2 = A(aL_2)^(1-alpha)(K_2)^alpha

So both sectors have similar Cobb-Douglas production functions. For the same amount spent on labor and capital (which means fewer physical hours of work in industry 2 since each gets a higher wage) the sectors have the same output so the two goods are sold for the same price which I set to 1. This makes everything relatively simple and means that the crude definitions like Y = Y_1+Y_2 make sense.

I want to keep things simple so I will assume that you can create 2 units of K from 1 unit of good 1 and one unit of good 2. This is an absolutely rigid no substitution allowed leontief type function. capital depreciates at rate delta. so

5) dK/dt = -deltaK + 2min(Y_1-C_1,Y_2-C_2)

This means that

6) dK/dt = -deltaK + (Y-C)

and the price of one unit of capital is 1, that is equal to the price of one unit of consumption good 1 which is equal to the price of one unit of consumption good 2.

I assume that delta is high enough that no one ever wants to convert K back to consumption goods (or that they can do that which is silly but standard). Also I assume either that delta is high enough that no one ever wants to take capital from sector 1 and add it to sector 2 or that this is possible. Again silly but standard.

Note that a shift of labor from sector 1 to sector 2 will increase demand for capital and will give a higher marginal product of capital r for the same total amount of capital K.

Finally I will make an assumption about tastes. For p_1/p_2 = 1 (which it must be given the supply side) C_1 and C_2 as a function of total consumption are given by figure 3. As C goes up from zero the ratio C_2/C_1 is constant for a while, then C_2/C_1 increases then it is constant with a higher C_2/C_1.

In the regions where C_2/C_1 doesn’t change with total consumption C, this model behaves just like a standard Solow model. However, when C_2/C_1 is higher it is as if there has been labor augmenting technological progress since labor is more productive in sector 2.

Recall I am assuming all workers divide their time and work in both sectors and have equal identical income. I also they are the saver/ investors and all have equal wealth so everyone always has the same income. Silly but standard in macro.

This means that, for the right a and assumptions about tastes, there can be two steady states again. In each steady state the real interest rate is equal to the rate of time preference r=rho, but in the good steady state C_2/C_1 is higher than in the bad steady state so the K/L needed to make r=rho is higher and output is higher and consumption is higher so, given tastes the ratio C_2/C_1 is higher.
Define the two steady state levels of capital as Kg and Kb with Kg>Kb and may use the subscripts the same way for other variables.

It is no longer possible to jump from one steady state to another. K is a state variable and changes slowly. C can jump.

There is a sunspot equilibrium where the economy spends much of it’s time near one of the steady states.

If K is just above the good steady state K (Kg) then rrho.

If it jumps the economy finds itself on the path of dC/dt and dK/dt which leads to a point with K just below bad steady state K. Once it gets real close to this point, the sunspot begins giving a jump up signal which arrives at rate p again. This makes consumption remain constant if the jump up signal doesn’t arrive at a point where r>rho so K below Kb.

If the jump up signal arrives then C jumps up to the region with high constant C_2/C_1 to exactly the point where the dC/dt and dK/dt equations lead it to the original point with K a little bit greater than Kb.

Also now the economy can have much more complicated dynamics. It is possible to make an equilibrium in which it can jump at any time and not just when close to a steady state.

update 3: I think everything below is totally confused

Note again two steady states. Also note that for the Pareto better steady state (with higher Y) dY/dpY >1. This means that this steady state is a stable steady state with Y = Yg (for good). if Y starts slightly above steady Yg then it will converge to steady Yg. This means that there is a sunspot equilibrium in which Y bounces around Yg just because people expect it too (rational animal spirits).

update 2: The assumption which I hate that, given permanent income, consumption decreases in the interest rate is not needed for there to be a stable steady state and sunspot equilibria. It is just needed so that the good steady state is stable. If, in contrast, consumption increases in the interest rate for given permanent income, then the bad steady state is stable. There is a very general result that when you pass from zero steady states to two (as in the figure) then one of the steady states is stable and one is unstable.

I will attempt to discuss the dynamics of Y near a steady state. I will use linear approximations (I have to do this to have any hope of writing out the explanation with plain ascii. That approximation is not necessary for the result.

There are two steady states Yg defined above and the one with lower Y which I now name Yb.

I will try to find the time derivative of Y, dY/dt for Y near a steady state. If d(dY/dt)/dY is negative then when Y is above the steady state Y falls down to the steady state. This means that the steady state is a sink, it is stable. It means that there are sunspot equilibria where the economy bounces around the steady state.

I will make an assumption about the utility function. First define total consumption

C = C_1+aC_2

note that C=Y.

Given the relative price P_2 = 2P_1, C_1 and C_2 are functions of C.

I assume that
the marginal utility of consumption of good 1 is equal to

4) U_1(C_1,C_2) = C^(-sigma)

This must be equal to (1/a) times the marginal utility of consumption of good 2.

Now consider the real interest rate r. Even though there is no saving and investment, there is a market clearing r such that no one wants to save or borrow.
Equation 4 implies that

5) (dC/dt)/C = (r-rho)/sigma = (dY/dt)/Y

(recall the national accounts identity is just C=Y).

now consider constant r (just for now)

permanent income at t (pY_t) is the integral as s goes from zero to infinity of exp(-rs)Y_(t+s)ds

Given 5 that equals the integral as s goes from zero to infinity of

exp(s(r(1-sigma)-rho)/sigma)Y_t so

6) pY_t = Y_t(-sigma/(r(1-sigma)-rho))= Y_t(sigma/(rho+(sigma-1)r)

if r = rho then pY_t = Y_t(sigma/sigma) = Y_t

if sigma is greater than 1, then pY_t/Y_t decreases in r. If sigma is less than one then pY_t/Y_t increases in r. Let’s assume that sigma>1 (the assumption I like). This means that Yb is a stable steady state.

OK so that wasn’t very hard, but the problem is that r_t changes as Y_t changes.

I’m working on it (update 3.1 not any more. I realize I was assuming that there was some way to save.

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