Suppose that monopoly power (as measured by the markup of price over marginal cost) has been increasing over time. What effect would that increase in monopoly power have on the equilibrium (real) rate of interest?

TL:DR the sign is right, but the magnitude looks far too small.

Let's start with a very simple model. Assume a technology: C+I=Y=F(K,L) and *d*K/*d*t=I-dK, where K is the stock of capital goods measured in physical units, I is the flow of new capital goods produced, d is the physical depreciation rate, so *d*K/*d*t is net investment, L is employment, C is consumption goods produced, and Y is total goods produced.

**It is important to understand that C+I=Y is not, in this case, merely an accounting identity. It is an engineering relationship that tells us that the PPF for producing capital and consumption goods is a straight line (with a slope of minus one).** That's because we are measuring C and I in physical units, not in value units.

Start with perfect competition. Let the price of the consumption good be the numeraire (we set it equal to one). The linear PPF between C and I determines the price of the capital good at one. Pk=1. Profit-maximisation ensures the equilibrium (real) rental rates on labour and capital goods are equal to their marginal products: W = MPL (= *d*Y/*d*L) and R = MPK (= *d*Y/*d*K). And the (real) interest rate is given by: r= R/Pk - d = MPK - d.

Now introduce monopoly power, so that price is a markup m over Marginal Cost: P = (1+m)MC = (1+m)W/MPL = (1+m)R/MPK. Assume that the degree of monopoly power (and so the markup of price over marginal cost) is the same for investment goods as consumption goods, so their relative price is unaffected, and remembering that the consumption good is numeraire, we get:

W = MPL/(1+m) and R = MPK/(1+m) and **r = MPK/(1+m) - d**

**For a given employment of capital and labour (and so a given MPK), an increase in the degree of monopoly power m across the whole economy causes a reduction in the (real) interest rate.** The intuition is simple: increased monopoly power causes lower (real) rentals on capital goods (just like it causes lower (real) wages), and in equilibrium the interest rate must fall to match the lower rate of return on buying capital goods and renting them out.

**Let's do a back-of-the-envelope estimate to see how big this effect might be.**

Assume initially d=10%, m=0%, r=5%, so MPK=15%. To reduce r to 0%, we would need to increase m from 0% to 50%. **That means it would take a very big increase in monopoly power to explain why real interest rates have fallen to near 0%.**

[Update: that same increase in monopoly power would also cause (global) wages to fall by 33%, which looks implausibly large. Plus, if markups had increased by that large an amount, I think I would have heard empirical IO economists shouting it from the rooftops.]

Let's try another one. Assume there is a 5% risk-premium, so a 5% safe interest rate translates into a 10% risky interest rate from buying capital goods. So initially d=10%, m=0%, r=10%, so MKP=20%. To reduce r to 5% (so the safe interest rate falls to 0%), we would need to increase m from 0% to 33.3%. That's a little bit more plausible, but still quite large.

[Update 2: Hugo Andre in comments says that 4% is a more plausible number for depreciation than the 10% I assumed, which makes the size of the increase in monopoly power needed to get a big enough effect on interest rates even more implausibly large.]

**What would it take to make the effect bigger?**

1. I have done my calculations taking employment of labour and capital as given (so MPK is given).

1a. If the labour supply curve slopes up, the fall in real wages due to increased monopoly power would reduce employment of labour, reducing MPK, which would mean a bigger fall in the rate of interest. But if the labour supply curve is very inelastic, this effect will be small.

1b. If saving (and hence investment) is a positive function of the rate of interest, the fall in the rate of interest due to increased monopoly power will reduce the stock of capital over time, increasing MPK over time, which would mean a smaller fall in the rate of interest. So that effect goes the wrong way.

2. We could assume that there is a bigger increase in monopoly power in the investment goods sector than in the consumption goods sector, so that the price of capital goods rises relative to consumption goods, which would reduce the interest rate by more. But that's a purely ad hoc assumption, which I have no independent reason for believing.

3. We could relax the assumption of a linear PPF between consumption and investment goods, and replace it with the standard assumption of a curved PPF where the price of capital goods is an increasing function of investment (so the supply curve of new capital goods slopes up). If saving (and hence investment) is an increasing function of the rate of interest, a fall in the rate of interest due to increased monopoly power would reduce Pk, which would partly offset that fall in the rate of interest. So that effect goes the wrong way too.

4. ?

Hmmm. **I think the answer to my question is a tentative partial "yes", but I don't think the effect of increased monopoly power is big enough to explain more than a part of the decline in equilibrium interest rates ("secular stagnation").**

[Addendum: I started writing this post after reading a Mark Thoma post about a Nick Bunker post about a Dietz Vollrath post. Then I read those posts.

Dietz says: *"If firms are gaining market power – meaning they can charge a higher markup – then this implies that they will use inputs less efficiently from a social perspective. Each individual firm is producing less than the amount they would under competition (with costs = marginal costs), and so we are not getting everything we can out of our inputs. If market power has increased, this exacerbates that issue, and so measured productivity – the efficiency of input use – will fall.*"

That is not correct in aggregate. Consider an economy that produces apples and bananas only. If apple producers gain market power, while banana producers stay competitive, so the price of apples rises relative to bananas, then it is true that too few apples and too many bananas will be produced. But if both apple and banana producers gain market power equally, then the relative price stays the same. It is true that real wages will fall, and so employment will fall too if the labour supply curve slopes up, but that says nothing about the social efficiency of production given that (inefficiently low) level of employment.

Though Dietz's *main* point in that post, about *measured* productivity growth, is I think correct and important.

Nick Bunker says: "*But Vollrath’s market power explanation for falling productivity growth, alongside the falling share of national income going to wage earners, is supported by some evidence. Work by Massachusetts Institute of Technology graduate student Matt Rognlie, for example, found evidence of higher markups.*"

On a quick skim, the only relevant bit of Matt's paper I could find is on page 16: "*The long-term U-shape in p may also indicate some changes in market power or the scope for monopoly profits. To the extent that the rents giving rise to pure profits can sometimes be shifted to workers, fluctuations in p could in principle reflect changes in worker/firm bargaining power or the role of unions. In practice, however, the U-shape path for p is difficult to interpret along these lines: p is lowest in the 1980s, a decade that certainly was not known for union strength.*" ("p" is supposed to be Greek "pi", or capital's share in income, and the

**bold**emphasis was added by me.)

Let's just say "maybe" to the idea that monopoly power has been increasing over time.]

This result would be truly ironic since (as PK among others has commented on) a recent BIS paper convincingly argues that lower interest rates have led to a reduction in bank profits. This is mostly because bank deposits are priced as a markdown on market rates, reflecting oligopoly power and transaction costs but deposit rates can't (or at least haven't) gone below zero. So the low interest rate have led to a lowering of profits due to oligopoly power.

Unfortunately, I think your assumption for the rate of depreciation is far too high. Piketty who has obviously spent a lot of time working with capital assumes a 3% rate of depreciation and even such a venerable institution as the Penn World Tables sets the rate to 4%. (I worked on a paper that required a reasonable number for this parameter value a little while back). If we use a 4% rate it takes a huge increase in monopoly power to explain the interest rate reduction (125% in the benchmark case, ~55% with the risk premia).

Posted by: Hugo André | October 03, 2015 at 11:12 AM

The intuition that increased monopoly power would reduce the demand for capital goods and therefore the rate of return on capital goods seems sound.

I can see couple of other factors that might add to this effect and lead to rate to get close to zero.:

- Most capital good are durable and it takes a ling time for a reduction in demand to be reflected in a reduction in supply and this would amplify the effect of increased monopoly power in the short run.

- If people are accumulating capital goods (directly or indirectly) as a way of savings for retirement then if they see a fall in interest rates that is expected to be permanent they may have no choice but to accumulate more to try and maintain their consumption level in retirement. More demand for capital goods as a form of savings and less demand for capital goods as means of production could drive interest rates very low (and give people like me who would like to retire one day a huge headache).

Posted by: Market Fiscalist | October 03, 2015 at 11:30 AM

Hugo Andre: thanks. I'm not very knowledgeable about plausible numbers for depreciation rates. But 4% sounds a bit low to me. That's not including land as part of the capital stock, is it?

MF: "Most capital good are durable and it takes a ling time for a reduction in demand to be reflected in a reduction in supply and this would amplify the effect of increased monopoly power in the short run."

I was holding K constant in my initial back of the envelope calculations.

"If people are accumulating capital goods (directly or indirectly) as a way of savings for retirement then if they see a fall in interest rates that is expected to be permanent they may have no choice but to accumulate more to try and maintain their consumption level in retirement."

OK, you are making the opposite assumption about savings that I made in my point 1b. Yep, that would amplify the effect.

Posted by: Nick Rowe | October 03, 2015 at 12:49 PM

Yes, it surprised me too. The result looks solid though, Here is a link that has PWT's discussion about how they decided on this rate. And no it does not include land, the capital stock is mostly made up of the "structures" category. The 7% rate of depreciation commonly used in textbooks appears to be a bit dated.

Posted by: Hugo André | October 03, 2015 at 02:24 PM

Great analysis. I agree that for any increase in markups of plausible magnitude (which would be a few percentage points, not 50%), we can't come anywhere close to matching the decline in r through this channel.

Indeed, I think there's another force that pushes in the other direction, and is probably larger in magnitude:

the PDV of profits is part of the supply of assets. Let's denote this PDV by Pi, and denote the overall supply of assets by W = Pi + K. W is a combination of physical capital and discounted profit. Along a balanced growth path with growth ‘g’, write Pi = pi/(r-g), where ‘pi’ is the flow of profits.Now, suppose that it’s overall savings W that’s exogenous, not just savings in the form of capital K. If the markup ‘m’ rises substantially, Pi will explode, and ‘r’ may actually need to rise so that — by decreasing the demand for capital and discounting profits more aggressively — K+Pi remains unchanged at its exogenous level.

More generally, let’s think about asset market equilibrium in (W/Y,r) space. I’ll leave the savings side unspecified - it’s just some relationship between the asset-output ratio W/Y and the return ‘r’. I’ll concentrate instead on the asset supply side, which is represented by a downward-sloping curve: higher ‘r’ decreases both K/Y (because less capital is demanded when the user cost increases) and Pi/Y (because the stream of profits is discounted at a higher rate).

If we want to understand the effect of markups, it’s natural to ask how this curve responds to a markup shock. In particular,

if there is a positive shock to ‘m’, does the curve move to the left or to the right? In other words, holding ‘r’ constant, does an increase in ‘m’ cause W/Y to increase or decrease?In general (unless the elasticity of savings W/Y with respect to r is extremely negative, which seems unlikely), ‘r’ will decrease in response to a positive shock to ‘m’ iff the curve moves to the left, i.e. if W/Y decreases in response to ‘m’, holding ‘r’ constant.

Is this true? Clearly K/Y is decreasing and Pi/Y is increasing in ‘m’, so this is a question of relative magnitudes. Write M=1+m. Then holding ‘r’ constant:

(A) The derivative of K/Y with respect to log M is -sigma*(K/Y), where sigma is the (local) elasticity of substitution between capital and labor.

(B) The derivative of Pi/Y with respect to log M is (1/(r-g))*(1/M). (Because a 1% increase in M means that roughly 1% of the fraction 1/M of aggregate income that was going to K and L now goes to profits, and to get the PDV of this increment in profits we divide by r-g.)

Combining (A) and (B), we see that W/Y is decreasing in ‘m’ iff sigma*(K/Y) > 1/(r-g)*1/(1+m)

And if we put numbers to it, this turns out to be extremely unlikely. Suppose that sigma = 1/(1+m), which seems roughly accurate (the elasticity of substitution is a little below one, and the share of income going to K and L rather than markups is a little below one). Then this inequality reduces to K/Y > 1/(r-g), which is crazy: the ratio of capital to output in the US private sector is roughly K/Y = 3, whereas even if r=10% and g=2% we have 1/(r-g) = 12.5.

So: quantitatively, it looks like a rise in markups leads to a net rise in W/Y, conditional on ‘r’. Hence the real interest rate actually has to

riseto achieve asset market equilibrium, and now our effect goes in the wrong direction! : )Conceptually, what’s happening is that the increase in asset supply from discounted profits is “crowding out” capital and forcing up interest rates. This effect is quite large and overwhelms the effect you discussed in the post, since that effect (as you observed) is relatively small.

Posted by: Matt Rognlie | October 04, 2015 at 06:02 AM

Some additional notes:

(1) More realistically, the savings side of a model would give a relationship between W/(Y-delta*K), the ratio of assets to

netoutput, and ‘r’. I ignored this above because it makes things messier and doesn’t seem to change the results very much.(2) All the above is assuming that the real interest rate used to discount profits is the same one that enters into capital demand, including (potentially) the same risk premium.

(3) Hugo and Nick, for the discussion about the rate of depreciation, this is a really tough question. If you just go to the BEA’s fixed asset accounts and divide line 1 of Table 1.3 (current-cost depreciation of fixed assets and consumer durables) by line 1 of Table 1.1 (current-cost net stock of fixed assets and consumer durables), you get an implied depreciation rate of 6.4%, which is closer to the traditional value.

If you restrict to private fixed assets (line 3), then it’s 5.5% instead, which splits the difference between 4% and 7%.

Even this rate, though, is heavily influenced by a few extremely high-depreciation assets, some of which could easily be classified as intermediate rather than capital goods if national accountants woke up on the other side of the bed; and it’s not clear that a simple arithmetic mean (weighted by the stock of each type of asset) is the right way to do aggregation here anyway. It really depends on what you’re trying to do.

For some purposes, you want to place a higher weight on the lower-depreciation assets. For instance, if you’re looking at the response of capital demand to a change in the real interest rate ‘r’, then since ‘r’ is a larger fraction of the user cost for low-depreciation assets, it turns out that the appropriate stand-in aggregate ‘delta’ here is actually lower than the simple average.

[This is intuitively clear if you take an extreme case. Let’s take a continuous-time model where there are two types of capital. 99% of the capital stock is in type A, which has a flow depreciation rate of 0%. 1% of the capital stock is in type B, which has a flow depreciation rate of 1000%. The average flow depreciation rate in this world is 10%.

Suppose that both types of capital have an elasticity of demand of ‘1’ with respect to their user cost r + delta. Then if the real interest rate ‘r’ goes from 5% to 10%, demand for type A will decrease by 50%, while demand for type B will barely be affected, leading to an overall decline in capital demand of about 49.5%. But if we ignored capital heterogeneity and looked at the ‘aggregate’ user cost r+delta, with delta=10%, then the aggregate user cost would go from 15% to 20%, and capital demand would decline by 25%, much less than the true amount. The problem here is that the average depreciation rate delta=10% is not an appropriate stand-in; instead, to get the right answer we’d need a delta much closer to 0%, with a much lower weight on capital type B.]

So… to sum up, it’s complicated. At least in the US, the average depreciation rate by most measures is more than 4%. But when we’re adopting the fiction of a single type of capital and a single depreciation rate for modeling purposes, we might want to pick something lower than the simple average anyway.

Posted by: Matt Rognlie | October 04, 2015 at 06:04 AM

"we are measuring C and I in physical units, not in value units."

What physical units are you using? Weight? Volume? Other?

Thanks.

Posted by: Min | October 04, 2015 at 06:28 AM

Also, with respect to my paper, at one point there is a decomposition that shows an expanded role for markups over time, though I would take it (like most work in this area) with a grain of salt.

The driving force is the fact that over time, the ratio of corporate "market" value (equity plus net financial indebtedness) to "book" value (nonfinancial assets) in the aggregate data has increased substantially. As a consequence (omitting some much more complicated intermediate steps), the decomposition allocates a larger share of today's corporate capital income to markups rather than a return on nonfinancial assets.

Is this appropriate? I'm not sure. The rise of market relative to book value could (and to some extent surely does) reflect an increase in unmeasured assets rather than markups -- for instance, organizational capital, IT capital that isn't fully measured in the data, etc. The magnitudes are a little implausible, though. It could also reflect problems with the data on both market and book value.

Or (and this would probably be the most popular explanation) it could reflect better corporate governance, where shareholders are now able to extract better returns. It this case, however, it might still be appropriate to model the improvement as a rise in "markups" over cost -- since the only way that bad governance can ultimately lower market value conditional on assets is if it increases costs or bungles the reinvestment of profits, which means a smaller long-term "markup" in some sense.

My main message is that the behavior of the net capital share in the corporate sector (in contrast to its steady rise in the residential sector) is bewildering and hard to reliably explain. : )

By the way, a more recent version of the paper is here: http://www.mit.edu/~mrognlie/brookings_final_spring2015.pdf

It's not the final proofed copy, which will be coming soon, but it does elaborate upon and fix some of the sloppier aspects of the original draft.

Posted by: Matt Rognlie | October 04, 2015 at 06:31 AM

If businesses can generate their capital through superprofits, then there is less demand for financing.

I think this has more intuitive appeal than your explanation about rental rates of capital goods: if I can enjoy monopoly pricing, won't physical capital yield higher profits per unit of capital?

I agree that the monopoly markup would have to be very very large to have a significant impact on equilibrium real interest rates.

I recently read that corporate profits as a share of GDP (USA) are at very high historic levels (about 10%). While it is plausible that this is the result of cost cutting, it would be consistent with the suggestion that many firms are enjoy monopoly pricing (I suggest pharmaceuticals and software/internet/electronics - Microsoft, Google, Facebook, Apple, as some potential areas where monopoly pricing seems obvious considering their very high profits to capital ratio and amount of idle capital.

A question though: rather than focusing on monopoly pricing per se, would you consider that the giant cash hordes of major firms should themselves have an effect on the real interest rate? Again, according to the logic that if firms are sitting on lots of cash, the aggregate demand for financing should be lower.

Posted by: Nathan W | October 04, 2015 at 07:39 AM

Matt R: "Indeed, I think there's another force that pushes in the other direction, and is probably larger in magnitude: the PDV of profits is part of the supply of assets."

Hmmm. Really interesting idea. It is closely related to what Fritz Machlup(?) called "The Junker Fallacy" (though it is not obvious that it is a fallacy). The Junker "fallacy" was the argument that Prussia had a low level of capital because the junkers (landowners) saved in the form of land instead of capital. (You sometimes hear a similar argument in the UK: that people are "investing" their savings in houses (i.e. the land on which houses are built) so there is too little investment in capital.) ((And maybe also applied to slavery in ante-bellum southern US??)) Fritz Machlup(?) thought it was a fallacy.

Take a model with labour only, and no capital, Y=L. Start in competition, where W=1, then suppose the government hands out monopolies in apples, bananas, carrots, etc. So wages drop to W=1/(1+m). The monopoly rights can be bought and sold. How will this affect the rate of interest?

In a simple infinitely-lived representative agent model it will have no effect on r, which is pinned down by the rate of time preference proper (plus the growth rate, if any). Even with heterogenous agents it's not obvious it has any effect; the monopoly rights simply duplicate what could be accomplished by consumption loans from the patient to the impatient (but maybe it's easier to enforce monopoly rights than claims on future earnings).

But in an OLG model, where people desire a stock of savings (for their old age) rather than a flow of saving, it would raise the rate of interest just like Samuelsonian money. Because the monopoly rights are functionally equivalent to a government debt financed by future taxes on the unborn, in a non-Ricardian world.

Neat!

Posted by: Nick Rowe | October 04, 2015 at 08:49 AM

Matt on depreciation: really good stuff. I never understood where to draw the line between capital and intermediate goods. Just like a trade-weighted exchange rate should really be weighted both by trade *and* by export-import elasticities. Ah, the perils of aggregation!

Posted by: Nick Rowe | October 04, 2015 at 09:19 AM

Min: there is a stock of chickens K. If you do nothing, that stock falls by dK each year, where d is a parameter. If you add your labour L you produce Y=F(K,L) new chickens each year, which can either be eaten C or kept as breeders I. The Marginal Rate of Transformation between C and I (the slope of the Production Possibilities Frontier between C and I) is always minus one.

It does not matter that the slope is minus one, because we can always change our units of measuring C or I so that the trade-off becomes minus one with those new units. But it does matter that the slope is a constant number. If you measure C and I in value units (dollars, or inflation-adjusted dollars), then anything that changes the relative price of C and I will make it look like C and/or I have changed (to the accountant), even though the engineer in charge of production sees no change in the physical numbers of C and I. If the PPF is a curve, then the MRT (the slope) will change as you move along it, and in competitive equilibrium MRT=Pk/Pc, so relative prices will change.

This was at the root of the Cambridge Capital Controversy.

Posted by: Nick Rowe | October 04, 2015 at 09:33 AM

Matt: on market vs book value. If book value is measured at historical cost, and there is a fall in r, we would expect to see a rise in market value over book value, at least temporarily until the old assets fully depreciate and are replaced. Could that be another explanation?

Posted by: Nick Rowe | October 04, 2015 at 09:43 AM

"In a simple infinitely-lived representative agent model it will have no effect on r, which is pinned down by the rate of time preference proper (plus the growth rate, if any)."

Yep. In the infinite-horizon representative agent case, the steady-state elasticity of net aggregate savings with respect to the interest rate is infinity, so there's no effect from marketable monopoly rights, or (for that matter) from the original monopoly power effect hypothesized in this post. But then again, there's no effect from anything at all except r = rho + sigma*g, where rho is time preference, g is growth and sigma is the inverse EIS.

And trying to explain the evolution of 'r' in terms of these constituents isn't that much fun. Nor is it too realistic - it doesn't seem credible that the elasticity of net aggregate savings is literally infinity. But once the elasticity is less than infinity, the Junker Fallacy is no longer a fallacy (!), and anything that increases net asset supply will push up equilibrium 'r' and crowd out other forms of investment.

(Caveat: the Junker Fallacy *is* still a fallacy to the extent that it claims that this crowding out is necessarily

bad. If there are incomplete markets and/or borrowing constraints, and households need net assets to smooth consumption, then they'll invest heavily in physical capital if no other net assets are available. But if you provide the households with some other asset to save in - say, land - then this will often make them better off even though capital investment is crowded out and their wages fall as a result. "Crowding out" isn't always bad; instead, by providing additional stores of value, you diminish the liquidity-providing role of the marginal unit of capital and thereby push down the optimal level of the capital stock. This is implicit in Aiyagari and McGrattan (1998)’s study of the optimum quantity of government debt; the debt’s liquidity-providing role is beneficial, though the optimal quantity is limited because the government needs to service it using distortionary and distributionally costly taxation. More generally, welfare analysis is complicated because in these kinds of models we’re so far from the first-best, but the point remains that crowding out is often a good thing.)My calculation above was trying to be agnostic about exactly what the elasticity of net aggregate savings is, and what its underpinnings are (since this is such a complicated issue), just assuming that it’s somewhere between zero and infinity. If that’s true, then a rightward shift in the net asset supply schedule will increase equilibrium ’r’, and a leftward shift will decrease it. The key question then is whether the net effect of a rise in monopoly power is to shift the net asset supply schedule rightward or leftward… and the rightward effect seems to dominate, because tradable monopoly rights are worth a lot.

Posted by: Matt Rognlie | October 04, 2015 at 09:54 AM

Nathan: "If businesses can generate their capital through superprofits, then there is less demand for financing.

I think this has more intuitive appeal than your explanation about rental rates of capital goods: if I can enjoy monopoly pricing, won't physical capital yield higher profits per unit of capital?"

Your first sentence, if true, would predict lower interest rates. (But it would only be true if the shareholders somehow forget about their higher profits, so total saving, including retained earnings, rises.)

Your second sentence, if true, would predict higher interest rates. (But it ignores the distinction between average return on historical capital and marginal rates of return on new investment.)

And you are using "capital" in at least two different senses.

Sorry, but it would take me too long to respond properly to your comment.

Posted by: Nick Rowe | October 04, 2015 at 10:04 AM

Matt: "My calculation above was trying to be agnostic about exactly what the elasticity of net aggregate savings is, and what its underpinnings are (since this is such a complicated issue), just assuming that it’s somewhere between zero and infinity."

Got it. Nice clean way to think about the general case.

Posted by: Nick Rowe | October 04, 2015 at 10:09 AM

"Even with heterogenous agents it's not obvious it has any effect; the monopoly rights simply duplicate what could be accomplished by consumption loans from the patient to the impatient (but maybe it's easier to enforce monopoly rights than claims on future earnings)."

Right, you usually need a little bit more to get a less-than-infinite elasticity of net savings -- some kind of incomplete markets or limitations on borrowing (so that claims on future earnings are limited, as you observe above).

The default model I have in mind that does this is a model in the Bewley-Aiyagari-Huggett tradition, where infinite-horizon households are hit by uninsurable shocks and face borrowing limits. (I guess the more traditional model to get a finite elasticity is an OLG model; this is the more computationally aggravating modern innovation!)

In such a model, steady-state net aggregate savings -- defined as aggregate household savings minus debt -- is strictly increasing in 'r', asymptoting to infinity as 'r' approaches the rate of time preference 'rho' of the most patient household. (The household wants to accumulate savings without bound if r=rho, because then the precautionary savings motive breaks the tie between time preference and interest rates in favor of saving.) Given finite net asset supply, in equilibrium 'r' will be somewhere below 'rho'.

One natural question to ask about this model is how far, when reasonably calibrated, it diverges from the representative agent model - in particular, what exactly is the steady-state elasticity of net aggregate savings? Strangely enough, it turns out that when the model is set up in the most natural and conventional way, the elasticity is still

really high. In partial equilibrium, increasing the interest rate by 3 percentage points might lead savings totriple. This is still so sensitive that the practical consequences aren't that different from the infinitely elastic case. This has been bugging me lately - I'm not sure whether it's saying something deep about the world, or whether it's just some kind of modeling artifact that I can't quite pin down. In the long enough run, is it perhaps true that aggregate savings are really, really, ridiculously elastic with respect to the rate of return? My intuition could go either way.Posted by: Matt Rognlie | October 04, 2015 at 10:22 AM

"Matt: on market vs book value. If book value is measured at historical cost, and there is a fall in r, we would expect to see a rise in market value over book value, at least temporarily until the old assets fully depreciate and are replaced. Could that be another explanation?"

The book value numbers I used are taken from the Integrated Macroeconomic Accounts, which measures them at current cost. I agree this would be the natural explanation if historical cost was used instead (as it usually is for "book value" as finance guys calculate it - the market/book ratios you see in finance data are crazy high compared to the ones calculated at current cost in macro data, where market/book is often <1!).

Posted by: Matt Rognlie | October 04, 2015 at 10:25 AM

Matt: " This has been bugging me lately - I'm not sure whether it's saying something deep about the world, or whether it's just some kind of modeling artifact that I can't quite pin down."

Think about the English landowning aristocracy, a dynastic model with primogeniture. Lots of things could go wrong to destroy a dynasty's wealth: no kids; all the heirs get killed in WW1 trenches; Lloyd George taxes inheritances; the heir has a gambling problem and blows the lot; the heir marries wrong; the heir joins a religious cult, or becomes a socialist, and gives it all away; someone rips it all off; the IRA burn it all down; etc. In other words, there are risks to your whole wealth that can't be diversified away and can't be insured against.

It's saying something deep about the world, but is leaving out multiplicative risks. Not exactly clogs to clogs in 3 generations, but sooner or later it will all get destroyed. So you might as well consume some of it now. (Hmm, maybe I should replace my MX6 with a Lotus, rather than let my kids blow it all.)

Just my twopenceworth; I may be missing the point.

Posted by: Nick Rowe | October 04, 2015 at 11:09 AM

More strictly, the multiplicative risks apply to your heirs' non-human wealth. God willing, they will still be able to work, wearing clogs.

Posted by: Nick Rowe | October 04, 2015 at 11:24 AM

@Matt You're absolute right, it IS complicated. I think a large part of the difference between BEA and PWT comes from the inclusion/exlusion of consumer durables. If you use BEA and divide line 2 "fixed assets" in 1.3 by line 2 in 1.1 you get a rate of 5.0%-5.2%. Because Nick's reasoning talks about depreciation in the context of MPK I think it makes sense to exclude durables. I'm not sure though and I may just be showing my ignorance here.

Now that I look more closely at the PWT data you're right that 4% is a bit low since Nick appears to be talking about rich countries (and they tend to invest a larger share of capital in computers that depreciate rapidly). To take a few examples, the depreciation rate is 4.1% for the US, 4.9% for Sweden and 4.6% for Switzerland so I guess the two measures are converging.

OT:

BTW I'm glad to see you commenting here since I used your work on Piketty as a source in the paper I mentioned earlier (which was about Piketty's second fundamental law).

Posted by: Hugo André | October 04, 2015 at 11:25 AM

Nick Rowe: "there is a stock of chickens K. If you do nothing, that stock falls by dK each year, where d is a parameter. If you add your labour L you produce Y=F(K,L) new chickens each year, which can either be eaten C or kept as breeders I. The Marginal Rate of Transformation between C and I (the slope of the Production Possibilities Frontier between C and I) is always minus one."

So K is a vector, with different units depending upon what is being measured?

Posted by: Min | October 04, 2015 at 07:07 PM

This was a very interesting analysis and NICK ROWE IS NOW ON TWITTER! Congratulations!

Posted by: rsj | October 05, 2015 at 09:10 AM

Nick,

Add intermediation / leveraged positions by the financial services industry and government borrowing to the discussion. Take your equation:

r = MPK (Marginal Product of Capital Goods) - d (Depreciation)

What capital good does the financial service industry offer? Or the federal government? In terms of total debt (courtesy of the U. S. Fed's Z1 report):

http://www.federalreserve.gov/releases/z1/Current/z1.pdf

Total Debt (Public and Private)

$59.2 Trillion

Financial Sector Debt

$15.2 Trillion

Federal Government Debt

$14.49 Trillion

I don't think it's monopoly power dragging down the real interest rate. I think it is enterprises offering a zero or near zero marginal product of capital goods dragging down the real interest rate.

Posted by: Frank Restly | October 05, 2015 at 08:06 PM