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Economy

Mechanisms, not numbers

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The NIESR claims that Johnson’s Brexit deal “would ultimately lead the UK economy to be about 3.5% smaller than it would have been had the UK remained in the EU.” Reaction to this, I fear, highlights an important way in which the media and public misunderstand economics. They pay too much attention to numbers (and seek a spurious precision in them) and too little to mechanisms.

To see what I mean, consider the NIESR’s reasoning. This starts from the fact that leaving the EU’s single market will throw a little sand into the wheels of trade by imposing trade barriers – “not the classic barriers of tariffs, but the insidious ones of differing national standards, various restrictions on the provision of services, exclusion of foreign firms from public contracts.” (I’m quoting Lady Thatcher, who called the single market “a fantastic prospect for our industry and commerce.”) A few days ago, Dominic Raab and Priti Patel claimed that Johnson’s deal was great for Northern Ireland because it gave the region “frictionless trade” with the EU. But if frictionless trade is good for them, its absence must be bad for us.

Now, you might object that these barriers will be small – and they will initially given that we start from the same regulations and product standards. But this is where another mechanism becomes absolutely crucial. My chart hints at what this is. It plots annual growth in world trade (as measured by the CPB) against UK GDP per worker-hour: the data starts in 1997 as this is when monthly GDP data begins. It’s obvious that there’s a strong correlation between the two.

Of course, a big reason for this is that the same things drive both, such as fluctuations in credit conditions, animal spirits and aggregate demand. But this is not the whole story. We’ve strong reasons to think that world trade drives productivity. One reason for this was pointed out in the first lines of the first book on modern economics. “The greatest improvement in the productive powers of labour” wrote Adam Smith “seem to have been the effects of the division of labour.” Greater world trade means greater division of labour and hence more efficiency. Also, freer trade encourages exporting firms to expand, which raises productivity as workers shift to these better firms. And then there’s the fact that more trade increases competition which spurs firms to raise efficiency and innovate. There’s also a knowledge diffusion channel; if managers visit suppliers and customers overseas, they should learn from them ways of increasing efficiency. As Danny Blanchflower says, you can learn a lot just by walking around.

These mechanisms are crucial. Over the long-run, they act as a multiplier. They cause apparently small barriers to trade to lead to significant losses of income because less trade means less productivity.

But we don’t know precisely how powerful these mechanisms are. Yes, fact and theory tell us their direction: the natural experiment of dividing Korea proves that openness creates prosperity. But their magnitude is uncertain; this is why the NIESR's estimate differs from that of the UK in a Changing Europe (pdf).

In fact, there’s another uncertainty. We don’t know what future trading arrangements will be. Brexiters claim the frictions in trading with the EU will be minimized by a deep free trade agreement, and compensated for by freer trade elsewhere. Many, though, are sceptical of this: as the old saying goes, a bird in the hand is worth two in the bush. Even 10% rises in goods exports to China or Japan would not compensate for a mere 1% loss of exports to the EU – and 10% is perhaps an optimistic estimate of what free trade deals typically achieve.

For these reasons, Julian Jessop is right to say there are big uncertainties around the NIESR’s estimate. This does NOT, however, justify philistine sniping at economists: “what do they know, ner, ner?” Instead, it directs our attention to the main issues: how strong is the link between trade and productivity? How likely are we to get meaningful trade deals? And what would these do to actually boost trade? It is possible to have a rational, intelligent debate about these issues – not that we’ll get one on the BBC.

Julian says these uncertainties cause him to “give little weight to long-term forecasts.” Some of us, however, draw the opposite inference: what is the point of introducing such uncertainties into the economy when there was no pressing need to do so?



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Economy

The Surplus Process

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How should we model surpluses and deficits? In finishing up a recent articleand chapter 5 and 6 of a Fiscal Theory of the Price Level update, a bunch of observations coalesced that are worth passing on in blog post form.

Background: The real value of nominal government debt equals the present value of real primary surpluses, [ frac{B_{t-1}}{P_{t}}=b_{t}=E_{t}sum_{j=0}^{infty}beta^{j}s_{t+j}. ] I ‘m going to use one-period nominal debt and a constant discount rate for simplicity. In the fiscal theory of the price level, the (B) and (s) decisions cause inflation (P). In other theories, the Fed is in charge of (P), and (s) adjusts passively. This distinction does not matter for this discussion. This equation and all the issues in this blog post hold in both fiscal and standard theories.

The question is, what is a reasonable time-series process for (left{s_{t}right} ) consistent with the debt valuation formula? Here are surpluses

The blue line is the NIPA surplus/GDP ratio. The red line is my preferred measure of primary surplus/GDP, and the green line is the NIPA primary surplus/GDP.

The surplus process is persistent and strongly procyclical, strongly correlated with the unemployment rate.  (The picture is debt to GDP and surplus to GDP ratios, but the same present value identity holds with small modifications so for a blog post I won’t add extra notation.)

Something like an AR(1) quickly springs to mind, [ s_{t+1}=rho_{s}s_{t}+varepsilon_{t+1}. ] The main point of this blog post is that this is a terrible, though common, specification.

Write a general MA process, [ s_{t}=a(L)varepsilon_{t}. ] The question is, what’s a reasonable (a(L)?) To that end, look at the innovation version of the present value equation, [ frac{B_{t-1}}{P_{t-1}}Delta E_{t}left( frac{P_{t-1}}{P_{t}}right) =Delta E_{t}sum_{j=0}^{infty}beta^{j}s_{t+j}=sum_{j=0}^{infty}beta ^{j}a_{j}varepsilon_{t}=a(beta)varepsilon_{t}% ] where [ Delta E_{t}=E_{t}-E_{t-1}. ] The weighted some of moving average coefficients (a(beta)) controls the relationship between unexpected inflation and surplus shocks. If (a(beta)) is large, then small surplus shocks correspond to a lot of inflation and vice versa. For the AR(1), (a(beta)=1/(1-rho_{s}beta)approx 2.) Unexpected inflation is twice as volatile as unexpected surplus/deficits.

(a(beta)) captures how much of a deficit is repaid. Consider (a(beta)=0). Since (a_{0}=1), this means that the moving average is s-shaped. For any (a(beta)lt 1), the moving average coefficients must eventually change sign. (a(beta)=0) is the case that all debts are repaid. If (varepsilon_{t}=-1), then eventually surpluses rise to pay off the initial debt, and there is no change to the discounted sum of surpluses. Your debt obeys (a(beta)=0) if you do not default. If you borrow money to buy a house, you have deficits today, but then a string of positive surpluses which pay off the debt with interest.

The MA(1) is a good simple example, [ s_{t}=varepsilon_{t}+thetavarepsilon_{t-1}% ] Here (a(beta)=1+thetabeta). For (a(beta)=0), you need (theta=-beta ^{-1}=-R). The debt -(varepsilon_{t}) is repaid with interest (R).

Let’s look at an estimate. I ran a VAR of surplus and value of debt (v), and I also ran an AR(1).

Here are the response functions to a deficit shock:

The blue solid line with (s=-0.31) comes from a larger VAR, not shown here. The dashed line comes from the two variable VAR, and the line with triangles comes from the AR(1).

The VAR (dashed line) shows a slight s shape. The moving average coefficients gently turn positive. But when you add it up, those overshootings bring us back to (a(beta)=0.26) despite 5 years of negative responses. (I use (beta=1)). The AR(1) version without debt has (a(beta)=2.21), a factor of 10 larger!

Clearly, whether you include debt in a VAR and find a slightly overshooting moving average, or leave debt out of the VAR and find something like an AR(1) makes a major difference. Which is right? Just as obviously, looking at (R^2)   and t-statistics of the one-step ahead regressions is not going to sort this out.

I now get to the point.

Here are 7 related observations that I think collectively push us to the view that (a(beta)) should be a quite small number. The observations use this very simple model with one period debt and a constant discount rate, but the size and magnitude of the puzzles are so strong that even I don’t think time-varying discount rates can overturn them. If so, well, all the more power to the time-varying discount rate! Again, these observations hold equally for active or passive fiscal policy. This is not about FTPL, at least directly.

1) The correlation of deficits and inflation. Reminder, [ frac{B_{t-1}}{P_{t-1}}Delta E_{t}left( frac{P_{t-1}}{P_{t}}right) =a(beta)varepsilon_{t}. ] If we have an AR(1), (a(beta)=1/(1-rho_{s}beta)approx2), and with (sigma(varepsilon)approx5%) in my little VAR, the AR(1) produces 10% inflation in response to a 1 standard deviation deficit shock. We should see 10% unanticipated inflation in recessions! We see if anything slightly less inflation in recessions, and little correlation of inflation with deficits overall. (a(beta)) near zero solves that puzzle.

2) Inflation volatility. The AR(1) likewise predicts that unexpected inflation has about 10% volatility. Unexpected inflation has about 1% volatility. This observation on its own suggests (a(beta)) no larger than 0.2.

3) Bond return volatility and cyclical correlation. The one-year treasury bill is (so far) completely safe in nominal terms. Thus the volatility and cyclical correlation of unexpected inflation is also the volatility and cyclical correlation of real treasury bill returns. The AR(1) predicts that one-year bonds have a standard deviation of returns around 10%, and they lose in recessions, when the AR(1) predicts a big inflation. In fact one-year treasury bills have no more than 1% standard deviation, and do better in recessions.

4) Mean bond returns. In the AR(1) model, bonds have a stock-like volatility and move procyclically. They should have a stock-like mean return and risk premium. In fact, bonds have low volatility and have if anything a negative cyclical beta so yield if anything less than the risk free rate. A small  (a(beta)) generates low bond mean returns as well.

Jiang, Lustig, Van Nieuwerburgh and Xiaolan recently raised this puzzle, using a VAR estimate of the surplus process that generates a high (a(beta)). Looking at the valuation formula [ frac{B_{t-1}}{P_{t}}=E_{t}sum_{j=0}^{infty}beta^{j}s_{t+j}, ] since surpluses are procyclical, volatile, and serially correlated like dividends, shouldn’t surpluses generate a stock-like mean return? But surpluses are crucially different from dividends because debt is not equity. A low surplus (s_{t}) raises  our estimate of subsequent surpluses (s_{t+j}). If we separate out
 [b_{t}=s_{t}+E_{t}sum_{j=1}^{infty}beta^{j}s_{t+j}=s_{t}+beta E_{t}b_{t+1}  ] a decline in the “cashflow” (s_{t}) raises the “price” term (b_{t+1}), so the overall return is risk free. Bad cashflow news lowers stock pries, so both cashflow and price terms move in the same direction. In sum a small (a(beta)lt 1) resolves the Jiang et. al. puzzle. (Disclosure, I wrote them about this months ago, so this view is not a surprise. They disagree.)

5) Surpluses and debt. Looking at that last equation, with a positively correlated surplus process (a(beta)>1), as in the AR(1), a surplus today leads to  larger value of the debt tomorrow. A deficit today leads to lower value of the debt tomorrow. The data scream the opposite pattern. Higher deficits raise the value of debt, higher surpluses pay down that debt. Cumby_Canzoneri_Diba (AER 2001) pointed this out 20 years ago and how it indicates an s-shaped surplus process.  An (a(beta)lt 1) solves their puzzle as well. (They viewed (a(beta)lt 1) as inconsistent with fiscal theory which is not the case.)

6) Financing deficits. With (a(beta)geq1), the government finances all of each deficit by inflating away outstanding debt, and more. With (a(beta)=0), the government finances deficits by selling debt. This statement just adds up what’s missing from the last one. If a deficit leads to lower value of the subsequent debt, how did the government finance the deficit? It has to be by inflating away outstanding debt. To see this, look again at inflation, which I write [ frac{B_{t-1}}{P_{t-1}}Delta E_{t}left( frac{P_{t-1}}{P_{t}}right) =Delta E_{t}s_{t}+Delta E_{t}sum_{j=1}^{infty}beta^{j}s_{t+j}=Delta E_{t}s_{t}+Delta E_{t}beta b_{t+1}=1+left[ a(beta)-1right] varepsilon_{t}. ] If (Delta E_{t}s_{t}=varepsilon_{t}) is negative — a deficit — where does that come from? With (a(beta)>1), the second term is also negative. So the deficit, and more, comes from a big inflation on the left hand side, inflating away outstanding debt. If (a(beta)=0), there is no inflation, and the second term on the right side is positive — the deficit is financed by selling additional debt. The data scream this pattern as well.

7) And, perhaps most of all, when the government sells debt, it raises revenue by so doing. How is that possible? Only if investors think that higher surpluses will eventually pay off that debt. Investors think the surplus process is s-shaped.

All of these phenomena are tied together.  You can’t fix one without the others. If you want to fix the mean government bond return by, say, alluding to a liquidity premium for government bonds, you still have a model that predicts tremendously volatile and procyclical bond returns, volatile and countercyclical inflation, deficits financed by inflating away debt, and deficits that lead to lower values of subsequent debt.

So, I think the VAR gives the right sort of estimate. You can quibble with any estimate, but the overall view of the world required for any estimate that produces a large (a(beta)) seems so thoroughly counterfactual it’s beyond rescue. The US has persuaded investors, so far, that when it issues debt it will mostly repay that debt and not inflate it all away.

Yes, a moving average that overshoots is a little unusual. But that’s what we should expect from debt. Borrow today, pay back tomorrow. Finding the opposite, something like the AR(1), would be truly amazing. And in retrospect, amazing that so many papers (including my own) write this down. Well, clarity only comes in hindsight after a lot of hard work and puzzles.

In more general settings (a(beta)) above zero gives a little bit of inflation from fiscal shocks, but there are also time-varying discount rates and long term debt in the present value formula. I leave all that to the book and papers.

(Jiang et al say they tried it with debt in the VAR and claim it doesn’t make much difference.  But their response functions with debt in the VAR, at left,  show even more overshooting than in my example, so I don’t see how they avoid all the predictions of a small (a(beta)), including a low bond premium.)

A lot of literature on fiscal theory and fiscal sustainability, including my own past papers, used AR(1) or similar surplus processes that don’t allow (a(beta)) near zero. I think a lot of the puzzles that literature encountered comes out of this auxiliary specification. Nothing in fiscal theory prohibits a surplus process with (a(beta)=0) and certainly not (0 lt a(beta)lt 1).

Update

Jiang et al. also claim that it is impossible for any government with a unit root in GDP to issue risk free debt. The hidden assumption is easy to root out. Consider the permanent income model, [ c_t = rk_t + r beta sum beta^j y_{t+j}] Consumption is cointegrated with income and the value of debt. Similarly, we would normally write the surplus process [ s_t = alpha b_t + gamma y_t. ] responding to both debt and GDP. If surplus is only cointegrated with GDP, one imposes ( alpha = 0), which amounts to assuming that governments do not repay debts. The surplus should be cointegrated with GDP and with the value of debt.  Governments with unit roots in GDP can indeed promise to repay their debts.



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Economy

Campos: The Trump Delusion—Noted

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Paul Campos: The Trump Delusion https://www.lawyersgunsmoneyblog.com/2020/06/the-trump-delusion: ‘How is it that, despite everything, 40% of America continues to support Donald Trump? I’ve suggested that Trump’s supporters can be sorted into a few broad categories, with many of those supporters belonging to more than one of these groups: White nationalists…. Alienated burn it all down anti-establishment types…. Upper class Republicans who want big tax cut…. Religious conservatives, overwhelmingly white evangelicals…. Low information voters who always vote Republican out of tribal habit. These people have the most fantastical ideas about Trump, such as for example that he’s a “successful businessman,” rather than a “politician,” which is why he manages to “get things done.” This last group in particular includes a lot of overlap with the more cultish strain of religious conservatives…. Relatively few people are capable of maintaining a genuine lesser of two evils attitude toward the leader of an essentially charismatic—to use Weber’s typology—political movement. Almost everyone in the movement must eventually embrace the delusion that the leader is actually a good person, despite all evidence to the contrary. For example, the following message has gone viral on social media over the last few days. The text is headed by the photo at the top of this post:

Anonymous: 'Let’s look at this man for one damn second!!!! A 74-year-old man is coming back home from work at 2 AM while most men his age are retired in their vacation homes. He comes back after a long day that probably started before the sun rose and gets back home exhausted with his tie open and hat in his hand, feeling that an accomplished day is finally over…

…This amazing man is in the age range of many people’s grandfathers, great grandfathers, or my grandfather when he passed away, but this man just came back home from work, for me, for you. This man left his massive gold-covered mansion where he could retire happily and play golf all day long. But this man put his wealth aside and went to work for free, for $1 a year, for me, for you, for us, for AMERICA.

While other presidents became rich from the presidency, this man LOST over 2 billion dollars of his wealth during this short 4 years of his life. He put aside his amazing retirement lifestyle for getting ambushed every single day by the media and the Radical Left Democrats that trash this man who works for them until 1 AM for free!

No, he doesn’t do it for money or power, he already had it. He is doing it so their houses will be safe, so their schools will get better, so they will be able to find jobs or start a new business easier, so they will be able to keep few dollars in their pockets at the end of the month.

Look at this picture again, that man is at the age of your fathers, grandfathers or maybe YOU! Where is your respect? Honor? Appreciation? Are you THAT BLIND? THAT BLIND to not see a thing this man is doing for you and for your family? THAT BLIND that after all his work for minority groups in America you keep calling him a racist? I am the son of an Auschwitz Survivor and someone who lost 99% of my family to the camps and ovens of Nazi Germany. And I’m no fool! DONALD TRUMP IS NO RACIST OR ANTI-SEMITE!

Are you THAT BLIND to not see how much this country developed in last 4 years? President Donald J. Trump, I want to thank you with all my heart. I am so sorry for blind hatred you have been made to endure. You are a good and generous man. I KNOW THIS.

What I don’t know and think about often is what kind of people is it who can be so hateful in their hearts to spew such hate and evilness, not just at you but at your family too? Or people mocking and making jokes of you because you’re not a professional politician groomed in speech making and straight faced lying. Or how about them attacking your wife and young son? How awful that must make you feel.

People are sure they have not been manipulated. People believe their hatred is their own. But for why, they can’t articulate. What kind of people are these? WHAT KIND OF PEOPLE ARE THESE?? People not realizing they have been manipulated and brainwashed by such a deep-rooted EVILNESS MOTIVATED BY AN EVIL MEDIA AND DEMOCRAT PARTY. The American People are in a bad place right now….in their hearts and souls. God help us…Trump is not the problem.

This level of frankly delusional thinking is, I believe, far more common than either an enthusiastic embrace of anyone resembling the actual Donald Trump, or the sort of arms-length transactional support of people who recognize him for what he is, but have concluded that Paris is worth a mass.

Which is a fancy way of saying that a lot of his supporters are, at this point, basically insane.

…Meanwhile:

Aaron Rupar: 'This morning, Trump retweeted a QAnon account, thanked supporters of his who were filmed yelling “white power,” and issued a misleading non-denial of a story about him turning a blind eye while Russia offered bounties for US troops. All before 9 am…

.#noted #2020-07-10



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Economy

Dorothy Theresa Sawchak Mankiw

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Above is a picture of my mother as a young woman. I would like to tell you about her.

My mother was born on July 18, 1927, the second child of Nicholas and Catherine Sawchak.

Nicholas and Catherine were immigrants from Ukraine. They came to the United States as teenagers, arriving separately, neither with more than a fourth-grade education. Catherine was from a farming area in western Ukraine. She left because her family wanted her to marry an older man rather than her younger boyfriend, who had been conscripted into the army. Her first job here was as a maid. Nicholas was from Kiev, where he had been trained to be a furrier. In the United States, he worked as a potter, making sinks and toilettes. When Nicholas and Catherine came to the United States, they thought they might return home to Ukraine eventually. But World War I and the Russian Revolution intervened, causing a change of plans. Catherine’s boyfriend died in the war. Nicholas and Catherine met each other, married, and settled in a small row house in Trenton, New Jersey, where they lived the rest of their lives.

Catherine and Nicholas had two children, my uncle Walter and my mother Dorothy. When my mother was born, her parents chose to name her “Dorothy Theresa Sawchak.” But because Catherine spoke with a heavy accent, the clerk preparing the birth certificate did not understand her. So officially, my mother’s middle name was “Tessie” rather than “Theresa.” She never bothered to change it.

Nicholas and Catherine were hardworking and frugal. They saved enough to send Walter to college and medical school. He served as a physician in the army during the Korean war. Once I asked him if he worked at a MASH unit, like in the TV show. He said no, he worked closer to the front. He patched up the wounded soldiers the best he could and then sent them to a MASH unit to recover and receive more treatment. After the war, he became a pathologist in a Trenton-area hospital. He married and had two daughters, my cousins.

My mother attended Trenton High School (the same high school, I learned years later, attended by the economist Robert Solow at about the same time). She danced ballet. She water-skied on the Delaware River. She loved to read and go to the movies.

In part because of limited resources and in part because of the gender bias of the time, my mother was not given the chance to go to college. Years later, her parents would say that not giving her that opportunity was one of their great regrets. Instead, my mother learned to be a hairdresser. She was also pressured to marry the son of some family friends.

The marriage did not work. With my mother pregnant, her new husband started “running around,” my mother’s euphemism for infidelity. They divorced, and she kicked him out of her life. But the marriage did leave her with one blessing—my sister Peg.

My mother continued life as a single mother. Some years later, she met my father, also named Nicholas, through social functions run by local Ukrainian churches. They both loved to dance. He wanted to marry her, but having been burned once, she was reluctant at first. Only when she realized that he had become her best friend did she finally accept.

In 1958, nine months after I was born, Mom, Dad, Peg, and I left Trenton for a newly built split-level house in Cranford, New Jersey. My father was working for Western Electric, an arm of AT&T, first as a draftsman and then as an electrical engineer. He worked there until his retirement. One of his specialties was battery design. When I was growing up, I thought it sounded incredibly boring. Now I realize how important it is.

My mother then stopped working as a hairdresser to become a full-time mom. But she kept all the hairdresser equipment from her shop—chair, mirrors, scissors, razors, and so on—in our basement. She would cut the hair of her friends on a part-time basis. When I was a small boy, she cut my hair as well.

I attended the Brookside School, the public grade school which was a short walk from our house. When I was in the second or third grade, my mother was called in to see the teacher. The class had been given some standardized aptitude test. “Greg did well,” the teacher said. “We were very surprised.”

At that moment, my mother decided the school was not working out for me. I was talkative and inquisitive at home but shy and lackluster at school. I needed a change.

She started looking around for the best school she could find for me. She decided it was The Pingry School, an independent day school about a dozen miles from our house. She had me apply, and I was accepted.

The question then became, how to pay for it? Pingry was expensive, and we did not have a lot of extra money. My mother decided that she needed to return to work.

She started looking for a job, and an extraordinary opportunity presented itself. Union County, where we lived, was opening a public vocational school, and they were looking for teachers. She applied to be the cosmetology teacher and was hired.

There was, however, a glitch. The teachers, even though teaching trades like hairdressing, needed teacher certification. That required a certain number of college courses, and my mother had not taken any. So she got a temporary reprieve from the requirement. While teaching at the vocational school during the day, she started taking college courses at night to earn her certification, all while raising two children.

My mother taught at the vocational school until her retirement. During that time, she also co-authored a couple of books, called Beauty Culture I and II, which were teacher’s guides. From the summary of the first volume: “The syllabus is divided into six sections and includes the following areas of instruction: shop, school, and the cosmetologist; sterilization practices in the beauty salon; scalp and hair applications and shampooing; hair styling; manicuring; and hairpressing and iron curling.” I suppose one might view this project as a harbinger of my career as a textbook author.

When my parents both retired, they were still the best of friends. They traveled together, exploring the world in ways that were impossible when they were younger and poorer. During my third year as an economics professor, I was visiting the LSE for about a month. I encouraged my parents to come over to London for a week or so. They had a grand time. I believe it was the first time they had ever visited Europe. When I was growing up, vacations were usually at the Jersey shore.

My father died a few years later. My mother spent the next three decades living alone. She was then living full-time at the Jersey shore in Brant Beach on Long Beach Island. The house was close to the ocean and large enough to encourage her growing family to come for extended visits. Two children, five grandchildren, four great-grandchildren. The more, the merrier. Nothing made her happier than being surrounded by family.

My mother loved to cook, especially the Ukrainian dishes she learned in her childhood. Holubtsi (stuffed cabbage) was a specialty. Another was kapusta (cabbage) soup. One time, the local newspaper offered to publish her kapusta soup recipe. They did so, but with an error. Every seasoning that was supposed to be measured in teaspoons was printed as tablespoons. The paper later ran a correction but probably to no avail. I am not sure if anyone ever tried the misprinted recipe and, if so, to what end.

During her free time in her later years, my mother read extensively, played FreeCell on her computer, and watched TV. A few years ago, when she was about 90 years old, I was visiting her, and I happened to mention the show “Breaking Bad.” She had not heard of it. She suggested we watch the first episode. And then another. And another. After I left, she binge-watched all five seasons.

As she aged, living alone became harder. When she had trouble going up and down the stairs, an elevator was added to her house. But slowly her balance faltered, and she fell several times. She started having small strokes, and then a more significant one. She moved into a nursing home. Whenever I visited, I brought her new books to read. Her love of reading never diminished.

This is, I am afraid, where the story ends. Last week, Dorothy Theresa Sawchak Mankiw tested positive for Covid-19. Yesterday, she died. I will miss her.



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