Emily Blunt recalled how Taylor Swift almost made her oldest daughter, Hazel, faint. Get the details.

Emily Blunt Thought Her Daughter Was 'Going to Faint' While Meeting Taylor Swift: 'She's the Nicest'

Vampire singer is heady mix of raw vulnerability and self-confidence with not a single bum note on display

Olivia Rodrigo in Dublin review: Watch out Taylor Swift. This bona fide pop superstar is coming for you – The Irish Times

Emily Blunt shared the anecdote of when Taylor Swift paid a compliment to her 10-year-old daughter.

Emily Blunt shares heartwarming gesture Taylor Swift did for her kids

Find the best deals, gift guides, product reviews and even freebies.

Taylor Swift's 'The Tortured Poets Department' Might Be Sold Out In Stores, 
but It's Still Online at Amazon and Target

Raise your hand if you have a hard time keeping track of your phone? Read ahead for more on this Taylor Swift-approved phone case and wallet.

Shop the Crossbody Phone Case Taylor Swift & Donna Kelce Love — Under $100

Anissa Tinnin says she and her 4-year-old granddaughter were watching the Taylor Swift &quot;Eras&quot; tour film when a nightmare began. Out of nowhere, she says, an intruder appeared in the dining room of her Albuquerque home. Cops say the intruder was fleeing police after crashing a stolen vehicle.

Woman Shoots Home Intruder Who Interrupted Taylor Swift Movie

Taylor Swift couldn&#x27;t help but cringe over her boyfriend Travis Kelce&#x27;s over-the-top antics at Patrick Mahomes’ charity event over the weekend, according to an expert lip reader.

Taylor Swift cringes at Travis Kelce shouting 'Viva Las Vegas' at gala

Taylor Swift and her partner Travis Kelce are both set to attend the 2024 Miami Grand Prix this weekend. The celebrity couple will likely be in the paddock just days after the singer released a song potentially relating to a relationship with Fernando Alonso.

Taylor Swift set for awkward meet with Alonso? Singer to attend Miami GP

Glasgow Clyde College (GCC) has launched a Taylor Swift masterclass aimed at teaching parents and plus-ones about The Eras Tour set list and the singer&#x27;s evolving wardrobe and hairstyles.

Taylor Swift Eras tour course offered by college for parents and carers ahead of sold-out Edinburgh shows | UK News | Sky News

Taylor Swift&#x27;s Eras Tour is returning soon. Here&#x27;s the updated dream setlist including songs from &#x27;The Tortured Poets Department.&#x27;

Here's How Taylor Swift Should Update Her Eras Tour Setlist to Include 'The Tortured Poets Department'

Teylor

teylor

Hey Hackers! I’m and I’m the Patrick CEO @ Teylor. First of all, I’d like to say a huge thank you to the community for recognizing us as the winner of: HackerNoon Startup of the year 2021 in Zürich Reporter: What does it mean for you to win this title? HackerNoon I have been reading Hackernoon for many years, so it's a really great feeling to be recognized by a publication that I have been reading since long before I started Teylor. Also, Hackernoon is a place where many smart tech minds come together, so it’s fantastic to be recognized as a leading startup by this community. How do you or your company intend to embrace the responsibility of this title in 2022? Our goal for 2022 is to become not just Zurich’s fastest-growing startup, but Europe's fastest-growing startup. Teylor’s mission is to build the lenders of tomorrow. So building up on our 2021 successes, we aim to bring our business to the next level by scaling our existing operations but also expanding into other European markets. With that said, and in the spirit of Hackernoon, scaling is not just about getting bigger but also about getting better. We will keep building and improving our platform, adding more features, and ensuring our tech can create the change we’re looking for. What are some of the goals you’re looking forward to accomplishing in 2022? We use our lending technology to provide fast and convenient loans to SMEs and we provide it as a SaaS solution to banks to help them digitize and automate their own lending processes. In 2022, we want to keep making progress in both our lending business and our technology business. That means providing more financing to SMEs and onboarding more banks to our technology platform. Our goal in 2022 is to become the leading SME lending platform in Europe. We are also massively growing our team, so finding smart minds and people that fit with our culture will be a key focus in 2022. That’s also a shout-out to the Hackernoon community: If you want to contribute to our mission of making SME lending better and faster for those who need it the most, get in touch with us. As you look at the variety of concerning trends in the world around you, what solutions can you think of? Name three trends and three solutions. Be as brief or as detailed as you like. Three core challenges for our generation are digitization, global inclusion, and climate change. I believe that we are seeing progress on all three fronts. Digitization has become a major theme since the onset of the pandemic and that won’t change after the pandemic is over. More people work remotely from home offices, the rise of cryptocurrencies, online shopping, telemedicine, and online learning - just to name a few examples. All of these things are a huge chance for our economy and society, but they also provide challenges. It will require smart technologies, but also social and behavioral change, to steer us into the new digital era. Second is global inclusion: The world has grown together in the past decades but it seems as if it recently has also drifted apart again. Examples are the geopolitical conflicts in eastern Europe and Asia-pacific. I also believe technology can make a difference here, by connecting people, by making sure they get the right information instead of falling for false narratives. It will be a major challenge, especially for the media industry to become better at that. And the third is climate change. A lot has to be done to solve this problem, both on the technological as well as the social side. However, what makes me positive is that climate change is finally getting a lot more attention and decisions are being made in the right direction. It’s not enough, but at least there is movement. On that note, what makes me proud of Teylor is that we are contributing to all three challenges. Teylor helps make finance digital, inclusive, unbureaucratic, and reduces the CO2 footprint of the financial industry. After participating in this journey, is there anything that you would like the team to know? HackerNoon Keep doing what you do. We like your platform, we like the community spirit and we appreciate your contribution. What is something special about you or your company that you’ve rarely had the chance to properly share but really wanted to? (Give this one a proper think, I’ll know if you didn’t!) What makes Teylor truly special is the team. When you run a startup the media focus is often on the business model, the financials, and the senior management. The truth is, it's the people on the ground making it all happen. The developers creating a world-class technology platform, the salespeople putting themselves out there everyday onboarding customers. Unfortunately, in the modern media world, the people that work hard every day and are doing their jobs well get way too little appreciation. Thank you for everything that you do! – Team. HackerNoon P.S. If you enjoy getting interviewed (AKA, the ol’ intellectual stimulation, check out some more interviews !) here For more information: Read the FAQs about or . To see all of the winners you may visit , pick any city and view the name that appears at the top. Or follow for posts with the list of names released over the course of the week of February 14, 2022. Startups here here this site this account Thank you to all of our sponsors! We truly appreciate your contribution to the tech world and thus the future! : Startups , , , , , , , , , , Blockster Brex FreeCodeCamp LaunchDarkly Coil Udacity Udemy Vercel Asia Pacific Transgender Network Bybit Everscale Network

Teylor has Been Named Startup of the Year 2021 in Zürich

HackerNoon Reporter: Please tell us briefly about your background. My name is Patrick Stäuble. I am Swiss but grew up all over the world. - I was born in Holland, then moved to Colombia, Malaysia, Brunei, and Oman. I returned to Switzerland at the age of 18 and joined the Swiss military, where I completed peacekeeping missions in Kosovo. After this, I studied Economics in Zurich, started and sold my first startup, and then worked as a product manager at a Swiss mobile banking fintech, where I eventually became Head of Business Development and Sales. During that time, the idea of Teylor was born. I left my old job at the end of 2017 and launched Teylor in 2018. What's your startup called? And in a sentence or two, what does it do? Our company is called “Teylor”and we have developed a modular technology platform to build, automate, deliver and scale credit products. In other words: We are building the lenders of tomorrow. What is the origin story? At the previous job, I dealt with many banks in Germany. I quickly realized that they lack digital tools in the realm of SME financing. This was amazing to me because for the last ten years, everyone had talked about digitizing banking. But somehow, only a few institutions had focused on the SME lending segment. That’s when I decided to seize this massive opportunity and jump in headfirst. We launched the Teylor Loan Platform in 2018 with a fully automated SME loan with a German bank, and it was an immediate success. We saw the demand for automated lending not only from SMEs but also from banks, and we saw that our solution works. As several banks came to us and inquired about our technology, we expanded the Teylor loan platform to a full-fledged technology platform. We now offer modular Software-as-a-Service solutions for banks to digitize and automate their lending processes. What do you love about your team, and why are you the ones to solve this problem? Our team comes from diverse backgrounds. We have engineers that worked for technology companies or other fintech before and traditional finance people that come out of investment banking, SME finance, and consulting. It’s fascinating to see how many synergies there are between people with completely different backgrounds and how everyone really pushes towards our common goal of revolutionizing SME lending. If you weren’t building your startup, what would you be doing? I would be building another startup. To me, there is absolutely nothing else I would rather be doing. It’s what I’m good at, and it’s what I want to do. There are so many problems out there that need to be solved, and that’s why there will always be a good idea for another startup. At the moment, how do you measure success? What are your core metrics? As a technology company we mainly look at two things: First, how good is our technology and does it solve the problem we have set out to solve. Today, I can confidently say that we have the best digital lending technology in the European market. No other company has accomplished what we have. The challenge is to ensure that we remain the technology leader. Second, we run a business, and even the best technology will fail if it doesn’t sell. So it’s paramount for our growth that we can quickly onboard more banks to our platform and grow our revenues. At the moment that is going great, and our numbers have been exploding since the beginning of the year. What’s most exciting about your traction to date? There has been an enormous push for more digitization in the last two years. One reason was the pandemic, but also the need to fight climate change and to keep up with new competitors from Asia. That has played into our hands as more and more banks see the need to digitize their credit processes. What technologies are you currently most excited about and most worried about? And why? One of the fascinating technologies to me is machine learning. We just launched a  new version of our OCR tool, “FinCR'' that we use to convert financial data from annual reports into machine-readable formats. Banks and other businesses use this tool to avoid manual data entry, for example, to avoid copying and pasting financial information from annual reports to their systems. We used machine learning to improve FinCR’s accuracy, and it’s absolutely astonishing how good our tool has become in a very short time. As for the second part of your question: I’m not really worried about any technology. I’m just worried about how humans may use it. Take the example of artificial intelligence. You can use it to read financial statements, or you can use it to build intelligent weapons. I’m a fan of the first, not so much of the second. What drew you to get published on HackerNoon? What do you like most about our platform? I’ve been a long-time reader! HackerNoon is a place where bright brains discuss new technologies and innovations. Where you can say what you think without fingerpointing, on HackerNoon, you can throw your ideas out there, and the community will challenge them without judging. I think that’s great. What advice would you give to the 21-year-old version of yourself? I think I would tell myself to check out this little thing called bitcoin and invest a few dollars when it launches! What is something surprising you've learned this year that your contemporaries would benefit from knowing? I think the pandemic has shown how fragile the world is. We think we have it all figured out, and then a virus turns up and puts the world upside down. The good news is: It teaches us to be flexible, both personally and in business. That was one of the key lessons for me this year and last year. Teylor is contesting as . the startup of the year in Zurich

Startup Interview with Patrick Stäuble, Teylor CEO & Founder 

213 Stories To Learn About Writing Prompts

240 Stories To Learn About Startups Of The Year

206 Stories To Learn About Digital Transformation

198 Stories To Learn About Founder Stories

148 Stories To Learn About Founder Interview

44,000 startups. 4,553 cities. “4 billion votes” It was such a blessing to get to know so many of your ideas and innovations from every corner of the world in 2021. The startup community amazed all of us with your hard work and persistence despite all of the challenges that we faced globally throughout last year. Not gonna lie, the competition got our hearts racing like Eurovision. We won’t let you wait for much longer, below are long-awaited results! ✨✨✨ To see who is a winner in each respective city, visit and pick a city. The name at the top of the list is the winner. (Compromised awards are under review.) startups.hackernoon.com The will announce the winners per region in their respective cities. Startups Winners Announcement Series 🔥 Central Europe Central Europe has always caught our attention with the talented pool of people in tech community. But were they able to get through 2021 despite the pandemic? Let’s cheers to the most outstanding amongst 4,500+ startups who have been working hard this year: Basel, Switzerland An overwhelming vote for ~ in their mission of helping machinery companies drive more service revenues from their installed base MachiQ Software Berlin, Germany We have with a whooping 538 votes! Let’s support Plan A with its mission to help companies to measure, monitor and reduce their environmental footprint. Plan A Düsseldorf, Germany It doesn’t get any easier with Kicking off in helping people organize online schedules more easily. EasyWeek! Geneva, Switzerland Manage code quality? We have to give props to for their startup! SonarSource Hamburg, Germany Taking over Hamburg is , an acoustic device to reduce noises in vehicle cabins. No more annoying sounds on the road 🚗 ReCalm Köln, Germany Small business? Hoping to manage your marketing campaign? is winning the vote! rankingCoach Lausanne, Switzerland is a Swiss-based company focused on cryptocurrency management solutions built for financial institutions. Metaco Leipzig, Germany empowers companies to operate and utilize drones for the most effective industrial data applications. FlyNex Mainz, Germany is growing sustainable digital mobility startups! Next Mobility Labs Munich, Germany wins for top-line, high-security international Identity Management! ORY Stuttgart, Germany Remote desktop application? Ask anywhere. AnyDesk Vienna, Austria , an IT company enabling privacy-preserving big data applications in the digital age. Mostly AI Zug, Switzerland Way to go 1762 votes for the first provably live pure Proof-of-Stake (PoS) CBC-Casper blockchain designed for real-world applications. CasperLabs! Nyon Calling for robust data solutions for impact-driven organizations! Havuta LLC Zürich is revolutionizing small business lending to give easy access to capital to those who need it most: SMEs ✨ Teylor Heidelberg, Germany is an innovative HR startup providing a solution for fast and efficient recruiting of employees with simple job profiles. Talk'n'Job Stuttgart, Germany Congratulations for winning this title! AnyDesk Moving on to our buddies in... 🔥 Eastern Europe Eastern Europe is home to many familiar big names in our start-up world. Trace history for Miro, Telegram, and Grammarly and you know where they are from. Who is the next generation for Eastern European startups? Budapest, Hungary is a platform for mobile app development, enabling tens of thousands of developers to automate integration, testing, and deployment. Bitrise I__ __ stanbul, Turkey Leading here is - a brand new software intelligence platform- helps organizations deliver faster, high-quality products by connecting their DevOps tools. Oobeya Kraków, Poland is a human cloud platform that connects global enterprises with European software houses, offering tech talent as a service. Talent Alpha Kyiv, Ukraine 281 votes for , an AI powered app where people can swap faces in the videos, GIFs and images in just a few seconds. Reface Sofia, Bulgaria 387 votes for __ __378, leading Blockchain and DLT solutions company that specializes in helping enterprises and blockchain startups design, build and scale innovative solutions using Blockchain & DLT! LimeChain Warsaw, Poland is a next-generation blockchain document management software (DMS). DoxyChain Yerevan, Armenia Whoever clicked to vote for was sure a vote, because it ended up winning as a full-service software provider delivering Artificial Intelligence & Machine Learning solutions for businesses. SmartClick smart Almaty Just like the name suggested, provides one pass to dozens of gyms and studios. 1FIT Debrecen got all the coders ranked figured out! CodersRank Kharkiv is complete solution digital assets management! Pics.io Minsk, Belarus with 243 votes is the first blockchain-powered sports media platform. Scorum Moscow, Russian Federation 1 e-grocery company in Russia with 26 votes and #1 on our HackerNoon board for Minsk. SberMarket Saint Petersburg, Russian Federation leads this area as a personal finance application that rewards responsible spending and savings. CoinKeeper Kazan, Russian Federation Smelling a sweet victory? 3d App designers for flower bouquets is in the house! Flowito Western Europe 🔥 Brexit? Pandemic? It seems like startups in Western Europe still keep their spirits regardless. Startups in Western Europe that stood out in votes this year go to: Antwerp is on a streak! As a bug bounty platform, it ensures online security through a crowd-sourced platform for computer experts. intigriti Belfast A record-breaking 10,003 votes? is an independent coffee shop. Come grab a coffee! Bobs & Berts Birmingham What’s happening in Birmingham? We have , an innovative UK-based advanced lithium battery developer aiming to accelerate the global shift to cleaner. Aceleron Bordeaux 30 is an open-source web framework for building modern apps and websites with Vue.js. NuxtJS Boulogne-billancourt is a software company that provides an in-store analytics platform to optimize operations. DiGEiZ Brighton In for some jam? offers musical education services. BIMM Bristol Another tasty victory for Western Europe is enabling the strongest digital encryption on the internet. KETS Quantum Security Brussels A fashionable win for , a social fashion network. You scan barcodes in fashion shops and see every influencer wearing those outfits! Share your wishlist with your friends and earn cashback. SCANDALOOK Cardiff is a technology company developing end-to-end digital connectivity for the point-of-care diagnostics market. Bond Digital Health London A whooping number of 1085 vote__ __ offers hardware based cyber-security solution for authentication and encryption in a wired local network. Crypto Quantique Edinburgh Hopping up on the crypto movement, just launched last year but they are already rocking it at making crypto accessible to everyone. Zumo Ghent 5379 votes for ! A Belgian InsurTech startup that helps insurance providers all over the world to better connect with their digital customers. WeGroup Glasgow specializes in single-cell analysis transforming the life sciences. Biomage Paris Show some love in the city of love for , a sports platform that is connected to a mobile sports coaching application. AirFit Aberystwyth enables artificial intelligence, data science & digital innovation solutions for the agriculture, biotech & life sciences industries. Agxio Norwich Congratulations for the amazing win of 1124 votes! WATR Charleroi Congratulations for your amazing win! Hirering Gibraltar Give a round of applause to Fortknoxter! Waltham Abbey is ruling the area with its amazing 72 votes! RPAIR Antwerpen Rocking up Antwerpen area is the rising startup ! Willow Egham What’s happening in Egham? Proudly announcing winning the Noonies this year! ATA.ONE Leicester Leicester is calling the name of as the winner for the region! DeadHappy Macclesfield Who else but ? Congratulations on dominating the award! AeroCloud Northern Europe 🔥 Want to listen to some music? Spotify. Need some games to kill time? Candy Crush. What about a quick call to someone you haven’t seen in a long time? Skype. These are all startups that originated from Northern Europe! Noonies, let’s take a look at the upcoming generation of startups from Northern Europe that hopefully will see an even bigger and brighter future. Gothenburg, Sweden Shipping was never made for small shippers, so took on the mission to built an end-to-end logistics platform to empower small businesses to spend less time and money on shipping and more time on growing their businesses. Sendify The Netherlands provides a virtual platform for many kinds of conferences and events. Let's Get Digital Helsinki, Finland A curated collection of public basketball courts in the Helsinki Metropolitan Area called . Everyone is welcome to submit to their local court. Koripallopaikat Lund, Sweden develops an antibiotic-free, safe and effective treatment of vaginal fungal infections. Gedea Biotech Luxembourg, Luxembourg 🚀 is a job-matching platform that connects developers and tech experts with CTOs and team leads from the best tech companies across Europe. 👾 nexten.io Malmö, Sweden Feeling studious, Noonies? __ __e is a tool for Google Classroom, that enables formative matrix assessment towards course learning objectives directly. StudyBe Oslo, Norway is the platform supports a unified API across all connected device categories, removing significant friction for app developers. Domos Labs Reykjavík, Iceland __ __s is a mobile-first technologically advanced neo-bank. Indo service #S__ __ tockholm, Sweden is the biggest blockchain start-up in Sweden. Double winner of the most prestigious nomination in Nordics 'NyTeknik 33-listan', winner of the Deloitte Fast 50, and now Noonies Award! ChromaWay AB T__ __ allinn, Estonia Passing 1000 votes is a fully digital and decentralized ecosystem for all actors in supply chain. OBORTECH Vilnius, Lithuania is an easy-to-use money app for cards and accounts. Via Genome, users can open a business or a merchant account or a personal one with dedicated IBANs, access transfers, exchange currency, process payments, and manage the funds. Genome Aalborg Digura A legaltech offering tenants fast, digital and reliable legal assistance Aarhus Partisia Market Design provides secure, simple, efficient solutions for auctions and exchanges realizing the potential of the market. Helsingborg is a hi-tech company developing cardiac occlusion devices and other vascular implants. Occlutech Tallinna is an AI-powered system for marketing automation in social networks. POSTOPLAN 🔥 Southern Europe Southern Europe’s economy went through a hard time this year due to a huge reduction in tourism. But our startups are of course always headstrong, from Spain to Italy and all the way through Greece. Who are the heroes rising amidst changes of Southern Europe? Skopje is a versatile software development company that solves the business challenges of SMBs, big brands, and brave hearts with awesome ideas. CodeChem Sliema Sounds like just a relaxing exhale. is a holistic health and wellbeing app built to allow users work on the development of their physical, mental and spiritual health Few Tajonar is a company of experienced business & industry entrepreneurs offering solutions to industries based on cutting-edge technologies. das-Nano Tarragona is a technology-based company focusing on the production of integrated microsystems. IMICROQ Terni is an ICT-based integrated solution that is able to modulate, respond, assist and learn to control most home activities. Mind Tirana is a video content and e-commerce platform for the Balkans built on top of an Albanian language specialized search engine. Gjirafa Torino develops and commercializes algorithms based on Artificial Intelligence to increase the autonomy of small satellite missions. AIKO T__ __ orres Buy online groceries like you would buy offline with personalized picking as fast as in 4 hours with Vedras 360hyper! Trento is a music-based platform built around streaming platforms that allow users to discover new emerging artists and make new friends. Brots 2021, what a YEAR. Congratulations to all of the startups that made it to the top winning list for Europe! Stay tuned for Startup awards in 2022 and let’s cheers to a new year of more innovative ideas, bold innovations, and robust creations from you, Europe 🌷 More Information: If you’re a winner, please answer ! If you’re a reader, participant, just a curious individual, consider answering one of our interviews . this interview here Read the FAQs about or . Startups here here To see all of the winners you may visit , pick any city and view the name that appears at the top. this site If you’re interested in partnering with us, we invite you to look through some of these or reach out to our business department. unique sponsorship options Perhaps creating your very own tickles your fancy or maybe you have a whole new idea for how we can partner! writing contest Thank you to all of our sponsors! We truly appreciate your contribution to the tech world and thus the future! The Startups Awards are sponsored by: , , , , , , , , , , Blockster Brex FreeCodeCamp LaunchDarkly Coil Udacity Udemy Vercel Asia Pacific Transgender Network Bybit Everscale Network This post was written by An Le and edited by Natasha Nel. If you have any questions or concerns or would like to report a discrepancy please email ellen@hackernoon.com.

Meet the Winners of Startups in Europe

Author:
(1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility G Further Keynesian Models The first two subsections lay out two Keynesian models that come up in the course of the discussion in Section 10. The aim is not to solve them out fully but to provide the reader with background. The last part verifies claims made in Section 10.1 concerning unusual policy settings. The idea is that firms reoptimize the price periodically. The difference with Calvo is that they know the length of contract for sure, in advance. Contracts overlap so that in each period the same fraction come up for renewal. This set up seems intuitively appealing. G.1 Taylor Pricing The Taylor [1979] model of staggered contracts is one of the most popular in applied macroeconomics. The idea is that firms re-optimize the price periodically. The difference with Calvo is that they know the length of contract for sure, in advance. Contracts overlap so that in each period the same fraction come up for renewal. This set up seems intuitively appealing. The objective function for a resetting firm is just (24) summed over the length of the contract M. Assume for simplicity at the outset that M = 2, as prominent in the text. The focus is on the small noise limit around ZINSS O( √ ε). For clarity, I will focus on the non-stochastic expansion. It is clear that the price change will be the weighted average of the expected changes in marginal cost and aggregate price level throughout the contract. The price level is simply Lagging this relationship leads to Hence, (352)-(353) (350)-(351) evaluates to lagging, substituting in, then simplifying yields As argued in Section 9.3, inflation is determined half by its lag and half by its lead. Furthermore, the terms occur in difference pairs, so it is clear that the sum of the real coefficients on marginal costs is zero, in keeping with real neutrality. These results extend over to contracts of general length where before a similar substitution process yields the Phillips curve There is a powerful link back to generalized stochastic mean back in Appendix B.2 and crystallized in Proposition 27. The property that the optimal reset price is a weighted average of all the flexible prices extends to inflation determination, if uncertainty and discounting (efficient or otherwise) vanish. This is a universal result. This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility G Further Keynesian Models The first two subsections lay out two Keynesian models that come up in the course of the discussion in Section 10. The aim is not to solve them out fully but to provide the reader with background. The last part verifies claims made in Section 10.1 concerning unusual policy settings. The idea is that firms reoptimize the price periodically. The difference with Calvo is that they know the length of contract for sure, in advance. Contracts overlap so that in each period the same fraction come up for renewal. This set up seems intuitively appealing. G.1 Taylor Pricing The Taylor [1979] model of staggered contracts is one of the most popular in applied macroeconomics. The idea is that firms re-optimize the price periodically. The difference with Calvo is that they know the length of contract for sure, in advance. Contracts overlap so that in each period the same fraction come up for renewal. This set up seems intuitively appealing. The objective function for a resetting firm is just (24) summed over the length of the contract M. Assume for simplicity at the outset that M = 2, as prominent in the text. The focus is on the small noise limit around ZINSS O( √ ε). For clarity, I will focus on the non-stochastic expansion. It is clear that the price change will be the weighted average of the expected changes in marginal cost and aggregate price level throughout the contract. The price level is simply Lagging this relationship leads to Hence, (352)-(353) (350)-(351) evaluates to lagging, substituting in, then simplifying yields As argued in Section 9.3, inflation is determined half by its lag and half by its lead. Furthermore, the terms occur in difference pairs, so it is clear that the sum of the real coefficients on marginal costs is zero, in keeping with real neutrality. These results extend over to contracts of general length where before a similar substitution process yields the Phillips curve There is a powerful link back to generalized stochastic mean back in Appendix B.2 and crystallized in Proposition 27. The property that the optimal reset price is a weighted average of all the flexible prices extends to inflation determination, if uncertainty and discounting (efficient or otherwise) vanish. This is a universal result. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Understanding Keynesian Models and Taylor Pricing

Author:
(1) David Staines. Table of Links Abstract 1 Introduction 2 Mathematical Arguments 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 5 Existing Solutions 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.3 Literature Comparison 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.3 Discussion 9 Bifurcation Analysis 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 11 Policy Rule 12 Conclusions and References Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.3 De Rham Cohomology F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.3 Trend Inflation Volatility 11 Policy Rule Here I prove the anti-Taylor policy rule and briefly outline its significance. Theorem 9. There exists polydromy if and only if aπ < 1, otherwise there is no solution at all. this expands out to which simplifies to It is clear that λ = 1/α is a root- which will lie outside the unit circle. The equation factorizes to Together Theorem 3 and Proposition 16 stipulate that a solution exists if and only if the quadratic contains precisely one root inside the unit circle. This rules out complex roots. The sign pattern rules out negative roots. Therefore, the cut-off must be associated with λ = 1 which is clearly associated with aπ = 1. Rouche’s theorem implies that when aπ is sufficiently large there will be no roots inside the unit circle. Hence the desired configuration must be associated with aπ < 1.[102] The extension to |ε| is straightforward. Its eigenvalue equation is This simplifies to Plugging in reveals that there is one root inside the unit circle at λ = α and another outside at λ = 1/α. When the polynomial is factorized, the other roots come from the same quadratic equation as in the √ ε limit in (180). There have to be two roots inside the unit circle overall, so the parameter condition is identical. There might be a wider lesson for general equilibrium theory and mathematical economics. Equilibrium existence is demanding a contraction on destabilizing general equilibrium effects. To see how, consider the following decomposition of the lagged inflation coefficient informed by Decomposition 1. Decomposition 2. where GE represents the general equilibrium impulse from the cost channel whilst the unitary terms come from the wall of the crossing (A) and the α comes from the singular component (D) and (F) from Decomposition 1. Turning to the New Keynesian field in particular, the following observation should prove interesting for empirical specification design, as well as theoretical consideration of the property of structural disturbances. Remark 35. Note that down the |ε| limit ∆ detaches from the rest of the system and behaves as an exogenous variable, like the cost-push shocks previously used to drive the Phillips curve. I anticipate the policy rule results will generalize widely and may extend out to moderate rates of trend inflation. To reaffirm, in economic terms the blow up is unproblematic and even instructive. On the one hand, the economy is stable under inactive settings, when the Taylor rule represents a realistic shorthand description of monetary policy-making. On the other hand, the model explodes under active policy, precisely when following a simple rule is most unrealistic. Successful central banks employ macroeconomists and other skilled practitioners in order to implement a semblance of optimal monetary policy. Best practice appears to be inertial. In time, I expect the theory of monetary policy to catchup.[103] Finally, there is a direct connection between ex post constraints implied by the existence conditions and ex ante constraints breaking Divine Coincidence. The policies that are ruled out ex post, which seek to raise real interest rates immediately, are precisely those that fail to engage with the ex ante restrictions that give the economy inertia. Elaborate mathematical machinery serves to rule out evidently unsustainable macroeconomic policies. I anticipate this will become a universal lesson in New Keynesian economics. This paper is available on arxiv under CC 4.0 license. Author: (1) David Staines. Author: Author: (1) David Staines. Table of Links Abstract Abstract 1 Introduction 1 Introduction 2 Mathematical Arguments 2 Mathematical Arguments 3 Outline and Preview 3 Outline and Preview 4 Calvo Framework and 4.1 Household’s Problem 4 Calvo Framework and 4.1 Household’s Problem 4.2 Preferences 4.2 Preferences 4.3 Household Equilibrium Conditions 4.3 Household Equilibrium Conditions 4.4 Price-Setting Problem 4.4 Price-Setting Problem 4.5 Nominal Equilibrium Conditions 4.5 Nominal Equilibrium Conditions 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.6 Real Equilibrium Conditions and 4.7 Shocks 4.8 Recursive Equilibrium 4.8 Recursive Equilibrium 5 Existing Solutions 5 Existing Solutions 5.1 Singular Phillips Curve 5.1 Singular Phillips Curve 5.2 Persistence and Policy Puzzles 5.2 Persistence and Policy Puzzles 5.3 Two Comparison Models 5.3 Two Comparison Models 5.4 Lucas Critique 5.4 Lucas Critique 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6 Stochastic Equilibrium and 6.1 Ergodic Theory and Random Dynamical Systems 6.2 Equilibrium Construction 6.2 Equilibrium Construction 6.3 Literature Comparison 6.3 Literature Comparison 6.4 Equilibrium Analysis 6.4 Equilibrium Analysis 7 General Linearized Phillips Curve 7 General Linearized Phillips Curve 7.1 Slope Coefficients 7.1 Slope Coefficients 7.2 Error Coefficients 7.2 Error Coefficients 8 Existence Results and 8.1 Main Results 8 Existence Results and 8.1 Main Results 8.2 Key Proofs 8.2 Key Proofs 8.3 Discussion 8.3 Discussion 9 Bifurcation Analysis 9 Bifurcation Analysis 9.1 Analytic Aspects 9.1 Analytic Aspects 9.2 Algebraic Aspects (I) Singularities and Covers 9.2 Algebraic Aspects (I) Singularities and Covers 9.3 Algebraic Aspects (II) Homology 9.3 Algebraic Aspects (II) Homology 9.4 Algebraic Aspects (III) Schemes 9.4 Algebraic Aspects (III) Schemes 9.5 Wider Economic Interpretations 9.5 Wider Economic Interpretations 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10 Econometric and Theoretical Implications and 10.1 Identification and Trade-offs 10.2 Econometric Duality 10.2 Econometric Duality 10.3 Coefficient Properties 10.3 Coefficient Properties 10.4 Microeconomic Interpretation 10.4 Microeconomic Interpretation 11 Policy Rule 11 Policy Rule 12 Conclusions and References 12 Conclusions and References Appendices Appendices A Proof of Theorem 2 and A.1 Proof of Part (i) A Proof of Theorem 2 and A.1 Proof of Part (i) A.2 Behaviour of ∆ A.2 Behaviour of ∆ A.3 Proof Part (iii) A.3 Proof Part (iii) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B Proofs from Section 4 and B.1 Individual Product Demand (4.2) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.2 Flexible Price Equilibrium and ZINSS (4.4) B.3 Price Dispersion (4.5) B.3 Price Dispersion (4.5) B.4 Cost Minimization (4.6) and (10.4) B.4 Cost Minimization (4.6) and (10.4) B.5 Consolidation (4.8) B.5 Consolidation (4.8) C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C Proofs from Section 5, and C.1 Puzzles, Policy and Persistence C.2 Extending No Persistence C.2 Extending No Persistence D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D Stochastic Equilibrium and D.1 Non-Stochastic Equilibrium D.2 Profits and Long-Run Growth D.2 Profits and Long-Run Growth E Slopes and Eigenvalues and E.1 Slope Coefficients E Slopes and Eigenvalues and E.1 Slope Coefficients E.2 Linearized DSGE Solution E.2 Linearized DSGE Solution E.3 Eigenvalue Conditions E.3 Eigenvalue Conditions E.4 Rouche’s Theorem Conditions E.4 Rouche’s Theorem Conditions F Abstract Algebra and F.1 Homology Groups F Abstract Algebra and F.1 Homology Groups F.2 Basic Categories F.2 Basic Categories F.3 De Rham Cohomology F.3 De Rham Cohomology F.4 Marginal Costs and Inflation F.4 Marginal Costs and Inflation G Further Keynesian Models and G.1 Taylor Pricing G Further Keynesian Models and G.1 Taylor Pricing G.2 Calvo Wage Phillips Curve G.2 Calvo Wage Phillips Curve G.3 Unconventional Policy Settings G.3 Unconventional Policy Settings H Empirical Robustness and H.1 Parameter Selection H Empirical Robustness and H.1 Parameter Selection H.2 Phillips Curve H.2 Phillips Curve I Additional Evidence and I.1 Other Structural Parameters I Additional Evidence and I.1 Other Structural Parameters I.2 Lucas Critique I.2 Lucas Critique I.3 Trend Inflation Volatility I.3 Trend Inflation Volatility 11 Policy Rule Here I prove the anti-Taylor policy rule and briefly outline its significance. Theorem 9. There exists polydromy if and only if aπ < 1, otherwise there is no solution at all. Theorem 9. There exists polydromy if and only if aπ < 1, otherwise there is no solution at all. this expands out to which simplifies to It is clear that λ = 1/α is a root- which will lie outside the unit circle. The equation factorizes to Together Theorem 3 and Proposition 16 stipulate that a solution exists if and only if the quadratic contains precisely one root inside the unit circle. This rules out complex roots. The sign pattern rules out negative roots. Therefore, the cut-off must be associated with λ = 1 which is clearly associated with aπ = 1. Rouche’s theorem implies that when aπ is sufficiently large there will be no roots inside the unit circle. Hence the desired configuration must be associated with aπ < 1.[102] The extension to |ε| is straightforward. Its eigenvalue equation is This simplifies to Plugging in reveals that there is one root inside the unit circle at λ = α and another outside at λ = 1/α. When the polynomial is factorized, the other roots come from the same quadratic equation as in the √ ε limit in (180). There have to be two roots inside the unit circle overall, so the parameter condition is identical. There might be a wider lesson for general equilibrium theory and mathematical economics. Equilibrium existence is demanding a contraction on destabilizing general equilibrium effects. To see how, consider the following decomposition of the lagged inflation coefficient informed by Decomposition 1. Decomposition 2. Decomposition 2. where GE represents the general equilibrium impulse from the cost channel where GE represents the general equilibrium impulse from the cost channel whilst the unitary terms come from the wall of the crossing (A) and the α comes from the singular component (D) and (F) from Decomposition 1. whilst the unitary terms come from the wall of the crossing (A) and the α comes from the singular component (D) and (F) from Decomposition 1. Turning to the New Keynesian field in particular, the following observation should prove interesting for empirical specification design, as well as theoretical consideration of the property of structural disturbances. Remark 35. Note that down the |ε| limit ∆ detaches from the rest of the system and behaves as an exogenous variable, like the cost-push shocks previously used to drive the Phillips curve. Remark 35. Note that down the |ε| limit ∆ detaches from the rest of the system and behaves as an exogenous variable, like the cost-push shocks previously used to drive the Phillips curve. I anticipate the policy rule results will generalize widely and may extend out to moderate rates of trend inflation. To reaffirm, in economic terms the blow up is unproblematic and even instructive. On the one hand, the economy is stable under inactive settings, when the Taylor rule represents a realistic shorthand description of monetary policy-making. On the other hand, the model explodes under active policy, precisely when following a simple rule is most unrealistic. Successful central banks employ macroeconomists and other skilled practitioners in order to implement a semblance of optimal monetary policy. Best practice appears to be inertial. In time, I expect the theory of monetary policy to catchup.[103] Finally, there is a direct connection between ex post constraints implied by the existence conditions and ex ante constraints breaking Divine Coincidence. The policies that are ruled out ex post, which seek to raise real interest rates immediately, are precisely those that fail to engage with the ex ante restrictions that give the economy inertia. Elaborate mathematical machinery serves to rule out evidently unsustainable macroeconomic policies. I anticipate this will become a universal lesson in New Keynesian economics. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Stories, News & Mentions about TEYLOR

Teylor’s Stories on HackerNoon

Teylor's News & Mentions