I like this model. It’s cute. Unrealistic, but cute. Environment: Intermediate sector. A unit measure of firms $i \in [0,1]$. Firm $i$ produces differentiated variety $Y_i$ with linear technology: $$ Y_i = A_i L_i $$ We assume $A_i\equiv A$ for now. Final sector. A competitive firm aggregates varieties via CES: $$ Y = \left(\int_0^1 Y_i^{\frac{\sigma-1}{\sigma}}\, di\right)^{\frac{\sigma}{\sigma-1}}, \qquad \sigma > 1. $$ Consumer. Representative agent with preferences: $$ U = \frac{C^{1-\gamma}}{1-\gamma} - \frac{L^{1+\varphi}}{1+\varphi}. $$ Equilibrium Prices $(\{P_i\}_i, W, P)$ and allocations $(\{Y_i, \Pi_i, L_i\}_i, C, Y, L)$ such that: ...

Alchemist 101: don’t create gold. Define what you created as gold. Motivation: find model to predict economic facts Kaldor facts —capital-output ratio and factor shares are roughly constant over time. Kuznets facts — the sectoral composition of employment and consumption shifts systematically: Agriculture (A) down, Manufacturing (M) stable, while Services (S) goes up. Can a single neoclassical growth model deliver both sets of facts simultaneously? Yes, if we throw in the cauldron a non-homothetic preferences (Engel’s law). ...

All models are wrong, but some are useful. — George E. P. Box And, some model would have particularly peculiar assumptions that makes them easy and handy. Motivation The neoclassical growth model was designed to explain several facts observed in developed economies over long periods (the “Kaldor facts” from data from the U.S. (1900-2000)): Output per capita grows at roughly constant rate Capital-output, investment-output, and consumption-output ratios are roughly constant Interest rates are roughly constant Factor shares (capital vs labor income) are roughly constant Uzawa’s Theorem Consider $(K_t, L_t, C_t, Y_t)_t$ governed by: ...

Images for Orchestra L. 122 and La Mer. Chicago audience are warm, welcoming. We give a lot of standing ovations: Salonen conducts CSO for Debussy’s La mer. De ja vu: NY Phil opens Shanghai Summer Music Festival with French Impressionism Works See the post here Fun facts: Debussy, Turner, and Impressionism We also know that Debussy greatly admired Turner’s work [the English painter]. His richly atmospheric seascapes recorded the daily weather, the time of day, and even the most fleeting effects of wind and light in ways utterly new to painting, and they spoke directly to Debussy. (In 1902, when Debussy went to London, where he saw a number of Turner’s paintings, he enjoyed the trip but hated actually crossing the channel.) The name Debussy finally gave to the first section of La mer, From Dawn to Noon on the Sea, might easily be that of a painting by Turner made sixty years earlier, for the two shared not only a love of subject but also of long, specific, evocative titles. ...

Motivation Baumol (1967): Differential productivity growth across sectors drives “uneven growth” and structural transformation. As sectors experience different rates of technological progress, this causes relative prices to change and labor to reallocate. Question: Can this supply-side mechanism generate structural change while maintaining balanced growth (Kaldor facts)? Model Household utility: $$ \begin{align*} & \int_0^\infty \exp(-\rho t) \frac{c(t)^{1-\theta} - 1}{1-\theta} dt\cr &s.t. \ \dot K(t) = r(t)K(t) - w(t)\bar L -\sum_{i \in \lbrace A, M, S\rbrace}p^iC^i(t) \ \cr & \text{where, }c(t) = \left(\sum_{i \in \{A,S,M\}} \eta^i C^i(t)^{(\sigma-1)/\sigma}\right)^{\sigma/(\sigma-1)}. \end{align*} $$ Note that $\sigma$ represents elasticity of substitution (constant across sectors) ...

Motivation Paradox: Standard productivity-based theories (see Supply Side Structural Change in Macroeconomics) predict that sectors with higher productivity growth should have declining employment shares. This is counter-intuitive, and false in reality: Japan (1960-1990): Rapid manufacturing productivity growth → increased manufacturing GDP share A lot of empirical evidence (common knowledge…) shows, countries with faster manufacturing TFP growth did not experience faster manufacturing employment decline. And cross-country data shows no negative correlation between manufacturing productivity and manufacturing employment. [Source pending] ...

Late January and early Feburary is an intriguing time for birthdays… There’s Mozart (Jan 27 1756 - Dec 5 1791), Schubert (Jan 31 1797 - Nov 19 1828) and Felix Mendelssohn (Feb 3 1809 - Nov 4 1847) I had the fortune of playing the Midsummer Night’s Dream Overture when I was even too young to understand the beauty before internalizing it (16). But the spirit lives on. Happy birthday to one of my favorite composers! ...

The Hidden Weight of GMM Consistency Conditions Consider estimating parameter $\theta\in \Theta$ from data $\lbrace w_i\rbrace_{i \in [N]}$ Assume: Parameter space $\Theta\in \R^K$ is compact. The criterion function of GMM $$ > s_N(\theta) = s(\vec w; \theta) > $$ is continuous in $\theta$ $\forall, \vec w$. $s_N(\cdot)$ well behaves: $$ > \sup_{\theta\in \Theta}|s_N(\theta) - s_\infty(\theta) \xrightarrow{p}0. > $$ $s_\infty(\theta)$ has a unique minimum at $\theta_0$. Then $\hat \theta_{GMM} \xrightarrow{p}\theta$. The proof is essentially a topological argument — uniform convergence of continuous functions on a compact set, plus a unique minimum, pins down the limit of the minimizers. It is clean, elegant, and almost suspiciously general. ...

In heterogeneous agent macroeconomics models, the way to compute (solve) a steady state equilibrium of the market is quite interesting. Consider a unit mass of agents, whose utility functions are the same $u:\R \to \R$. In a continuous time world, at each ifinisimmo time point they receive (stochastic) income and capital rents from their historial savings, consume and save more (sometimes saved capital are also called as ‘assets’ that generate returns, but per the no-arbitrage principle, the return of assets and captial rents net capital depreciation should be the same). So it’s being abstracted as a stochastic continuous time markov system, the state is the asset that the agent hold, action is consumption, plus random income shocks. ...

“Don’t try, just be.” — E Pahud comment on playing Schubert. Franz Schubert was born today 229 years ago. He composed more than 1500 works in his short career (He died 1828, aged 31) Courtesy of Wikipedia. I don’t own copyright! Some of my favourites include Die Forelle, and the Impromptus (No. 3 in G Flat Major). Schubert’s foremost contribution is his Lieders (art songs). The Winterreise was composed just one year before his pass away. Listen to Thomas Quasthoff & Daniel Barenboim performing Gute Nacht. ...