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Well, according to what I see, let us take each part of the value function by its own and then $$\underbrace{\pi((t_1,t_1)|G)}_{1/2}\underbrace{\sigma((a_1,a_2)|(t_1,t_1),G)}_{3/5\times 3/5=9/25}\underbrace{u_1((a_1,a_2),G)}_{8}=36/25$$ $$\underbrace{\pi((t_1,t_2)|G)}_{1/2}\underbrace{\sigma((a_1,p_2)|(t_1,t_2),G)}_{3/5\times 2/5=6/25}\underbrace{u_1((a_1,...


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This can be easily done by taking price index and by dividing each index value by the index value in the new base year you selected and then multiplying by 100. For example, if you have two indexes with different base year (2001 and 2002 respectively): Index 2000 2001 2002 I_1 75 100 125 I_2 50 75 100 You can create new ...


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Following Galí (2015, p.3) the RBC model rests in three major assumptions: i) the efficiency of business cycle, ii) technology shocks as source of economic fluctuations, iii) the limited role of monetary factors. The main flaws may come from i) and iii), in a nutshell iii) implies that monetary policy (and broadly monetary factors) are fully neutral in ...


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Taking a Taylor expansion of $e^\hat x$ gives around $0$ gives: $$ e^\hat x \approx e^0 + e^0 \hat x = 1 + \hat x. $$ Substitution gives: $$ \begin{align*} &(c + \alpha g) + c \hat c_t + \alpha g \hat g_t \approx \frac{1}{\lambda}(1 - \hat \lambda_t),\\ \iff & \lambda(c + \alpha g) + \lambda c \hat c_t + \lambda \alpha g \hat g_t \approx 1 - \hat \...


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This will still count towards GNP as it would be counted as US export to Korea. GNP is given by: $$GNP=C+I+G+X-M + NFI$$ Where $C$ is consumption , $I$ investment l, $G$ government spending, $X-M$ are net exports an NFI is a net factor income, that is factor income of foreign residents minus factor income of non-residents (eg see the definition in Todaro &...


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At every time period, we have: $$ y_t = c_t + i_t + g_t $$ Long run steady state gives: $$ y = c + i + g $$ so, taking differences we have: $$ \begin{align*} &y_t - y = c_t - c + i_t - i + g_t - g,\\ \iff &\frac{y_t - y}{y} = \frac{c}{y}\frac{c_t - c}{c} + \frac{i}{y}\frac{i_t - i}{i}+ \frac{g}{y} \frac{g_t - g}{g} \end{align*} $$ Set $s_c = \dfrac{c}...


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Try taking derivative of x* w.r.t. py (which is 0) And verify with Slutsky decomposition.


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The key here is to distinguish between ex ante and ex post concepts of saving and investment. These concepts were defined by the economist Gunnar Myrdal as follows (quoted here): An important distinction exists between prospective and retrospective methods of calculating economic quantities such as incomes, savings, and investments; and [...] a ...


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It is because these two examples use the word investment in different meaning. In the top example, investment means all investment whether it is investment into something productive or inventory investment. In the second graph, the word “investment” is applied to investment minus inventory investment. This is why in the second picture the difference between ...


4

We have that: $$ \dot K = s K^\alpha L_0^b e^{nbt} $$ Rewriting the differential equation gives: $$ K^{-\alpha} \frac{dK}{dt} = s L_0^b e^{nbt} $$ Integrate both sides with respect to $t$ from $0$ to $T$ gives: $$ \frac{1}{b} [K^b]^T_0 = s L_0^b \frac{1}{nb}[e^{nbt}]^T_0 $$ So: $$ K^b_T = K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT} $$ Equivalently:...


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The series <I + A + A^2 + A^3 + ... + A^N> converges on the Leontief inverse (I-A)^(-1) as N approaches infinity. In this format, I can be thought of as initial demand for a given product, A represents the first tier inputs in the supply chain required to produce I, A^2 represents the second tier inputs in the supply chain needed to produce the first ...


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Speaking from a US centric view, there is either physical currency in a vault, or on an account with a US bank. These are called correspondent accounts or nostro / vostro accounts. Therefore, nothing is hidden. What is hidden is who the ultimate beneficiary may be but that doesn't matter for macroeconomic policies. It matters for tax purposes. The Fed is not ...


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My best guess is that they detrended by computing the discounted value. (Although I'm not sure if this is really what they've done.) Let $r = 0.015$ be the growth rate and let $t$ be the time period and $t_0$ the base year, (e.g. $t = 2000$ and $t_0 = 1982$). If $Y_t$ is GDP in period $t$ then the detrended GDP is given by: $$ \frac{Y_t}{(1+r)^{t-t_0}}. $$ ...


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In a graph you linked, the durables spending FRED series PCEDG is 1968 versus 1550 (Aug 2021 versus pre-pandemic Feb 2020, a span of 18 months). That is +27%. The PCE price level series PCEPILFE that excludes food and energy shows levels of 118.1 versus 113.2, so +4.3% over 18 months. The U.S. is not a net importer of food and energy. Currently there is ...


3

Here's my guess. Let use the notation $$ \tilde x_t \approx \ln(x_t) - \ln(x) \approx \dfrac{x_t - x}{x}. $$ If we take logs on both sides we get: $$ \ln(G_t) = \frac{1}{1 -\rho} \ln(p_t) + \ln(y_t) - \ln(1 + p_t^{\frac{\rho}{\rho-1}}) $$ Subtracting the steady state gives: $$ \tilde G_t = \frac{1}{1 - \rho} \tilde p_t + \tilde y_t - \left[\ln(1 + p^{\frac{\...


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The direction of $\frac{\partial c^{*}}{\partial n}$ is not ambiguous. An easy way to show this is taking derivative of $c^*=(1-s)f(k^*)$ so that $\frac{\partial c^{*}}{\partial n}=(1-s)f'\frac{\partial k^{*}}{\partial n}$ and because $f'>0$ and we can prove $\frac{\partial k^{*}}{\partial n}<0$ we thus have $\frac{\partial c^{*}}{\partial n}<0$. ...


2

Let's do the maths in a simple way: Take profit function: $B = p · f(L,K) - w·L - r·K$ Where $B$ is profit, $L$ is labour and $K$ means capital. $f(L,K)$ is the product function, $p$ is price and $r$ is the rental price of capital and $w$ is wage. Take $PMg(K)$ as marginal product of $K$ and know that $df(K,L)/dK = PMg(K)$ In order to optimize $B$ with $K$ ...


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Intuitively it works like this: There is an infinite number of firms competing for the capital in the economy. If the rental price of capital $r$ is below its marginal product , then a firm can increase its profit by hiring more capital. In order to do this a $firm_1$ can offer a slightly higher rate than $r$ and immediatly all capital in the entire economy ...


4

You should try to avoid mixing data sets unless you have a good way to bridge them. There are very precise methodologies used to construct each data set and mixing them can lead to nonobvious problems. You might consider looking at the BEA's IO tables at a more disaggregated level. For example, the Use table shows Personal Consumption Expenditures at a ...


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As you have been already answered on therms of the quantitative equation of money, I will give you others approachs between the realtionship of these ways of funding and real variables. First, more printed money lead to more assets and therefore more demand coming as effect of this new financial wealth, that could lead to inflation as well, depending on the ...


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Okay, unfortunately the term "hyperinflation" has been thrown around by people like Peter Schiff just far too often, and as a result it has been oversimplified. One of the questions you are asking is, why did Zimbabwe experience it and Weimar, etcetera. Imagine we are working on a master thesis and you are being told the topic you want to cover is ...


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If a dollar depreciates then: Financial assets and liabilities (loans, bonds) denominated in US dollars are still worth the same number of US dollars. Financial assets and liabilities (loans, bonds) denominated in foreign currencies are both worth more US dollars (because it costs more US dollars to get a foreign dollar). Tangible assets (e.g. real estate) ...


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Since the OP asked for a rigorous proof, here is one. By Acemoglu's inequality in the first part of his proof, we can separate $\{x(t)\}_{t=0}^{\infty}$ into two subsequences, an increasing subsequence $\{x(t_{i})\}_{t_{i}\in I}$ bounded above by $x^{*}$ and a decreasing subsequence $\{x(t_{j})\}_{t_{j} \notin I}$ bounded below by $x^{*}$. Indeed, we define $...


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