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Let $L$ be the loan amount, then $L=P\frac{1-(1+i)^{-n}}{i}$ The balance at time $t$ is defined as the present value of the remaining payments, $B_t=P\frac{1-(1+i)^{-(n-t)}}{i}$ By using the following equations $I_k+C_k=P$ and $I_k=iB_{k-1}$ where $I_k$ is the interest paid at time $k$ we derive the following equation $C_k=\frac{P}{(1+i)^{n-k+1}}$ and ...

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The simplest way to understand this is to write out the numbers for a particular loan. You can track the balance, and interest over time. Very easily done in a spreadsheet. The amount of principal repaid in each payment is increasing as time passes, as the amount of interest paid falls (since the interest is proportional to a falling principal balance).

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Let $P$ be the CPI of the Home country, $P^*$ the CPI of the Foreign country, $P_H$ and $P_F$ the price of Home and Foreign intermediate goods expressed in domestic currency, and $P^*_H$ and $P^*_F$ be the price of Home and Foreign intermediate goods expressed in foreign currency. The log-linear version of the real exchange rate is given by \begin{gather} \...

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Giskard is right lexicographic preferences would make the job being dicsontinuous, but the problem is to find relations with closed contour sets. Maybe an idea would be to start from constructing a preference relation for which the only possible upper and lower contour sets are $X$ and $\emptyset$. In that case, you take the indiscrete topology, and all ...

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What you are trying to do here is estimate the demand curve of a firm. Broadly speaking, even if you got data on price and quantity, you would not be able to estimate this because your right-hand-side variable (quantity) is also determined by the supply curve. This is what people call the 'simultaneity' form of the problem of endogeneity. The way to ...

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(Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.) General Fact Let $W$ be standard Brownian motion with respect to filtration $( \mathscr F_t )_{t \in [0,T]}$. Consider $(L_t)_{t \in [0,T]}$ defined by $$\frac{dL_t}{L_t} = \psi_t dL_t, \; L_0 = 1.$$ In general, \$L_t = e^{\int_0^t \psi_s ...

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