--- date: '2024-04-12' description: Economics for engineer, a guide. id: Finals modified: 2026-06-05 15:08:34 GMT-04:00 tags: - eng3px3 title: Economics for engineer, a guide. created: '2024-04-12' published: '2024-04-12' pageLayout: default slug: thoughts/university/twenty-three-twenty-four/eng-3px3/Finals permalink: https://aarnphm.xyz/thoughts/university/twenty-three-twenty-four/eng-3px3/Finals.md generator: quartz: v4.6.0 hostedProvider: Cloudflare baseUrl: aarnphm.xyz full: https://aarnphm.xyz/llms-full.txt --- ## samples. 4.b 11.e 12.c 13.d 14.b 15.a 16.e 17.c 18.c 19.b 20.c 21.b 22.c 23.b 24.a 25.a 26.e 27.a 28.e 29.a 30.a --- ## [[thoughts/university/twenty-three-twenty-four/eng-3px3/Net Value Function|net value function]] $$ \text{NVF} = \text{benefit} - \text{cost} $$ ## conversion factors ![[thoughts/university/twenty-three-twenty-four/eng-3px3/Conversion Factors#marginal value change.|Conversion Factors]] ## optimisation ![[thoughts/university/twenty-three-twenty-four/eng-3px3/Optimization#model-based|Optimization]] Linear optimization: ![[thoughts/university/twenty-three-twenty-four/eng-3px3/Linear Optimization#^linops|Linear Optimization]] ## time value of money ### interest Interest $I$ is the compensation for loaning money. > \[!tip\] interest rate > > $i = \frac{I}{P}$. Thus $F = P(1+i)$ > \[!tip\] Simple interests > > $I_{\text{each}} = P \times \frac{i}{\text{year}}$, total interest $I = I_{\text{each}} \times N_{\text{year}}$ > > $F_n = P(1 + ni)$ > \[!tip\] Compound interests > > $F_n = P(1+i)^n$ > \[!tip\] nominal interest rates > > $r$ is the equivalent yearly rate if interest is withdrawn so it doesn’t compound. (i.e: $r=mi$ where $m$ is the number of compounding periods per year) > \[!tip\] effective annual interest rates > > $i_{\text{eff}} = (1 + \frac{r}{m})^m - 1$ > \[!tip\] effective interest rates > > how much interest do you accrue after a year if nominal rate is 12%? > $F=P(1+i)^m=P(1+\frac{r}{m})^m$ > \[!tip\] continuous compounding > > $F = P e^{ry}$ ### net present value $$ \text{NPV} = \text{CF}_0 + \sum_{n=1}^{N}{\frac{\text{CF}_n}{(1+i)^n}} $$ where $\text{CF}_0$ is the initial cash flow, $\text{CF}_n$ is the cash flow at the end of the $n^{th}$ period, $i$ is the _effective interest rate_ > \[!tip\] discount rate > > Present value $PV = \frac{\text{CF}_t}{(1+r_d)^t}$, where $\text{CF}_t$ is cash flow happening in $t$ years in the future, and $r_d$ is the discount rate. > > sources: opportunity cost, inflation, risk, time preference, inflation, option premium regular deposit: Future value $FV = A \sum_{k=0}^{n-1}(1+i)^k = A \frac{(1+i)^n - 1}{i}$ where $A$ is the monthly, or time period, deposit. fraction of last payment that was interest was $\frac{i}{1+i}$, principal of the last payment is $A = F_{\text{last}}(1+i)$ > \[!tip\] geometric series > > $$ > \sum_{k=0}^{n-1}r^k = \frac{1-r^n}{1-r} > $$ ### inflation > \[!tip\] real vs. nominal > > nominal value refers to actual cash flow at the time it hapens, real value refers to equivalent amount of value at reference time, converted using inflation rates. > > real dollar $R = \frac{\text{CF}_n}{(1+r_i)^n}$, where $\text{CF}_n$ is the nominal cash flow at time $n$, and $r_i$ is the effective yearly inflation rate. > \[!tip\] internal rate of return > > the discount rate that results in a NPV of zero (break-even scenario) > > $$ > \text{CF}_0 + \sum_{n=1}^{N}{\frac{\text{CF}_n}{(1+r_{\text{IRR}})^n}} = 0 > $$ > \[!tip\] minimum acceptable rate of return > > a rate of return set by stakeholders that must be earned for a project to be accepted > > real vs. nominal MARR: real MARR is MARR if returns are calculated using real dollars, whereas nominal MARR is MARR if returns are calculated using nominal dollars. > > $\text{MARR}_{\text{real}} = \frac{1+\text{MARR}}{1+f} - 1$ where $f$ is the inflation rate ## risk management and stochastic modelling > Convert to dollar/wk to base calculation on same unit uncertainty, evaluating likeliness and potential impact, organize to risk matrix, determine expected impact, then propose mitigation strategies ![[thoughts/university/twenty-three-twenty-four/eng-3px3/most-critical-risk.webp]] > \[!tip\] expected impact > > the chance it happens multiplied by the impact it will have if it happens. > $\text{E[NPV]} = \sum_{i}{\text{NPV}(x_i)p(x_i)}$ > > Then use this to create necessary mitigation ### NPV with risk and uncertainty > \[!note\] probability distribution > > $p(x)$ of a discrete random variable $x$: Normalization requires that $\sum_{i}{p(x_i)} = 1$ > > PDF (probability density function) $f(x)$ of a continuous random variable $x$: Normalization requires that $\int{p(x)dx} = 1$ > \[!tip\] expected value for calculating stochastic to deterministic > > of function $f(x)$ is $\text{E}[f] = \sum_{i}{f(x_i)p(x_i)}$ for discrete random variable $x$ with probability distribution $p(x)$ > > of function $f(x)$ is $\text{E}[f] = \int_x{f(x)p(x)dx}$ for continuous random variable $x$ with PDF $p(x)$ > \[!note\] Normal distribution > > $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ > > `NORM.DIST(x, mean, stddev, cumulative)`: cumulative is `1` for CDF, `0` for PDF > `NORM.INV(RAND(), 0.5, 0.05)`: draw values from a normal distribution with mean 0.5 and stddev 0.05 ### non-linear deterministic and stochastic models mean value $\mu_{x}$ of a random variable $x$ is its own expected value $\text{E}[x]$, variance $\sigma^2_{x}$ is the expected value of the squared deviation from the mean $\text{E}[(x-\mu_x)^2]$, and stddev $\sigma_x$ > \[!tip\] central limit theorem > > sample size becomes large enough, the distribution of the sample mean will be approximately normally distributed, regardless of the distribution of the population, using [[thoughts/Monte-Carlo|Monte-Carlo]] simulation > Expected value of linear and nonlinear functions: suppose $x$ and $y$ are independent random variables with means $\mu_x$ and $\mu_y$, and variances $\sigma^2_x$ and $\sigma^2_y$, then $E[x^{2}] = \sigma_x^2 - \mu_x^2$, $E[xy] = \int \int xyp_xp_ydxdy=\int xp_xdx \int yp_ydy=\mu_x \mu_y$ Dealing with 12 months per year: saying outcomes over a year should be **normally distributed** (CLT), with a mean given by expected value of monthly outcome and stddev given stddev of outcome divided by square root of the # of rolls ($\sqrt{12}$) --- ## project management and CPM - scope, cost, time to maximize quality WBS (work breakdown structure): hierarchical decomposition of the total scope of work CPM (critical path method): determine the longest path through the network, the critical path, and the shortest time to complete the project ![[thoughts/university/twenty-three-twenty-four/eng-3px3/cpm.webp|cpm.webp]] crashing a project means using additional resources to shorten a specific task ## supply and demand market equilibrium: where supply and demand curves intersect, quantity demanded equals quantity supplied. shift to right: greater demand, higher price, higher quantity. shift to left: lower demand, lower price, lower quantity. factors of production: land, labour, capital, entrepreneurship determinants of demand: - price: quantity demanded $Q_d$ falls when price $P$ rises and vice versa - prices of related goods: substitutes and complements determinants of supply: - price: quantity supplied $Q_s$ rises when price $P$ rises and vice versa - factors of productions - fiscal policies, taxes, regulation > \[!tip\] elasticity: how responsive quantity demanded or supplied is to a change in price. > > Surplus when $Q_s > Q_d$, shortage when $Q_s < Q_d$. > > Elasticity of demand: $E_d = \frac{\% \Delta Q_d}{\% \Delta P} = \frac{\mid \frac{P}{Q_D} \mid}{\mid \frac{dP}{dQ_D} \mid}$ > > Elasticity of supply: $E_s = \frac{\% \Delta Q_s}{\% \Delta P} = \frac{\mid \frac{P}{Q_S} \mid}{\mid \frac{dP}{dQ_S} \mid}$ > > higher slope corresponds to lower elasticity: inelastic, lower slope corresponds to higher elasticity: elastic Demand elasticity: $E_D <1$ means if price increases by 5% then demand will decrease by less than 5%, inelastic. $E_D >1$ means if price increases by 5% then demand will decrease by more than 5%, elastic. > \[!tip\] taxes > > arbitrary lower the equilibrium quantity, > > price seen by consumers vs. suppliers changes depends on relative elasticities of demand and supply: more price change will end up on consumer side > > quantities change depends on total elasticities of demand and supply: more elastic means more quantity change. > \[!tip\] subsidies > > arbitrary increase the equilibrium quantity, > > price seen by consumers vs. suppliers changes depends on relative elasticities of demand and supply: more price change will end up on consumer side > > quantities change depends on total elasticities of demand and supply: more elastic means more quantity change. ## behavioural economics invisible hand of the market: self-interest of individuals leads to the best outcome for society as a whole, in a free market economy, as rational actors are motivated by incentives. perfect competition: wheat (control of price none, low barrier to entry, high # of producers, products are identical) monopolistic competition: restaurants (control of price low, low barrier to entry, high # of producers, products are similar) oligopoly: airlines (control of price high, high barrier to entry, few producers, products are similar) monopoly: utilities (control of price high, high barrier to entry, one producer, unique product) game theory, most notable [[thoughts/The Prisoner's Dilemma|The Prisoner's Dilemma]] anti-trust legislation: prevent monopolies, promote competition, protect consumers > behavioural economics: + psychology to look at reasons people make _irrational_ decisions > > “bounded rationality”: you don’t have perfect information, and understand there’s an opportunity cost to get it law of demand and _ultimatum game_: people will pay less for a good if they can get it elsewhere for less, even if they value it more than the price they pay. [[thoughts/Cooperation|Cooperation]]: R. Axelrod’s _The Evolution of Cooperation_ propose a “strategy”, what you do dependent on what the other person does. PPF (production possibility frontier): trade-offs between two goods, given a fixed amount of resources. risk aversion: people prefer a certain outcome to a risky one, even if the expected value of the risky one is higher. ⇒ assume that the given investment is loss, then calculate based on margin gains ## tax, incentives and depreciations _income, corporate, property, sales_ personal income tax: progressive tax rate corporate tax: flat tax rate, regardless of income level → net income: subtracting expenses from gross income. profit on investments will be tax. If yields loss, then offset the loss against the profits from another to pay less tax overall. [[thoughts/university/twenty-three-twenty-four/eng-3px3/Optimization|optimization]] strategies: minimize liabilities, timing of expenditures → incorporate into financial models, do sensitivity analysis before-tax MARR: set MARR high enough to include taxes that need to be paid ⇒ for investment’s gross profit after-tax MARR: if tax is explicitly accounted for in the cash flows of the project, then MARR should be lower ⇒ for final investment decisions $$ MARR_{\text{after-tax}} = MARR_{\text{before-tax}} \times (1 - \text{corporate tax rate}) $$ _incentives_: tax credits, tax reliefs, programs to encourage certain activities _depreciation_: due to use-related physical loss, technological obsolescence, functional loss, market fluctuation. > Deprecation is a non-cash expense, but reduces the taxable income of a business. > Can deduct annually by spreading the cost of an asset over its useful life. affects NPV (net present value), IRR (internal rate of return), and payback period calculation _Market value_: actual value of the asset can be sold for, estimated _Book value_: deprecated value of the asset, using a depreciation model _Salvage value_: estimated value of the asset at the end of its useful life > \[!tip\] value calculations > > Depreciation in year $n$ $D(n)$ is the decline in book value over that year: $BV(n) = BV(n-1) - D(n)$ > > Salvage value $SV$ is the book value at object’s EOL: $SV = BV(N) = MV(0) - \sum_{n=1}^{N} D(n)$ > \[!note\] Straight-line depreciation > > spreads uniformly over useful life, SLD of a period $D_{\text{sl}}(n) = \frac{\text{Purchase price}-\text{Salvage value after N periods}}{\text{N periods of useful life}}$. > > book value at end of $n^{th}$ year: $BV_{\text{sl}}(n) = P - n \times \frac{P-S}{N}$ > \[!note\] Declining-balance depreciation > > different assets are classified into classes: $D_{\text{db}}(n) = BV_{\text{db}}(n-1) \times d (\text{depreciation rate})$, such that book value at the end of a period $BV_{\text{db}}(n)$ is $BV_{\text{db}}(n) = P(1-d)^n$ > > given salvage value $S$ and period of useful life $N$, depreciation rate $d = 1 - \sqrt[N]{\frac{S}{P}}$ > \[!note\] Sum-of-years-digits depreciation > > $D_{\text{syd}}(n) = \frac{N-n+1}{\sum_{i=1}^{N} i} \times (P-S)$ > \[!note\] Unit of production depreciation > > $D_{\text{uop}}(n) = \frac{\text{units produced of period}}{\text{life in \# of units}} \times (P - S)$ > > assumes a SLD but vs. # of units rather than time.