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======= ENGM90011 - Economic Analysis for Engineers =======
{{tag>notes unimelb economics}}
====== Consumer choice ======

===== Commodities =====

A commodity is a good or service.
Unless otherwise specified, the quantity of a good is non-negative, real number: \(x\in\mathbb{R}^+\)
Quantities tend to be specified with respect to time, but the overall models tend to be atemporal, meaning we can not borrow from today to benefit tomorrow.
An intertemporal choice is one where time is taken into account.
Durable goods are those that are consumed over multiple time periods.
We can measure the amount of a durable good consumed every year with depreciation.
We can calculate depreciation with straight line valuing, opportunity cost (rental equivalence).
===== Bundles =====

We can represent a set of choices as a bundle: \(x=(x_1, x_2, x_3, ..., x_n)\)
We can represent the consumption set as the total set of all bundles: \((X\in\mathbb{R}^{2+})\)
Consumers rank bundles in terms of their preferences alone. 

We can represent preference as \(A\succ B\), and indifference as \(A\sim B\).
Preferences are transitive, i.e. \(A\succ B\) and \(B\succ C\) so \(A\succ C\).
If preferences are not transitive, we can get cycling and may not be able to decide a choice.

Preferences are monotonic if \(\forall x, x'\in X\) and \(x\geq x'\), then \(x\succsim x'\).
A stricter rule can be applied as \(\forall x, x'\in X\) and \(x>>x'\), then \(x\succ x'\).
The comparison is element wise so every element in \(x\) is preferred over the corresponding element in \(x'\).
Strong monotonicity is when \(x, x'\in X\) and \(x\geq x'\) and \(x\neq x'\), then \(x\succ x'\).
===== Indifference Curves =====

An Indifference Curve (IC) is the set of bundles having the same ranking, i.e. the consumer is indifferent among the set of bundles.
ICs exist for every set of bundles and ICs further from the origin are preferred to those closer.
ICs cannot intersect as that would cause a contradiction of indifference, as all points on an IC are indifferent to each other.
We draw ICs as inward bowed curves, which are both monotonic and convex.

The Upper Contour Set is the set of bundles that are at least as good as x: \(UC(x)={y\in X|y\succsim x}\).
This includes all bundles on the same IC as x, as well as all above x.
The Lower Contour Set can be defined similarly for all bundles at least as bad as x.

A set S, is a Convex Set if \(\forall x, y\in S\) and \(\forall\alpha\in[0,1]\) then \(\alpha x+(1-\alpha)y\in S\).
A preference is convex if \(\forall x\in X\), the Upper Contour Set is a convex set.
A preference is concave if \(\forall x \in X\), the Lower Contour Set is a convex set.
An IC bowed inward forms a convex curve.
An IC bowed outward forms a concave curve.

A set is strictly convex if \(\forall x,y\in S,\forall a\in(0,1)\), then \(\alpha x+(1-\alpha)y\in\) interior S.
If preferences are strictly convex, a combination of two preferences is strictly preferred to the preferences.
Strictly convex preferences mean consumers prefer averages to extremes.
We add the assumption of convexity as empirically we observe consumers moving to averages and it provides desirable properties when solving the consumer's choice problem.
===== Marginal Rate of Substitution =====

The Marginal Rate of Substitution (MRS) is the negative of the slope of the indifference curve.
\[MRS=-\left.\frac{dy}{dx}\right\rvert_{IC}\]
It represents the amount of one good a consumer is willing to forfeit for another.
===== Utility function =====

A utility function, \(U:X\to R\), a real-valued function over the consumption set that satisfies two rules:
  - \(x_1\sim x_2\implies U(x_1)=U(x_2)\)
  - \(x_1\succ x_2\implies U(x_1)>U(x_2)\)
Thus utility is a representation of an individual's preferences.
Utility functions need not be unique, it is an ordinal measure not a cardinal measure.

Goods are perfect substitutes if the consumer is just as happy to consume one as the other.
Goods are perfect compliments if the consumer likes to consume them in fixed proportions, such that more of one without the other does not make the consumer happier (e.g. shoes).
The consumer's utility can be represented by the Leontief utility function:
\[U=\min\{x_1,x_2\}\]
This satisfies monotonicity.

There is also the Cobb-Douglas utility function:
\[U(x,y)=x^\alpha y^{(1 -\alpha)},\alpha\in(0,1)\]
This gives the MRS of:
\[\frac{\alpha}{(1 -\alpha)}\frac{y}{x}\]
Cobb-Douglas functions are strictly concave in 3d, but give strictly convex indifference curves (level sets/contours).

If the utility function is differentiable, we can define the marginal utility of a good x as:
\[MU_x=U_x=\frac{\partial U}{\partial x}\]
Marginal utility is non-negative for monotonic cases, and strictly positive for strongly monotonic cases.
We can find MRS by:
\[MRS=\frac{MU_x}{MU_y}\]
For perfect substitutes, MRS is always a constant.

There is also the CES utility function which can be used to approximate the others:
\[U(x,y)=(x^\rho+y^\rho)^\frac{1}{\rho},\rho\leq 1, \rho\neq 0\]
As \(\rho=1\), we get perfect substitutes.
As \(\rho\to0\), we get a resemblance to Cobb-Douglas.
As \(\rho\to-\infty\), we get perfect compliments.
In addition to being more flexible, CES makes less assumptions than Cobb-Douglas.

Utility functions must be continuous as preferences exist on a continuity (by assumption).
There is monotonicity of preferences, so utility must be an increasing function (\(x>y\implies U(x)>U(y)\)).
A utility function is said to be quasi-concave iff every upper contour set is a convex set.
It is strictly concave iff the upper contour set is strictly convex.
(Strictly) concave functions are (strictly) quasi-concave but not necessarily the converse.
We generally assume that the utility function is differentiable (actually assume utility is twice differentiable).
We define the Hessian to be a matrix of second order derivatives.
Assuming \(U\in\mathcal{C}^2\), means that the hessian is a symmetric \(n\times n\) matrix and thus a quadratic form.
===== Budget set =====

So far we have looked at all bundles, but consumers are restricted to those they can afford.
Thus we are left to question, what makes a bundle affordable?
We have \(\mathbf{p}=(p_1,...,p_n)\in\mathbb{R}^n_{++}\) as the prices of the N goods and M as the consumers budget.
These are parameters, which we use to find the budget set.
The budget set is the set of affordable bundles: \(\mathcal{B}(\mathbf{p},M)\).
We have the budget constraint that you cant spend more than you have: \(p\cdot x\leq M\).
This lets us define the budget set as:
\[\mathcal{B}(p,M)=\{x\in\mathbb{R}^N_+|p\cdot x\leq M\}\]
The budget line, \(\mathcal{B}'(p,M)\) is the set of bundles that have a total expenditure equal to income M.
\[\mathcal{B}'(p,M)=\{x\in\mathbb{R}^N_+|p\cdot x= M\}\]

The budget set is a closed set, with the budget line and axes forming the boundary points.
It is also bounded as is sits within the open ball with radius marginally larger than the largest axial intercept.
This means the budget set is compact.
==== Optimisation problem ====

The optimisation problem is to find the value in a set which maximises or	minimises a function.
The function is called the objective function, and the set is the constraint set.
The values of x are the choice variables and \(\theta\) denotes a parameter that the objective function or constraint set may depend on.

Weierstrass' theorem states that for \(\mathcal{D}\subset\mathbb{R}^n\) being a compact set and \(f:\mathcal{D}\to\mathbb{R}\) being continuous, there is a maximum and minimum value.
\[\exists x_{max}\in\mathcal{D} :\forall x\in\mathcal{D}, f(x_{max})\geq f(x)\]
\[\exists x_{min}\in\mathcal{D}:\forall x\in\mathcal{D},f(x_{min})\leq f(x)\]
==== Mathematical definitions ====

An open ball of radius r about x, \(B_r{x}\) is the points within the distance r of x, i.e.
\[B_r(x)=\{y\in\mathbb{R}^n|d{x,y}<r\}\]
If \(x\in\mathbb{R}\), then \(B_r(x)\) is on the open interval \((x-r,x+r)\).
In two dimensions it is an open circular disk and in three it is an open ball.

If it is possible to chose \(x\in S\subseteq\mathbb{R}^n\) and \(r>0\) such that \(B_r(x)\subset S\), then x is said to be an interior point of S, \(x\in\text{int}S\).
If for any r, \(B_r(x)\) will contain some points in S and some outside, x is said to be a boundary point of S.
==== Open, closed, bounded and compact sets ====

If \(\forall x\in S\subseteq\mathbb{R}^n\) and \(x\in\text{int}S\), S is said to be an open set.
S is a closed set of the compliment of S in \(\mathbb{R}^n\) is an open set.
That is, a closed set contains all its boundary points.

A set is said to be bounded if \(\exists r>0\) such that \(S\subset B_r(0)\).
A set is said to be compact if it is closed and bounded.
==== Consumer's problem ====

The consumer's problem is to choose the most preferred affordable bundle.
This solution is \(x_{max}\).
\[x_{max}\in\mathcal{B}(p, M)\text{ and }\forall x\in\mathcal{B}(p, M),x_{max}\succeq x\]
We assume consumer's preferences are complete, transitive and continuous, which means there is always a continuous utility function representing the consumer's preferences.
\[\max U(x):x\in\mathcal(p, M)\]
Here utility is the objective function, the budget set is the constraint set, price and income are parameters, quantities of goods are the choice variables.
Additionally utility can depend on parameters, such as \(\rho\) in the CES utility function.
Weierstrass' theorem tells us that the solution always exists.
If the preferences are monotonic, then any bundle not on the budget line always has a bundle preferred to it so the solution is on the budget line.
This makes the problem:
\[\max U(x):x\in\mathcal{B}'(p,M)\]
The problem is now a constrained maximisation problem with an equality constraint.
Solutions can be at the boundary points (boundary or corner solutions), or between the boundary (interior solutions).
We can express the interior of the budget line as:
\[x=t\left(\frac{M}{p_1},0\right)+(1-t)\left(0,\frac{M}{p_2}\right),t\in(0,1)\]
This is all the points on the budget line that are strictly positive, i.e. excluding bounding points.
=== Method of Lagrange ===

The Method of Lagrange is a way of finding solutions to constrained optimisation problems.
We are going to look at the method when there are equality constraints.
This method only works for interior points, hence why we excluded the boundary points above.
Often it is obvious that the solution will be an interior solution.
If we don't know this, we can use Lagrange with inequality constraints, or we can compare the solution with equality constraints to the boundary points.

If we express the equality constraint as a function, we can solve the constraint by substitution for two variables.
Alternatively we can use the method of Lagrange, which requires an objective function and k equality constraints expressed as functions, such that both the constraints and objective functions are \(\mathcal{C}^1\).
We can then express the constraint set as \(\mathcal{D}=U\cap\{x\in\mathbb{R}^n|g(x)=0\}\), U is an open set so the intersection removes any boundary points from the set.
We introduce Lagrange multipliers \(\lambda_i\) for each constraint to allow us to incorporate the constraint into the objective function, creating the Lagrangian function.
\[\mathcal{L}(x,\lambda)=f(x)+\sum_{j=1}^k\lambda_jg_j(x)\]
Setting the first order partial differential equations of the choice variables (\(x,\lambda\)) to zero gives n+k First Order Necessary Conditions (FONC).
\[\frac{\partial\mathcal{L}(x,\lambda)}{\partial x_i}=\frac{\partial f(x)}{\partial x_i}+\sum_{j=1}^k\lambda_j\frac{\partial g_j(x)}{\partial x_i}=0\]
\[\frac{\partial\mathcal{L}(x,\lambda)}{\partial\lambda_j}=g_j(x)=0\]
A critical point is one where the first order partial derivatives are equal to 0.

We let \(Dg(x)\) be the matrix of the derivatives of g with respect to x.
Where we have only one constraint, this is the grad of g.
\(Dg(x)\) is a \(k\times n\) matrix, and if the rank is equal to k, we say it is of full rank.
The theorem of Lagrange states that if there is a point, \(x^*\), that is a constrained maximum or minimum that satisfies the constraint qualification \(\text{rank}(Dg(x))=k\), then there exists a \(\lambda^*\in\mathbb{R}^k\) such that \((x^*,\lambda^*)\) satisfies the Lagrangian FONC.
So if \(x^*\) is a global constrained optima and the constraint qualification holds there, \((x^*,\lambda^*)\) is a critical point of the Lagrangian.
In order to distinguish between maximum, minimum and saddle points, we need to use Second Order Necessary Conditions (SONC).

When solving for the consumers problem, the FONC will form the MRS and the budget line.
Hence the solution will be where the budget line and IC intersect at the maximum IC.
The method ignores non-negativity constraints, so the solution must be a boundary point when a negative value is produced.
If the boundary points can't be ruled out as global optima, compare the utility at the boundary to the utility found by Lagrange.
It is important to note that differentiation is a local function, so it will find local maxima and minima in addition to global maxima and minima, so all found points will need to be evaluated.
The method of Lagrange fails at points that are non-differentiable, so they may need to be included manually after the fact.
If the solution requires inequality constrains, a different method is needed with different FONC called the Kuhn-Tucker conditions.

Monotonic, but not convex preferences can cause the Method of Lagrange to yield local maxima and minima in addition to global maxima and minima.
This can even happen at the same time.
Non-convex preferences can also yield multiple global maxima far apart from each-other.
==== Convex preferences ====

Preferences are said to be convex if \(\forall x\in X\), the upper contour set of \(x\) is a convex set.
Convex preferences are equivalent to a quasi-concave utility function.
If we add convexity to our assumptions about preferences (complete, transitive, continuous and monotonic), then indifference curves are tangent to the budget line only at global maxima.
There still may be multiple global maxima, but they will form a convex set.
Strictly convex preferences are strictly quasi-concave utility functions, and will yield a unique global maxima.
===== Demand =====

Assuming preferences are complete, transitive, continuous, monotonic and strictly convex.
The consumer's most preferred affordable bundle is unique for any given set of parameters.
Since there is a single \(x_i\) for a set of parameters, \(x_i(p,M)\) is a function called the demand function for good \(i\).
We can say the following about demand functions:
  - \(p\cdot x(p,M)=M\)
  - If \(x(p,M)\) is an interior solution, \(MRS(x(p,M))_{ij}=\frac{p_i}{p_j}\forall i,j,...,N,i\neq j\)
  - \(x_i(p,M)\) are homogeneous of degree 0 in \((p,M)\): \(x_i(tp,tM)=x_i(p,M),\forall t>0\)

Comparative statics compares two equilibria: before and after change in a parameter.
It is static in that we are not concerned with the process of moving from one equilibrium to the other.
In doing this we invoke the "ceteris paribus" assumption, that everything stays the same.
Additionally, instead of working directly with the changing variables, we work with elasticity measures which equate the relative change in one quantity compared to the change in another.
If there is a finite change between two points, we use the arc elasticity: \(\frac{\Delta x}{\Delta y}\frac{y}{x}\).
Typically we use the average values of \(x\) and \(y\).
For elasticity at a point, we use the point elasticity: \(\frac{\partial x}{\partial y}\frac{y}{x}\), although this does assume demand is differentiable.
==== Income ====

When income changes, the intercepts of the budget line change, but the slope stays the same.
A good is said to be a normal good if \(\frac{\Delta x_i}{\Delta M}>0\), that is the amount demanded moves with income.
A good is said to be a inferior good if \(\frac{\Delta x_i}{\Delta M}<0\), that is the amount demanded moves counter to income.
Goods can change from normal to inferior and back as income changes through different ranges.
Homothetic preferences states that all points on a ray from the origin have the same MRS, i.e. \(MRS(x(p,M))=MRS(kx(p,M))\).
When combined with the fact that income doesn't change elasticity (\(MRS(x(p,M))=MRS(x(p,tM))\)), we can get that \(x(p,tM)=kx(p,M)\) as they are both the sole most affordable bundle on the budget line.
With homothetic preferences, income elasticity of demand is unit elastic.
It also follows that the share of expenditure on the good is also unchanged.

Unfortunately goods tend not to be homothetic, so we group normal goods into necessities and luxuries.
Necessities are goods who's income elasticity is between 0 and 1.
Luxuries are goods who's income elasticity is greater than 1.
Necessary goods are income inelastic while luxury goods are income elastic.
Necessary goods have a decreased share of income when income increases, where as luxury goods have an increased share of income.
==== Cross price ====

When the price of a good changes, the budget line rotates, so the optimal position on the IC changes.
A good is a substitute when an increase in price of a good causes in increase in quantity of the good: \(\frac{\Delta x_i}{\Delta p_j}>0\).
Conversely, a good is a compliment when the increase in price of a good causes a decrease in quantity of the good: \(\frac{\Delta x_i}{\Delta p_j}<0\).
We measure this with the cross price elasticity of demand:
\[\epsilon_{ij}=\frac{\%\Delta x_i}{\%\Delta p_j}=\frac{\Delta x_i}{\Delta p_j}\frac{p_j}{x_i}\]
==== Own price ====

An ordinary good is one whose price and demand are negatively correlated: \(\frac{\Delta x_i}{\Delta p_i}\leq0\).
A Giffen good is one whose price and demand are positively correlated: \(\frac{\Delta x_i}{\Delta p_i}>0\).
Giffen goods are very rare in the real world.
We define the own-price elasticity of demand to be:
\[\epsilon_i=-\frac{\%\Delta x_i}{\%\Delta p_i}=-\frac{\Delta x_i}{\Delta p_i}\frac{x_i}{p_i}\]
We add the negation due to the rarity of Giffen goods.
The larger the elasticity, the larger the change in demand for a given change in price.
If \(\epsilon=0\), we say demand is perfectly inelastic.
If \(\epsilon<1\), we say demand is inelastic.
If \(\epsilon=1\), we say demand is unit elastic.
If \(\epsilon>1\), we say demand is elastic.
If \(\epsilon=\infty\), we say demand is perfectly elastic.
The elasticity of demand is determined by the closeness of substitutes, time period and the definition of a market.
==== Demand curve ====

The most important relationship for demand is between quantity of a good and its price.
When we plot the demand curve, we actually plot the inverse demand curve, with price on the vertical axis and quantity on the horizontal one.
A change in demand is a shift of the demand curve, where as a change in quantity demanded is a shift along the curve.
An increase in demand shifts the curve to the right, and a decrease shifts to the left.
Anything that changes the budget set, or preferences, can cause a change in demand.
A change in tastes can also change demand.
Expectations of future prices and incomes can also change demand, as consumers will wait until the price is better for them.
There are other less obvious factors affecting demand (weather, current events, etc.).

Perfectly elastic demand is a horizontal line, and perfectly inelastic demand is a vertical line.
Elasticity of demand is the derivative of the demand curve.
For a linear demand curve (constant derivative), the elasticity is less than one for low prices and high quantities, and greater than 1 for high prices and low quantities.

The market demand curve is the sum of each consumer's demand curves.
A change in the number of buyers can change the market demand.
==== Income and substitution effect ====

When the price of a good changes, its intercept with the budget line shifts.
This affects the slops of the budget line and can shift demand.
When the price of a good increases, this effectively decreases the real income of the consumer, as the same nominal income can now buy less goods.
\[\left(\frac{M}{p_1}\right)_o=1,\left(\frac{M}{p_1}\right)_n=0.5\]
There is also the obvious change in the relative price of goods.
We try to separate the change in demand into the component due to change in real income (income effect) and change in relative price (substitution effect).
\[\Delta x_1=x_1^n-x_1^0=\Delta x_1^M+\Delta x_1^s\]
It is important to know this so that in designing policy we can alter consumer behaviour without affecting their incomes too much.
=== Hicks Method ===

Given the new prices, change income such that demand lies on the old IC.
This is done by finding the utility of the original demand, then parameterising the new demand in terms of income, and finding the income such that the new demand's utility is equal to the old demand.
The change in demand from the new demand to the budget compensated demand is the change due to the income effect, and the remainder is due to the substitution effect.
The substitution effect always goes in the opposite direction to the change in price, as we assume the IC is convex and monotonic.
\[\frac{\Delta x^s}{\Delta p_x}\leq 0\]
For a normal good, an increase in price causes a decrease in real income.
\[\frac{\Delta x^M}{\Delta p_x}<0\]
So for a normal good, the quantity demanded unambiguously changes in the opposite direction of price changes.
\[\frac{\Delta x}{\Delta p_x}=\frac{\Delta x^M}{\Delta p_x}+\frac{\Delta x^s}{\Delta p_x}\leq 0\]

For an inferior good, a decrease in income causes an increase in consumption.
\[\frac{\Delta x^M}{\Delta p_x}>0\]
So for an inferior good, the quantity demanded moves ambiguously in response to a price change.
As long as \(|\Delta x^s|>|\Delta x^M|,\frac{\Delta x}{\Delta p_x}<0\), an inferior good will behave like a normal good.
When \(|\Delta x^s|<|\Delta x^M|,\frac{\Delta x}{\Delta p_x}>0\), an inferior good will behave like a Giffen good.
From this we can see that a Giffen good must be inferior.
This requires a large income effect, usually from goods which hold a large portion of income, although these goods tend to be normal (housing, food, etc.).

This effect tells us that demand curves slope down, since price and quantity are negatively correlated.
This is with the exception of Giffen goods.
=== Slutsky Method ===

This determines the demand at the new prices if the consumer was given sufficient income to afford the original bundle at the new prices.
\[M'=p_1^nx_1^0+p_2x_2^0\]
With the new budget line, there is a different demand to the left of the original point, as the price increased.
We can find the substitution effect from the shift along the IC, and the income effect is the rest.
The advantage of the Slutsky method is that it uses observable data, but it tends to overestimate.
=== Kaldor Method ===

This is the least used method.
It is the converse of Hicks method, in that it asks what is the demand for a new income at the old prices, such that utility is equal to the new utility.
This is keeping utility constant at the new level and finding an income such that the old budget line is shifted onto the new curve.
The Hicks method finds the minimum income needed to spend at the new prices to obtain the same utility as the old bundle.
The Kaldor method finds the minimum income needed to spend at the old prices to obtain the same utility as the old bundle.
Both are a minimum expenditure constraint subject to constant utility.
===== Welfare =====

We can measure the change in welfare (in dollars) from a price change by:
  - Change in consumer surplus (\(\Delta CS\))
  - Equivalent Variation (EV)
  - Compensating Variation (CV)
==== Consumer surplus ====

The consumer surplus is the excess value derived from buying a unit for less than willing to pay for it.
This is the area between the demand curve and the price paid for the good.
\[CS=\int_p^\infty x(p)dp\]
So the change in surplus is the change in area due to a shift in prices.
\[\Delta CS=\mp\int_{p_1}^{p_2}x(p)dp\]
This method uses observables to find the change, so is useful in practice.
==== Compensating and Equivalent Variation ====

\[CV=e(p^0,U^0)-e(p^n,U^0)=M-e(p^n,U^0)\]
Compensating Variation is the differences in expenditure from the old level to the new level on the old IC.
By holding utility constant, we are able to work with real numbers, making it nice to use theoretically.

Equivalent Variation is similar to Compensating Variation, but working a the new IC.
\[EV=e(p^0,U^n)-e(p^n,U^n)\]
From the Hicks method we get \(M'=e(p^n,U^0)\), and from the Kaldor we get \(M'=e(p^0,U^n)\).
==== Relationship between measures ====

The change in quantity demanded from a price change includes both the income and substitution effects.
The Hicks and Kaldor methods measure substitution effects only, so for a normal good \(\left|\frac{\Delta x}{\Delta p}\right | >\left |\frac{\Delta x ^ S} {\Delta p }\right|\).
\[CV\leq\Delta CS\leq EV\]
The Hicks and Kaldor demands are income compensated (utility and real income holds constant so substitution effect is only at play).
The consumer surplus uses ordinary demand (Marshalian demand), which includes both the substitution and income effects.
If the good has no income effect, then all three are concordant.
==== Quasi-Linear utility ====

Suppose there are many goods, but we are only interested in one.
We can define \(y=M-px\), as the expenditure on all other goods.
The budget line can then be expressed as \(px+y=M\), and we can think of \(y\) as a good with price 1.
If the utility of \(x\) is independent of the other goods, we can represent utility as \(U=v(x)+y\).
This is a quasi-linear utility function in functional form.
For this, we find that there is no income effect, meaning demand is independent of income.
This isn't necessarily true for all \(x\).
====== Producer theory ======

===== Introduction =====

We have outputs of a firm \(y\in\mathbb{R}_+\), and inputs (factors of production) \(x\in\mathbb{R}_+\).
The inputs consist of capital (K), land (T), labour (L), raw materials, etc.
In general there are multiple inputs and outputs, not necessarily the same number.
In general we want to measure the amount of input used in a period of time (flow), rather that the stock amounts of resources used.
===== Technology (production functions) =====

Technological constraints provide the combinations of inputs to produce outputs, constraining to possible combinations.
A production set is all possible production plans \((y,x)\).
If \((y,x)\) is feasible, then so is \((y,x')\) where \(x'\geq x\).
If \((y,x)\) is feasible, then so is \((y',x)\) where \(y'\leq x\).
The maximum output for a given level of inputs is the production function \(y=f(x)\).
The production function embodies the technological constraints.
We assume positive input prices in the production function.
==== Returns to scale ====

Returns to scale analyse the outcome of changing all inputs.
Technology exhibits increasing returns to scale (IRS) if \(f(tx)\geq tf(x),\forall t>1\).
Increasing returns to scale tends to be seen with specialisation.
Technology exhibits decreasing returns to scale (DRS) if \(f(tx)<tf(x),\forall t>1\).
This can be due to coordination problems.
Constant returns to scale (CRS) is when \(f(tx)=tf(x),\forall t\geq 0\).
This occurs when the production function is homogeneous of degree 1 in inputs \(x\).
An example is replication of an existing process.

Not technically returns to scale, but a common occurrence, is when an input is fixed so the full effect is not seen.
==== Production with multiple inputs ====

We generally want at least two inputs, with which we can characterise labour and capital.
There is often a direct trade off between labour and capital.
We define an isoquant \(Q(y)\) to be all combinations of inputs that produce a given level of output \(y\).
\[Q(y)=\{x\in\mathbb{R}_+^m:f(x)=y\}\]
We assume monotonicity of the production function (isoquants slope down, further from the origin means more output).
We also assume convexity for the production function.
Isoquants are analogous to indifference curves in demand.

The marginal product is the change in output per unit change in input, holding all other inputs fixed.
\[\frac{\Delta y}{\Delta x}=\frac{f(x_i+\Delta x_i,\overline{x_{-i}})-f(x_i,\overline{x_{-i}})}{\Delta x_i}\]
If the production function is differentiable, this becomes:
\[MP_i=\frac{\partial f(x)}{\partial x_i}=f_i\]
The marginal rate of technical substitution (MRTS) is the amount of input j that can be decreased while increasing input i to keep output at the same level.
For a differentiable function this is the negative of the slope of the isoquant.
\[MRTS=-\frac{dx_2}{dx_1}=\frac{f_1}{f_2}=\frac{MP_1}{MP_2}\]

In general, marginal product decreases as the input increases.
Additionally we can have homogeneous and homothetic production functions.
===== Cost curves =====

==== Short and long runs ====

The long run is the period where all inputs can vary.
\[y=f(x_1,x_2)\]
The short run is where at least one input is fixed.
\[y=f(x_1,\overline{x_2})\]
==== Long run cost minimisation ====

We can solve the consumer's problem as maximising utility subject to a budget constraint or minimising expenditure subject to a given level of utility.
Likewise we can solve the producer's problem as maximising profits or minimising costs subject to a given level of output.
The profit maximisation method depends on market structure (monopoly, perfect competition, oligopoly, etc.).
The cost minimisation method works regardless of market structure as every profit maximising firm will choose the minimum cost combination of inputs for a desired output.

Given m inputs \(x=(x_1,...,x_m)\geq 0\) and input prices \(w=(w_1,...,w_m)>>0\) which are assumed taken (firms cannot influence input prices), the total cost is \(C=w\cdot x\).
The firm's cost minimisation problem is to choose \(x\) to product \(y\) units of output at minimum total cost.
\[\min w\cdot x\text{ subject to }f(x)\geq y\]
Since the costs are greater than 0, \(f(x)=y\) at the cost minimisation inputs \(x\).
This can then be solved with a Lagrangian.
\[\mathcal{L}(x,\lambda)=w\cdot x+\lambda(y-f(x))\]
This gives:
\[w_i=\lambda\frac{\partial f(x^*)}{\partial x_i}=\lambda MP_i(x^*)\]
We can also get that \(\frac{w_i}{w_j}=MRTS_{ij}(x^*)\).
Hence we can get that the economic rate of substitution is equal to the technical rate of substitution.

We can define an isocost line as all combinations of inputs that have the same total cost.
For two inputs, labour and capital, this is;
\[C=wL+rK\]
Isocost lines further from the origin correspond to a higher cost.
The minimum cost production is where the slope of the isocost is equal to the slope of the isoquant.
For the example, it is where \(\frac{w}{r}=MRTS(L^*,K^*)\).

Conditional factor demands are those which depend on input prices \(w\) and output \(y\).
\[x^*(y,w)=(x_1^*(y,w),...,x_m^*(y,w))\]
This gives the long run total cost function as:
\[C(y,w)=w\cdot x^*(y,w)\]
Since the level of output is a choice variable by the firm, but the input prices are parameters, we often write \(C(y,w)\) as \(C(y)\).
The long run cost function is homogeneous of degree 1 with respect to \(x\), so functions for returns to scale.
=== Long run average and marginal cost ===

When determining the output decision \(y\), we often work with average and marginal cost curves, rather than total cost curves.
The long run average cost function is:
\[LRAC(y)=\frac{C(y,w)}{y}\]
The long run marginal cost is:
\[LRMC(y)=\frac{\partial C(y,w)}{\partial y}\]
The average cost follows the marginal cost.

With increasing returns to scale, the LRAC will decrease, being negative sloping.
With decreasing returns to scale, the LRAC will increase, being positive sloping.
With constant returns to scale, the LRAC will be constant, being a horizontal line.
The returns to scale may change over different amounts of output.
==== Short run cost curves ====

The short run is when there is at least one input that is fixed.
The short run total cost is:
\[C=C(y,\overline{x_i})=w_i\overline{x_i}+w_ix_i^*(y,w)\]
The total fixed costs are: \(TFC=w_i\overline{x_i}=F\).
The total variable costs are: \(TVC(y)=w_ix_i^*(y,w)\).
The total cost is comprised of variable and fixed costs.
\[C=TFC+TVC\]
In the long and short runs, the optimal production amounts may be different.
=== Short run average and marginal cost ===

The average total cost is:
\[AC=\frac{C}{y}\]
The average fixed cost is:
\[AFC=\frac{F}{y}\]
The average variable cost is:
\[AVC=\frac{TVC(y)}{y}\]
The average fixed cost is a hyperbola, approaching zero as output goes to infinity.
The average variable cost, because of diminishing marginal product, increases as output increases.
Average fixed cost is the sum of average fixed and variable costs, so we would expect a U shaped curve.
We define the short run marginal cost as \(MC=\frac{\partial C}{\partial y}\).
=== Link short run to long run cost curves ===

For every level of the fixed factor, there is a different short run average cost curve.
In the long run every factor is variable, so the average cost curve will be the one that minimises at a given level of output.
This results in the LRATC being equal to the minimum SRATC at every level of output.

Decreasing LRATC is also known as economies of scale, and increasing LRATC is diseconomies of scale.
Returns to scale is the relationship between output and all inputs.
Economies of scale is the relationship between output and costs.
When input prices are taken as given, returns to scale and economies of scale are equivalent.
If the firm can affect the price of goods, this equivalence need not hold.
===== Opportunity cost =====

Economic value of all inputs is the current market value, regardless of if actual money is paid or not.
The market value is the opportunity cost.

For a given output level \(q\), there is a downward sloping price (due to demand) for that output.
The revenue is \(R=p(q)q\) and the costs are \(C=w\cdot x\).
The profit is \(\pi=R-C=p(q)q-w\cdot x\), where costs are opportunity costs.
An entrepreneur receives accounting profits as return on their entrepreneurial skill.
Hence a firm can be making zero economic profits while still earning accounting profits.
Likewise a firm can make negative economic profits while making positive accounting profits.
Zero economic profits are normal.
The firm chooses inputs to maximise economic profits.
\[\max\pi=p(q)q-w\cdot x|f(x)\geq y\]
As prices are greater than 0, it is never profit maximising to produce more than would be sold, so \(y=f(x)\).
This lets us substitute the constraint into the optimisation problem, giving:
\[\max\pi=p(f(x))f(x)-w\cdot x\]
We can also write the problem in terms of the output and the cost function:
\[\max\pi=p(q)q-C(q,w)\]
Trying to solve this problem gives:
\[\frac{d\pi}{dq}=0\implies\frac{dR}{dq}=\frac{dC}{dq}\]
We can recall that \(\frac{dC}{dq}\) is the marginal cost of production and define \(\frac{dR}{dq}\) as marginal revenue.
This makes the solution to the problem one where marginal revenue is equal to marginal cost.

The output depends on the firm's demand, which depends on the market structure.
In perfect competition, the price does not depend on the firm's level of output, so the price is a parameter.
In a monopoly, there is only one firm, so its output is the market output and price depends on the amount of output.
==== Perfect competition ====

In a perfectly competitive market, there is a given trading price.
If an individual firm wanted to sell above that price, they wouldn't sell any goods as the other firms would be selling at a cheaper price.
If an individual firm were to sell below that price, they would capture the whole market demand, out-competing every other firm.
If the firm were to sell at that price, they would sell between 0 and the market demand of the good, creating a flat demand curve for the firm.

Here the marginal revenue is the price, so a profit maximising firm will have its marginal cost equal to the price.
\[p=MC(q)\]
As the output is dependent on the input prices, we can write the firm's supply function as:
\[q=q(p,w)\]
The long and short run supply functions are often different.

We can define the profit function \(\pi(p,w)=\max\pi\), and note that \(\pi_{SR}\leq\pi_{LR}\).
We can express the profit in terms of output:
\[\pi(p,w)=pq(p,w)-C(q(p,w),w)\]
By the envelope theorem, gives \(\frac{d\pi(p,w)}{dp}=q\).
The profit function is non-decreasing in output price \((p)\), non-increasing in input prices \((w)\), homogeneous of decree 1 in \((p,w)\), convex in \((p,w)\).
The first derivative and convexity together mean that the supply curve slopes up.
\[\frac{\partial q}{\partial p}=\frac{\partial^2\pi(p,w)}{\partial p^2}\geq 0\]

To characterise profit in terms of input, we get:
\[\pi=pf(x)-w\cdot x\]
This is because output has no effect on price.
This gives the first order necessary conditions of \(p\frac{\partial f(x^*)}{\partial x_i}=w_i\), which gives \(MP_i(x^*)=\frac{w_i}{p}\).
This produces the firm's factor demand functions \(x=x(p,w)\), giving:
\[\pi=pf(x(p,w))-w\cdot x(p,w)\]
By the envelope theorem, we get \(\frac{\partial\pi(p,w)}{\partial w_i}=-x_i\).
The second order necessary condition gives \(pD^2f(x^*)\) is negative semi-definite, while concavity ensures at all points and ensures the solution to the FONC is a maximum.
We can also find that the hessian of the production function is the inverse of the derivatives of the inputs with respect to price.
\[H_f^{-1}=Dx(w)=\begin{bmatrix}\frac{\partial x_1}{\partial w_1}&\frac{\partial x_1}{\partial w_2}\\\frac{\partial x_2}{\partial w_1}&\frac{\partial x_2}{\partial w_2}\end{bmatrix}\]
Has the hessian is negative semi definite, so is its inverse, which means demand slopes down.
=== Long run supply ===

The long run profit maximising output is \(q=q(p,w)\), where \(p=LMC(q)\).
In the long run the firm can exit the industry, so \(\pi=0\).
The firm will exit if it is earning negative economic profits, so;
\[\pi<0\implies pq<C(q,w)\implies p<\frac{C(q,w)}{q}=LRAC(q)\]
So the firm will exit if the price is less than the minimum LRAC, an produce if the price is equal to or greater than the minimum LRAC.

Hence for a U shaped LRAC, the supply curve is equal to the LMC for prices greater than the minimum LRAC, and 0 for prices less.
For a horizontal LRAC, the supply curve is 0 for prices less than the production price, and never goes above.
IRS is incompatible with perfect competition, so we get a horizontal line for the supply curve.
=== Short run supply ===

In the short run the profit maximising output is \(q=q(p,w)\), where \(p=MC(q)\).
Exiting is a purely long run decision, but in the short run, a firm can shut down, setting \(q=0\).
Fixed costs need to be paid, even when there is no production so \(\pi(q=0)=-F\).
So a firm will choose to shut down when \(\pi(q)<\pi(0)\), which is when \(pq<TVC(q)\), being \(p<AVC\).

So when the price is less than the minimum AVC, the firm will shutdown, and when it is greater it will produce equal to its marginal cost.
===== Short run profits and producer surplus =====

We can characterise the total cost as the product of the average cost and output.
We can then express profit as: \(\pi=(p-AC)q\).
The producer surplus is the difference between the price a producer receives for a unit of output and the price they were willing to produce it for (given by the supply curve).
The producer surplus is \(PS=p^*q^*-TVC(q^*)\), being amount of revenue after variable costs.
So profit is producer surplus less the total fixed costs:
\[\pi=PS-TFC\]
As a result, in the short run a change in profit corresponds to a change in producer surplus.
===== Market supply =====

Market supply is the aggregate of individual firm's supply.
The market supply is the output at every price.
The quantity supplied (\(Q^S\)) refers to the quantity at a particular price whereas supply refers to the curve.
Changes in price represent a movement along the supply curve, so the quantity supplied changes but the curve stays the same.
All factors other than price cause a shift in the supply curve.
A shift to the right is an increase in supply, and to the left is a decrease.
The factors affecting supply include:
  * **Number of firms:** A new firm increases the number of firms, causing the output at each price to increase, increasing supply.
  * **Change in factor prices:** Increased factor costs, increasing marginal costs, decreasing supply.
  * **Technological improvement:** Increased productivity corresponds to more being produced for the same input, decreasing MC, increasing supply.
==== Elasticity ====

The elasticity of supply is the sensitivity of the quantity supplied to changes in price.
\[\epsilon_s=\frac{\%\Delta Q^S}{\%\Delta p}=\frac{\partial Q^S}{\partial p}\frac{p}{Q^S}\]
Perfectly inelastic supply is a vertical line and perfectly elastic supply is a horizontal line.
====== Market equilibrium ======

===== Competitive markets =====

We get an individual demand from solving the producer choice problem and get market demand by summing individual demands.
We get a firm's supply by solving for max profit and get market supply by summing individual supplies.
The equilibrium price is the price such that the quantity supplied by the market is equal to the quantity consumed.
This is where the curves so the equilibrium price is where demand equals supply.
Demand increases cause an increase in price and quantity.
Supply increases cause a decrease in price and an increase in quantity.
If supply and demand both increase, quantity increases but price is uncertain.
If supply increases and demand decreases, price decreases but quantities is uncertain.

In the long run of a competitive, we assume there are no barriers to entry or exit.
Positive economic profits signal to entrepreneurs to enter the industry, increasing supply, decreasing prices and decreasing economic profits.
Firms will continue to enter the market until there are no economic profits and the price is equal to the average cost.
Likewise if there are negative economic profits, firms will exit until the price is equal to the average cost.
In the long run a firm will produce at the minimum LRAC.
===== Taxes and subsidies =====

A per unit tax (excise tax) increases the cost of a unit by the tax amount.
This causes a difference in the price paid by the buyer and the price received by the seller.
The tax being applied to the buyer or seller doesn't matter.
The tax reduces the consumer and producer surplus, and generates revenue equal to the tax amount multiplied by the quantity being traded.
The tax burden is determined by the elasticity of supply and demand.
With perfectly elastic supply, the seller holds on burden and with perfectly inelastic supply the seller holds all the burden.

A subsidy functions as the counterpart to a tax, where the government pays for the good.
Here the seller receives a greater price than the buyer pays.
===== Monopoly =====

The counterpart to perfect competition, where only a single firm exists, with high barriers to entry and no substitutes.
Sometimes the government can mandate the monopoly, serving as barriers to entry.
A monopoly seeks to maximise profits and has the usual cost curves, however its demand is market demand.
As a result it chooses its price rather than taking it, making a supply curve make less sense for a monopolistic market.
The monopolistic marginal revenue is \(MR=\frac{dR}{dQ}=p+\frac{dp}{dQ}Q<p\), as demand is downward sloping.
The monopolist seeks to maximise profits \(\max\pi=p(Q)Q-C(Q)\), which gives:
\[p+\frac{dp}{dQ}Q-C'(Q)=0\]
This is where marginal revenue is equal to marginal costs, like for competitive markets.
The monopoly, however, will set the price at what consumers will be willing to pay for the amount it produces rather than take the market price.
In the long run, a monopoly may make positive economic profits as no new firms can enter the market.
===== Efficiency =====

The net social benefit is the benefit to society as a whole.
We say allocation is allocatively efficient if it maximises social benefit.
This is where the social marginal benefit is equal to the social marginal cost.
The area between the two curves is the social benefit.
If the buyers and sellers are the only ones affected by a transaction (non externalities), then the social marginal benefit is the private marginal benefit (demand) and the social marginal cost is the private marginal cost (supply).
A competitive market with on externalities is allocatively efficient, with the social benefit being the sum of consumer and producer surpluses.

If the output is less than the efficient output, the decrease in social benefit is known as Dead Weight Loss.
A tax on an efficient market reduces consumption, reducing consumer and producer surplus but introducing government revenue and a dead weight loss (less overall consumption).
If the output is greater than the efficient output, there is also a reduced social benefit resulting in a dead weight loss.
A subsidy increases consumption, resulting in a dead weight loss due to increased government costs outweighing increased consumer and producer surplus.

A monopoly is allocatively inefficient as the monopolist seeks to maximise profits and does not produce enough.
This is done by reducing output and charging a higher price.
As a result there is a dead weight loss.

Competitive markets in the absence of externalities are allocatively efficient, maximising total surplus.
Competitive markets are technically efficient, producing at efficient scale and minimising costs.
===== Externalities =====

An externality is a benefit or cost to a party not directly involved in the transaction.
This can result in the SMB or SMC not being the same as the PMB or PMC respectively.

There can be production externalities, which can increase or decrease SMC above or below PMC.
There can be consumption externalities, which can increase or decrease SMB above or below PMB.
An example negative production externality is pollution, which can affect parties not directly involved in the transaction.
An example positive consumption externality is vaccination, which can help parties not directly involved in the vaccination.
====== Asset valuation ======

An asset is a good that provides a stream of services over time.
Firm's profits are paid out to shareholders, or retained and converted to capital gains.
As such, we can value a firm by its stream of profits.
\[V(\pi_0,\pi_1,...,\pi_T)=V(R_0,R_1,...,R_T,C_0,C_1,...,C_T)\]
Likewise we can value any asset from its future earnings less future costs.
We can't simply add the future revenues and deduct future costs as there is a time value of money, being that a dollar today is worth more than one tomorrow.
This requires us to discount future revenues and costs, with either the present value or future value methods.
===== Return =====

A period is a unit of time and a holding period is the period from the start of an investment to the end.
The rate of return is the difference in value over the holding period:
\[\text{Return}=\frac{\text{Ending value}-\text{Starting value}}{\text{Starting value}}\]
The Present Value (\(PV\)) is the value of an asset today.
The Future Value at time t (\(FV_t\)) is the value of an asset at time in the future t.
As different assets can have different holding periods, so it can be more useful to compare the return over an annual basis.

The simple annual return is the return multiplied by the number of periods in a year.
This assumes the returns are not reinvested.
Reinvesting gives compounding or effective return.
\[FV_n=PV*(1+r)^n\]
Compounding gives a greater effective annual return than simple annual return.
If we want to discuss the return in a number of years, compounding n times a year:
\[FV_t=PV*(1+r)^{nt}\]
Likewise, we can find the present value:
\[PV=\frac{FV_t}{(1+r)^{nt}}=FV(1+r)^{-nt}\]
The greater the interest rate, the lesser the present value of a future amount.

The effective annual return is a way of finding the compounding rate when an investment may compound many times in a year.
\[\text{Effective Annual Return}=(1+r)^n-1\]
===== Present value of future cash flows =====

An investment involves a future receipts (revenues) and payments (costs) called cash flows.
There are four parts of cash flows:
  - Is there a net receipt or payment
  - Amount of cash flow
  - Timing
  - Interest rate

A single cash flow can be trivially calculated:
\[PV=\frac{FV}{(1+r)^t}\]
A more interesting cash flow is a uniform cash flow.
That is where there is a flow that is the same every period.
\[PV=\sum_{t=1}^T\frac{c}{(1+r)^t}\]
If the payments were made at the beginning of the period, instead of the end, the payments would start at \(t=0\) instead of \(t=1\).
Assuming the cash flow went on indefinitely, it forms an infinite geometric series.
\[PV=\frac{\frac{c}{1+r}}{1-\frac{1}{1+r}}=\frac{c}{r}\]
Additionally there is a geometric cash flow, where the payment grows at a constant geometric rate.
\[PV=\frac{c}{1+g}\sum_{t=1}^T\left(\frac{1+g}{1+r}\right)^t\]
There is also a linear gradient cash flow, where the cash flow grows by a constant amount every period.
\[PV=\sum_{t=1}^T\frac{c}{(1+r)^t}+\sum_{t=1}^T\frac{(t-1)G}{(1+r)^t}\]
An irregular cash flow has cash flows that have no trend, and must be discounted individually.
==== Bonds ====

A bond makes coupon payments every period for \(T\) periods, then returns its face value \(F\).
The period \(T\) is the maturity date and typically payments are made semi-annually.
It is convention for the face value to be $100.
Pricing a bond is done by finding the present value of the coupons (uniform cash flow) and the present value of the face value.
The rate of return is needed to be found for the calculation.
==== Effective annual return ====

If a given simple annual interest rate is \(R\), and there are \(n\) periods in the year, the interest rate per period is \(r=\frac{R}{n}\).
We can then calculate the effective annual return as \(1-(1+r)^n\).
We can notice that the effective rate increases as the number of periods increases.
When taking the limit, we get:
\[\lim_{n\to\infty}(1+r)^n=\lim_{n\to\infty}\left(1+\frac{R}{n}\right)^n=e^R\]
So the effective annual return for continuous compounding is \(1-e^R\).
It is often useful to assume continuous compounding as growth tends to happen incrementally in quick succession.
We can empirically find R for a continuous compounding situation from:
\[R=\ln\left(\frac{FV_1}{PV}\right)\]
===== Project valuation =====

Project cash flows are made up of payments (cash outflows) and receipts (cash inflows).
We can incrementally add costs from each future year, discounting them accordingly to find the present value of the project.
==== Traditional payback method ====

The payback period is the time required to pay back the initial costs.
\[\text{Payback period}=\frac{\text{Initial Cost}}{\text{Uniform annual net inflow}}\]
The payback period needs to be less than some company required period.

This method ignores the time value of money (solved with the discounted payback method).
It also says nothing about profitability.
==== Minimum Acceptable Rate of Return (MARR) ====

There are three ways a firm can get funds for a project:
  * **Debt:** Borrowing from financial institutions or issuing bonds
  * **Equity:** Issuing shares
  * **Retained earnings:** Keeping earnings for future projects instead of paying out to shareholders
We assume these three costs are the same, and are the costs of capital.
The MARR on assets assumed to be the cost of capital \(r\).
Yield on government bonds is assumed risk free \(r_f\), as governments are assumed not to default on their debts.
Firms need to offer a greater rate, as firms are riskier than governments.
The difference in rates is the risk premium: \(r_p=r-r_f\).
Thus the cost of capital includes a premium for a normal amount of risk \(r=r_f+r_p\).
==== Net Present Value (NPV) ====

We can value assets in terms of the Net Present Value at time 0.
We can also value in terms of the Net Future Value at time t.
We can also use the Annual Equivalent, in terms of equivalent annual payments.
These methods also evaluate the profitability of projects.

For the NPV, a firm sets the MARR, which is used as the interest rate for the projects.
The project lifetime and net cash inflows need to be estimated.
The sum of the discounted net cash inflows over the project lifetime is the NPV of the project.
When the project lifetime is large, we can use the capitalised equivalence, which assumes the project lifetime is infinite, which gives an easier calculation for uniform payments.
The error from this method is \(CE-NPV_N=(1+r)^N*\frac{C}{r}\).

For evaluating single projects, we accept is \(NPV>0\), reject if \(NPV<0\) and are indifferent (but accept) if \(NPV=0\).
The higher the MARR, the lower the PV, so the fewer projects will be accepted.
For multiple projects, the worst we can do is nothing with \(NPV=0\).
If projects are mutually exclusive, we simply pick the one with the greatest NPV, however the size of the project doesn't matter as uninvested funds can be otherwise used.
Where we can pick multiple projects, we pick from the greatest \(NPV>0\) until we run out of funds.

The total approach (NPV) evaluates mutually exclusive projects separately.
The incremental approach is based on differences between projects.
==== Net Future Value (NFV) ====

Useful for investments where a long time is taken until revenues begin.
Does not affect analysis, the decision rule in accepting is the same as NPV.
==== Revenue vs service projects ====

If projects differ by their revenues, they are revenue projects.
We analyse the NPV of revenues less costs and want to maximise NPV, i.e. maximise profits.
If project revenues are the same, they are service projects.
We need only to look at the NPV of costs, and want to minimise NPV, i.e. minimise costs.
==== Break even interest rate ====

The break even rate is the rate needed for \(NPV=0\).
This is also known as the Internal Rate of Return (IRR).
So we accept projects when the MARR is less than the break even rate, as then the \(NPV>0\).
==== Asset life and project life ====

The life of an asset can differ to that of the project.
When an asset's life is greater than the life of the project, we need to estimate the salvage value of the asset at the end of the project.
We can deduct the salvage value at the end of the project in the calculation of the net present value.
When the asset's life is less than the project's, we need to find a replacement to complete the project.

In comparing mutually exclusive assets with different lives, we need to account for the lifetime cost of the asset, including salvage and replacement values in the cost calculation.
==== Sinking fund ====

A sinking fund is an interest bearing account where equal amounts are deposited each period.
It is commonly used to set up a fund to replace fixed assets.
We know that the future value is:
\[F=A(1+i)^{N-1}+...+A(1+i)+A=A\left[\frac{(1+i)^N-1}{i}\right]\]
This gives us equal payments of:
\[A=F\left[\frac{i}{(1+i)^N-1}\right]\]
The term in square brackets is the sinking fund factor.
==== Capital Recovery Factor ====

A sinking fund is the uniform cash flow equivalent to future value F.
A capital fund is the uniform cash flow equivalent to present value P.
Can be thought of as borrowing today and paying off the loan.
\[A=P\left[\frac{i(1+i)^N}{(1+i)^N-1}\right]\]
The term in the square brackets is the capital recovery factor (Crf).
==== Annual Equivalent Value ====

The annual equivalent value of a project is the uniform annual net cash flow that is equivalent to the NPV of the project.
\[\text{AE}=\text{AE Revenues}-\text{AE Costs}\]
If some revenues or costs are not measured annually, or are not uniform, we need to convert their NPV to an AE, \(\text{AE}=\text{NPV}*Crf\).
Many assets has a lifespan greater than a year, so we need to calculate the annual equivalent of the net capital cost.
This is usually called the capital recovery cost:
\[\text{CR}(i)=\text{NPV of net capital recovery cost}*Crf\]

The decision rule for AE is the same as for NPV, accept for \(AE>0\), reject for \(AE<0\), indifferent for \(AE=0\).
Likewise, for mutually exclusive projects we pick the revenue project with the highest AE and service projects with the lowest AEC.
The AE is used as companies are used to working with annual data.
It is also useful for unit profit and cost which are usually annual.

In cost analysis, frequently the AE of costs (AEC) will vary with a design variable (\(x\)).
To allow for the possibility that AEC can increase or decrease with \(x\), we set:
\[AEC(i)=a+bc+\frac{c}{x}\]
With \(a,b,c\) being parameters, and we choose \(x\) to minimise AEC, assuming a service project.
This gives \(x^*=\sqrt{\frac{c}{b}}\), and a positive second derivative if \(c>0\).
==== Internal Rate of Return ====

The IRR is a relative measure of value, compared to the absolute methods of NPV, NFV and AE.
It is easy to compare to MARR, which is a relative measure.
IRR calculation for mixed investments can be complicated.
Additionally an incremental approach is needed when evaluating mutually exclusive projects.

If the Unrecovered Project Balance in year n is \(PB_n\) is the amount a project owes a firm for a given rate of return, the IRR is the rate such that at the end of the project, \(PB_N=0\).
If the IRR is greater than MARR, we accept the project, if less we reject and if equal we are indifferent.
The calculation of IRR depends on the type of investment, being a pure or mixed investment.

A simple investment is one where all the early investments are outflows and all later investments are inflows.
For a simple investment at its IRR, the present value of net cash inflows is equal to the present value of net cash outflows.
As such, for a simple investment, the IRR is equal to the break even rate.
A simple investment has a unique break even interest rate, and thus a unique IRR.

A non-simple investment is one where cash flows can change between inflows and outflows.
There can be multiple break even interest rates, making it unclear what to use as the IRR.
The number of positive break even rates is less than or equal to the number of sign changes of cash flows.
The accumulated net cash flows is the running sum of cash flows.
If the initial investment is negative and the accumulated net cash flows has no sign change, there is no positive break even rate.
If the sign changes once, there is a unique break interest rate and for multiple changes there are multiple break even rates.
It can be useful to graph the NPV to find the break even rates.

A pure investment is one where the unrecovered project balance at the break even rate at every period is less than or equal to 0.
\[PB_n(i^*)\leq 0\]
An investment with at least one \(PB_n(i^*)>0\) is a mixed investment.
All simple investments are pure investments and all pure investments have a unique positive break even interest rate.
Not all investments with a unique positive break even interest rate are pure.
For pure investments, \(IRR=i^*\).

If \(PB_n(i^*)\), the investment is producing cash inflows that are being externally spent.
Typically the rate of return outside the project is less than the rate inside the project.
We normally let the external rate of return \(k\) be the cost of capital, which we also assume is the MARR.
So when the project balance at the start of the year is positive, we need to use \(k\) instead of \(i\), meaning that the IRR is not the break even interest rate.
Using this mixed interest rate provides the true IRR, also known as the Return on Invested Capital (RIC).

We can also use the Modified Internal Rate of Return (MIRR) instead of the true IRR.
\[MIRR=\left(\frac{FV_{Inflows}}{PV_{Outflows}}\right)^{\frac{1}{N}}-1\]
We accept if \(MIRR>MARR\).
===== Financial Statements =====

There are three main financial statements:
  * **Income statement:** Records revenues and costs, and thus profits
  * **Balance sheet:** Records assets and liabilities, and thus equity
  * **Statement of cash flows:** Records cash flows, including operating, financing and investing activities and excludes non-cash flows
==== Balance sheet ====

Current assets are assets which can be converted to cash within a year.
Examples are cash, accounts receivables and inventories.
Long term (fixed) assets are assets that take over a year to convert to cash.
Examples are property, plant and equipment.
Other assets encompasses anything not already covered.
Examples include intangible assets (IP, good will) and investments in other companies.

Property, plant and equipment are prone to depreciation, being a decrease in value over time.
We calculate the "book value" of an asset in year \(n\) as the gross value of the asset \(I\) less the cumulative depreciation.
\[B_n=I-\sum_{t=1}^nD_t\]

Current liabilities are debts that must be paid within a year.
Examples are accounts payable, notes payable, accrued expenses and advance payments.
Long term liabilities are debts that must be paid in over a year.
Examples include bonds, mortgages and long-term loans.

A firm can raise funds with debt financing, issuing long term debt.
Alternately equity financing is recorded as equity common or preferred stock.
Retained earnings are recorded in equity.
==== Income statement ====

Cost of revenue (cost of goods sold) is direct costs incurred in the production of goods and services.
Operating expenses are expenses indirectly required for production, e.g. selling general and administrative, R&D, depreciation (tangible) and amortisation (intangible), etc.
Interest payments are payments made on debt.
Taxes are payments made to the government.

Gross profit is revenue less cost of revenue.
Net income is gross profit less operating expenses, interest payments and taxes.
EBITDA is the Earnings Before Interest Taxes Depreciation and Amortisation.
The earnings per share is net income divided by the number of common stock shares.
==== Cash flows ====

Cash inflows come from reducing an asset or increasing a liability.
Cash outflows come from increasing an asset or decreasing a liability.
Net cash flows is the difference between cash inflows and outflows.

Cash flows can be separated into three types:
  * **Flows from operating activities:** Related to production and sales of goods and services, equal to net income with non-cash items less changes in working capital.
  * **Flows from investing activities:** Purchases and sales of property, plant and equipment, trading of short term financial assets and mergers and acquisitions.
  * **Flows from financing activities:** Financing with debt or equity
==== Depreciation ====

Straight line depreciation \(D_t\) is a linear depreciation between the investment cost \(I\) and salvage value \(S_T\) over the life of the asset \(T\).
\[D_t=\frac{I-S_T}{T}\]
Alternately depreciation can be done with a declining balance (diminishing value method).
We find \(\alpha=\frac{k}{T}\), where \(T\) is the life of the asset and \(k=1\) is the declining balance and \(k=2\) is the doubly declining balance.
The depreciation in a given year is \(D_t=\alpha B_{n-1}\).
This has depreciation decreasing over time, with the maximum loss in value occurring early.
When the book value is less than the salvage value, the company needs to stop depreciating the asset so that they equal.
When the book value is greater than the salvage value, there is unused depreciation.
Sometimes we can switch to a straight line depreciation when it has a higher depreciation value than the declining balance method.
Borrowing funds for a project can increase the NPV of the project compared to fronting the whole initial investment.
===== Risk =====

An outcome is uncertain if there is more than one possible outcome.
Risk is a measure of uncertainty based on the known probabilities of possible outcomes.
Project risk is the variability of the project NPV.
The NPV depends on future cash flows, which aren't known for certain.
There are 5 methods to determine the variability in the NPV for a project:
  - Statistical Measures
  - Risk-adjusted discount rate
  - Sensitivity analysis
  - Best and worst case scenario analysis
  - Break even analysis

==== Statistical methods ====

The two primary considerations for a project are the return and risk of the project.
If we can assign probabilities, we can measure average return as expected return and risk as variance.
\[E[NPV]=\sum_{i=1}^{N}p_iNPV_i\]
\[\sigma^2=\sum_{i=1}^Np_i(NPV_i-E(NPV))^2\]
The greater the risk, the greater the variance.
The coefficient of variation is a measure of the return compared to the risk.
\[CV=\frac{\sigma}{E(R)}\]
The lower the CV, the less relative risk there is.
==== Risk adjusted discount rate ====

A simpler and more popular method to account for the riskiness of a project is to add a risk premium to the normal project interest rate.
\[r=MARR+r_{premium}\]
The risk premium is the additional return required for the above normal risk.
==== Sensitivity analysis ====

Sensitivity analysis determines how much a result will change in response to a change in input affecting the result.
This requires the inputs, a base estimate on the output, and changing the inputs to see their effect on the result.
We can then determine which inputs are the most sensitive.