notes-on-microeconomic-theory

note: this page is still in development.

these definitions and theorems apply to intermediate and advanced microeconomics courses.

kuhn-tucker method

remark

consider a maximization or minimization problem for a function $f : \mathbb{R}^n \times \mathbb{R}^q \to \mathbb{R}$ that depends on $n$ decision variables and $q$ parameters, subject to:

$k$ equality constraints $h_i : \mathbb{R}^n \times \mathbb{R}^q \to \mathbb{R}$ , and
$m$ inequality constraints $g_j : \mathbb{R}^n \times \mathbb{R}^q \to \mathbb{R}$ .

maximization

example

\begin{align*} \max_{\{x_1, \dots, x_n\}} \quad & f(x_1, \dots, x_n; p_1, \dots, p_q) \\ \text{subject to} \quad & h_i(x_1, \dots, x_n; p_1, \dots, p_q) = 0, \quad \forall i = 1, \dots, k; \\ & g_j(x_1, \dots, x_n; p_1, \dots, p_q) \geq 0, \quad \forall j = 1, \dots, m. \end{align*}

lagrangian of the problem:

\mathcal{L}(x_1, \dots, x_n; p_1, \dots, p_q; \lambda_1, \dots, \lambda_k; \mu_1, \dots, \mu_m) = f(x_1, \dots, p_q) + \sum_{i=1}^k \lambda_i h_i(x_1, \dots, p_q) + \sum_{j=1}^m \mu_j g_j(x_1, \dots, p_q)

minimization

example

\begin{align*} \min_{\{x_1, \dots, x_n\}} \quad & f(x_1, \dots, x_n; p_1, \dots, p_q) \\ \text{subject to} \quad & h_i(x_1, \dots, x_n; p_1, \dots, p_q) = 0, \quad \forall i = 1, \dots, k; \\ & g_j(x_1, \dots, x_n; p_1, \dots, p_q) \geq 0, \quad \forall j = 1, \dots, m. \end{align*}

lagrangian of the problem:

\mathcal{L}(x_1, \dots, x_n; p_1, \dots, p_q; \lambda_1, \dots, \lambda_k; \mu_1, \dots, \mu_m) = f(x_1, \dots, p_q) - \sum_{i=1}^k \lambda_i h_i(x_1, \dots, p_q) - \sum_{j=1}^m \mu_j g_j(x_1, \dots, p_q)

stationarity conditions

proposition

\frac{\partial \mathcal{L}}{\partial x_i}(x_1^*, \dots, x_n^*; p_1, \dots, p_q; \lambda_1^*, \dots, \mu_m^*) = 0.

\frac{\partial \mathcal{L}}{\partial \lambda_i}(x_1^*, \dots, x_n^*; p_1, \dots, p_q; \lambda_1^*, \dots, \mu_m^*) = 0.

complementary slackness conditions

proposition

maximization:

\frac{\partial \mathcal{L}}{\partial \mu_j}(x_1^*, \dots, x_n^*; p_1, \dots, p_q; \lambda_1^*, \dots, \mu_m^*) \geq 0;

\mu_j^* \geq 0;

\mu_j^* h_j(x_1^*, \dots, x_n^*; p_1, \dots, p_q; \lambda_1^*, \dots, \mu_m^*) = 0.

minimization:

\frac{\partial \mathcal{L}}{\partial \mu_j}(x_1^*, \dots, x_n^*; p_1, \dots, p_q; \lambda_1^*, \dots, \mu_m^*) \leq 0;

\mu_j^* \geq 0;

\mu_j^* h_j(x_1^*, \dots, x_n^*; p_1, \dots, p_q; \lambda_1^*, \dots, \mu_m^*) = 0.

theoremenvelope theorem

let $V$ and $\mathcal{L}$ be continuously differentiable and $(x^*, \lambda^*, \mu^*)$ the solution of some optimization problem with parameters $(p_1, \dots, p_q)$ , then:

\frac{\partial V}{\partial p_h}(p_1, \dots, p_q) = \frac{\partial \mathcal{L}^*}{\partial p_h}.

utility function

definitioncompleteness

let $a, b \in A$ be any two alternatives, we say that a consumer has complete preferences if they are capable of saying whether they prefer $a$ over $b$ , $b$ over $a$ , or are indifferent.

definitiontransitivity

let $\mathcal{R}$ be a homogeneous relation on the set $X$ , we say that the consumer has transitive preferences if for any three alternatives $a\mathcal{R}b$ and $b\mathcal{R}c$ , then $a\mathcal{R}c$ .

definitionrationality

we say that consumer preferences are rational if they are complete and transitive.

definitionbasket

a basket is a vector $(a_1, \dots, a_n) \in \mathbb{R}^n_+$ that represents the consumption of goods $x_1, \dots, x_n$ .

definitionutility function

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ , this represents the utility of a consumer; for any two alternatives $(x_1, \dots, x_n')$ , $(x_1', \dots, x_n') \in \mathbb{R}^n_+$ , we have $u(x_1, \dots, x_n) < u(x_1', \dots, x_n')$ if and only if the consumer prefers basket $(x_1', \dots, x_n')$ over basket $(x_1, \dots, x_n)$ .

definitionindifference curves

given a utility function $u : H \to \mathbb{R}$ and $k \in \mathbb{R}$ , the set of level $k$ is defined as:

\mathcal{C}_k = \{ x \in H \mid u(x) = k \}.

if $H = \mathbb{R}^2$ , said set of level $k$ is known, in economics, as indifference curves.

definitionmarginal utility

marginal utility measures the change in the utility of a consumer given a marginal change in said good.

theoremtotal differential

given a utility function $u : \mathbb{R}^2 \to \mathbb{R}$ , the differential of $u$ evaluated at point $(x_0, y_0)$ is:

\Delta u(x_0, y_0) = \Delta x \frac{\partial u}{\partial x}(x_0, y_0) + \Delta y \frac{\partial u}{\partial y}(x_0, y_0),

where $\Delta x = x_1 - x_0$ and $\Delta y = y_1 - y_0$ .

definitionmarginal rate of substitution

the marginal rate of substitution $\text{MRS}(x, y)$ measures the number of units of good $Y$ that the consumer is willing to sacrifice to obtain one additional unit of good $X$ and maintain their utility constant, and is defined as:

\text{MRS}(x, y) = \frac{\frac{\partial u}{\partial x}(x, y)}{\frac{\partial u}{\partial y}(x, y)}.

definitionmonotone transformation

a function $w$ is a monotone transformation of $u$ if only if there exists a function $g(\cdot)$ such that it is strictly increasing in all:

w(X) = g(u(X)).

properties of the utility function

monotonicity

definitionweak monotonicity

a utility function $u : \mathbb{R}^n_+ \to \mathbb{R}$ is weakly monotone if for any $X \neq Y \in \mathbb{R}^n_+$ , such that $x_i \leq y_i$ , $\forall i \in \{1, 2, \dots, n\}$ , we have $u(X) \leq u(Y)$ .

definitionstrict monotonicity

a utility function $u : \mathbb{R}^n_+ \to \mathbb{R}$ is strictly monotone if for any $X \neq Y \in \mathbb{R}^n_+$ , such that $x_i \leq y_i$ , $\forall i \in \{1, 2, \dots, n\}$ , we have $u(X) < u(Y)$ .

quasiconcavity

definitionweak quasiconcavity

a utility function $u : \mathbb{R}^n_+ \to \mathbb{R}$ is weakly quasiconcave if for any $X \neq Y \in \mathbb{R}^n_+$ , such that $u(X) = u(Y)$ , we have:

u(\alpha X + (1 - \alpha)Y) \geq u(X),

for all $\alpha \in (0, 1)$ .

definitionstrict quasiconcavity

a utility function $u : \mathbb{R}^n_+ \to \mathbb{R}$ is strictly quasiconcave if for any $X \neq Y \in \mathbb{R}^n_+$ , such that $u(X) = u(Y)$ , we have:

u(\alpha X + (1 - \alpha)Y) > u(X),

for all $\alpha \in (0, 1)$ .

homotheticity

definitionhomotheticity

a utility function $u : \mathbb{R}^n_+ \to \mathbb{R}$ is homothetic if for any $X, Y \in \mathbb{R}^n_+$ , such that $u(X) < u(Y)$ , we have:

u(\lambda X) < u(\lambda Y),

for all $\lambda > 0$ .

theorem

given a differentiable utility function with monotone and homothetic preferences, it holds that:

\text{MRS}(\lambda x_i, \lambda x_j) = \text{MRS}(x_i, x_j),

for all $x \in \mathbb{R}^n_+$ , $i \neq j$ and $\lambda > 0$ .

marshallian demand

example

\begin{align*} \max_{\{x_1, \dots, x_n\}} \quad & u(x_1, \dots, x_n) \\ \text{subject to} \quad & \sum_{i=1}^n p_i x_i \leq I, \\ & 0 \leq x_1, \\ & \quad \vdots \\ & 0 \leq x_n. \end{align*}

marshallian model solutions

propositionoptimality conditions

given a utility function $u : \mathbb{R}^2 \to \mathbb{R}$ and suppose that, at optimum, $p_x x^* + p_y y^* = I$ , then:

if $x^*, y^* > 0$ , at optimum: $\text{MRS}(x^*, y^*) = \frac{p_x}{p_y}$ .
if $x^* = 0$ , at optimum: $\text{MRS}\left(0, \frac{I}{p_y}\right) \leq \frac{p_x}{p_y}$ .
if $y^* = 0$ , at optimum: $\text{MRS}\left(\frac{I}{p_x}, 0\right) \geq \frac{p_x}{p_y}$ .

definitionmarshallian demands

if the optimal solution of the problem $(X^*)$ is also known as marshallian demands and is usually denoted as:

x_i^* = X_i^M(P, I).

definitionindirect utility function

the indirect utility function $V(P, I)$ is the value function of the marshallian problem:

V(P, I) = u(X_1^M(P, I), \dots, X_n^M(P, I)).

theoremwalras' law

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ be a monotone utility function, then, in the optimal solution $(X^*)$ , it holds that:

\sum_{i=1}^n p_i x_i^* = I.

theoreminada conditions

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ be a monotone and differentiable utility function.

(i) $\lim_{x \to 0} \text{MRS}(x, y) = \infty \implies x^* > 0$ ,

(ii) $\lim_{y \to 0} \text{MRS}(x, y) = 0 \implies y^* > 0$ .

theoremuniqueness theorem

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ be a strictly quasiconcave function, then the solution to the marshallian problem is unique.

lemmaroy's identity

let $V(P, I)$ be differentiable, then:

X_i^M = -\frac{\frac{\partial V}{\partial p_i}}{\frac{\partial V}{\partial I}}, \quad \forall i = 1, \dots, n.

properties of the indirect utility function

proposition

(i) homogeneous of degree zero in prices and income, i.e. $V(\lambda P, \lambda I) = V(P, I)$ .

(ii) non-increasing in prices, i.e. $\frac{\partial V}{\partial p_i}(P, I) \leq 0$ .

(iii) non-decreasing in income, i.e. $\frac{\partial V}{\partial I}(P, I) \geq 0$ .

properties of marshallian demands

proposition

(i) homogeneous of degree zero in prices and income, i.e. $X^M(\lambda P, \lambda I) = X^M(P, I)$ .

(ii) roy's identities hold.

comparative statics

definitionelasticity

given a parameter $P \in \{p_1, \dots, p_n, I\}$ , the elasticity of good $X$ with respect to parameter $P$ is defined as:

\varepsilon_{X, P}(p_1, \dots, p_n, I) = \frac{\partial X^M}{\partial P} \cdot \frac{P}{X^M}.

definitionown-price elasticity

we say that good $X$ is:

(i) ordinary if $\varepsilon_{X, p_i} < 0$ .

(ii) inelastic if $\varepsilon_{X, p_i} = 0$ .

(iii) giffen if $\varepsilon_{X, p_i} > 0$ .

definitioncross-price elasticity

we say that good $X_i$ is:

(i) complement of $X_j$ if $\varepsilon_{X_i, p_j} < 0$ .

(ii) independent of $X_j$ if $\varepsilon_{X_i, p_j} = 0$ .

(iii) substitute of $X_j$ if $\varepsilon_{X_i, p_j} > 0$ .

definitionincome elasticity

we say that good $X$ is:

(i) inferior if $\varepsilon_{X, I} < 0$ .

(ii) neutral if $\varepsilon_{X, I} = 0$ .

(iii) normal if $\varepsilon_{X, I} > 0$ .

aggregations of the marshallian problem

definitionconsumer expenditure

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ be a monotone function, the consumer expenditure on good $X$ is defined as:

S_i = \frac{p_i X_i^M}{I}.

theoremengel aggregation

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ be a monotone utility function, then:

\sum_{i=1}^n S_i \varepsilon_{x_i, I} = 1.

theoremcournot aggregation

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ be a monotone utility function, then:

\sum_{i=1}^n S_i \varepsilon_{x_i, p_1} = -S_1.

theoremeuler aggregation

let $u : \mathbb{R}^n_+ \to \mathbb{R}$ be a utility function, it holds that:

\varepsilon_{x_i, I} + \sum_{i=1}^n \varepsilon_{x_i, p_i} = 0.

compensated demand

example

\begin{align*} \min_{\{x_1, \dots, x_n\}} \quad & \sum_{i=1}^n p_i x_i \\ \text{subject to} \quad & \bar{u} \leq u(x_1, \dots, x_n), \\ & 0 \leq x_1, \\ & \quad \vdots \\ & 0 \leq x_n. \end{align*}

definitionisocost curves

an isocost curve level $k$ is defined as those baskets that, with given prices, produce the same expenditure:

\text{IG}_k = \{ (x_1, \dots, x_n) \mid p_1 x_1 + \cdots + p_n x_n = k \}.

compensated model solutions

propositionoptimality conditions

given a utility function $u : \mathbb{R}^2 \to \mathbb{R}$ and suppose that, at optimum, $u(x, y) = \bar{u}$ , then:

if $x^*, y^* > 0$ , at optimum: $\text{MRS}(x^*, y^*) = \frac{p_x}{p_y}$ .
if $x^* = 0$ , at optimum: $\text{MRS}(0, y^*) \leq \frac{p_x}{p_y}$ .
if $y^* = 0$ , at optimum: $\text{MRS}(x^*, 0) \geq \frac{p_x}{p_y}$ .

definitioncompensated demands

if the optimal solution of the problem $(X^*)$ is also known as compensated demands or hicksian demands and is usually denoted as:

x_i^* = X_i^C(P, \bar{u}).

definitionminimum expenditure function

the minimum expenditure function, denoted $E(P, \bar{u})$ , is the value function of the compensated problem:

E(P, \bar{u}) = \sum_{i=1}^n p_i X_i^C.

theoremcompensated demand law

any compensated demand is non-increasing in its own price and non-decreasing in cross price.

i.e. $\frac{\partial X_i^C}{\partial p_i} \leq 0$ ; $\frac{\partial X_i^C}{\partial p_j} \geq 0$ , $\forall i \neq j$ .

lemmashephard's lemma

X_i^C = \frac{\partial E}{\partial p_i}, \quad \forall i = 1, \dots, n.

properties of the minimum expenditure function

proposition

(i) homogeneous of degree one in prices, i.e. $E(\lambda P, \bar{u}) = \lambda E(P, \bar{u})$ .

(ii) non-decreasing in utility, i.e. $\frac{\partial E}{\partial \bar{u}}(P, \bar{u}) \geq 0$ .

(iii) non-decreasing in prices, i.e. $\frac{\partial E}{\partial p_i}(P, \bar{u}) \geq 0$ .

(iv) concave in prices, i.e. $\frac{\partial^2 E}{\partial p_i^2}(P, \bar{u}) \leq 0$ .

properties of compensated demands

proposition

(i) homogeneous of degree zero in prices, i.e. $X^C(\lambda P, \bar{u}) = X^C(P, \bar{u})$ .

(ii) shephard's law holds, by obvious reasons.

(iii) symmetry holds in cross effects, i.e. $\frac{\partial X_i^C}{\partial p_j} = \frac{\partial X_j^C}{\partial p_i}$ .

(iv) there exists symmetry in cross effects, i.e. $\frac{\partial X_i^C}{\partial p_j} = \frac{\partial X_j^C}{\partial p_i}$ .

(v) $X_i^C(P, \bar{u}) = X_i^M(P, E(P, \bar{u}))$ .

marshallian-compensated duality

propositionduality relations

the relations between the marshallian and compensated models are known as duality relations.

(i) $E(P, V(P, I)) = I$ .

(ii) $X_i^C(P, V(P, I)) = X_i^M(P, I)$ .

(iii) $V(P, E(P, \bar{u})) = \bar{u}$ .

(iv) $X_i^M(P, E(P, \bar{u})) = X_i^C(P, \bar{u})$ .

(v) $\mu_{\bar{u}}^M = \left(\mu_{\bar{u}}^C\right)^{-1}$ .

proposition

let $u(X)$ be a homothetic and monotone utility function, then $x_1, \dots, x_n \in X$ are normal goods.

slutsky decomposition

theoremslutsky equation

let $u$ be a monotone utility function, then:

\underbrace{\frac{\partial X_i^M}{\partial p_\ell}(P, I)}_{\text{total effect}} = \underbrace{\frac{\partial X_i^C}{\partial p_\ell}(P, \bar{u})}_{\text{substitution effect}} - \underbrace{\frac{\partial X_i^M}{\partial I}(P, I) X_\ell^M(P, I)}_{\text{income effect}}

and

\underbrace{\frac{\partial X_i^M}{\partial p_j}(P, I)}_{\text{total effect}} = \underbrace{\frac{\partial X_i^C}{\partial p_j}(P, \bar{u})}_{\text{substitution effect}} - \underbrace{\frac{\partial X_i^M}{\partial I}(P, I) X_j^M(P, I)}_{\text{income effect}}

for all $i \neq j$ . in elasticity terms:

(i) $\varepsilon_{X_i^M, p_i} = \varepsilon_{X_i^C, p_i} - S_i (\varepsilon_{X_i^M, I})$ (direct effect).

(ii) $\varepsilon_{X_i^M, p_j} = \varepsilon_{X_i^C, p_j} - S_j (\varepsilon_{X_j^M, I})$ (cross effect).

definitionsubstitution effect

given some change in prices (i.e. $P$ to $P'$ ), the substitution effect is calculated, in algebraic form, as follows:

\text{ES} = X^C(P, V(P, I)) - X^M(P, I).

definitionincome effect

given some change in prices (i.e. $P$ to $P'$ ), the income effect can be calculated, in algebraic form, as follows:

\text{EI} = X^M(P, I) - X^C(P, V(P, I)).

theorem

between the marshallian and compensated demand the following relation holds:

\frac{\partial X_i^M}{\partial \bar{u}}(P, \bar{u}) = \frac{\partial X_i^C}{\partial I}(P, I) \mu_R.

welfare measures

definitioncompensating variation

the compensating variation measures the change in income that a consumer should receive before a change in prices so that, with final prices, they are capable of reaching the same initial utility level.

i.e. $\text{VC}(P, P', I) = E(P, V(P, I)) - I$ .

definitionequivalent variation

the equivalent variation measures the change in income that a consumer should receive after a change in prices so that, with initial prices, they are capable of reaching the final utility level.

i.e. $\text{VE}(P, P', I) = E(P, V(P, I)) - I$ .

theoremhicks theorem

\text{VC} = \int_{p_1^{(0)}}^{p_1^{(1)}} X_i^C(P, u_0) dp_1

\text{VE} = \int_{p_1^{(0)}}^{p_1^{(1)}} X_i^C(P, u_1) dp_1

where $u_i = V(p_0, I)$ .

definitionconsumer surplus

the consumer surplus is the difference between the price that consumers pay and the price they are willing to pay:

i.e. $\text{EC}(P, I) = \int_{P_1}^{\infty} X_i^M(P, I) dp_1$ .

likewise, we can define the change in consumer surplus as follows:

\Delta \text{EC} = \int_{p_1^{(0)}}^{p_1^{(1)}} X_i^M(P, I) dp_1.

walrasian demand

example

\begin{align*} \max_{\{x_1, \dots, x_n\}} \quad & u(x_1, \dots, x_n) \\ \text{subject to} \quad & \sum_{i=1}^n p_i x_i \leq \sum_{i=1}^n p_i \bar{x}_i, \\ & 0 \leq x_1, \\ & \quad \vdots \\ & 0 \leq x_n. \end{align*}

definitionwalrasian income

we define walrasian income as follows:

I^W = (P, \bar{X}) = \sum_{i=1}^n p_i \bar{x}_i.

walrasian model solutions

propositionoptimality conditions

given a utility function $u : \mathbb{R}^2 \to \mathbb{R}$ and suppose that, at optimum, $p_x x^* + p_y y^* = p_x \bar{x} + p_y \bar{y}$ , then:

if $x^*, y^* > 0$ , at optimum: $\text{MRS}(x^*, y^*) = \frac{p_x}{p_y}$ .
if $x^* = 0$ , at optimum: $\text{MRS}\left(0, \frac{I^W}{p_y}\right) \leq \frac{p_x}{p_y}$ .
if $y^* = 0$ , at optimum: $\text{MRS}\left(\frac{I^W}{p_x}, 0\right) \geq \frac{p_x}{p_y}$ .

definitionwalrasian demands

if the optimal solution of the walrasian problem $(X^*)$ is also known as walrasian demands and is usually denoted as:

x_i^* = X_i^W(P, \bar{X}).

definitionwalrasian indirect utility function

the walrasian indirect utility function, denoted $V(P, \bar{X})$ , is the value function of the walrasian problem:

V(P, \bar{X}) = u(X_1^W(P, \bar{X}), \dots, X_n^W(P, \bar{X})).

definitionnet demands

net demands are defined as:

X_i^N(P, \bar{X}) = X_i^W(P, \bar{X}) - \bar{x}_i.

properties of the walrasian indirect utility function

proposition

(i) homogeneous of degree zero in prices, i.e. $V^W(\lambda P, \bar{X}) = V^W(P, \bar{X})$ .

(ii) non-decreasing in endowment, i.e. $\frac{\partial V^W}{\partial \bar{x}_i} \geq 0$ .

(iii) the sign of the impact on prices of the indirect utility depends on the sign of the net demand, i.e. $\frac{\partial V}{\partial p_1} = -\mu p_1^W X_1^N(P, \bar{X})$ .

properties of walrasian demands

proposition

(i) homogeneous of degree zero in prices, i.e. $X^W(\lambda P, \bar{X}) = X^W(P, \bar{X})$ .

(ii) roy's identities hold, i.e. $X_i^N = \frac{\frac{\partial V^W}{\partial p_i}}{\frac{\partial V^W}{\partial \bar{x}_i}} P_i$ , $\forall i = 1, \dots, n$ .

marshallian-walrasian duality

propositionduality relations

the relations between the marshallian and walrasian models are known as duality relations.

(i) $I^W = I$ .

(ii) $X_i^W(P, \bar{X}) = X_i^M(P, I)$ .

(iii) $V^W(P, \bar{X}) = V(P, I)$ .

theoremslutsky equation

let $u$ be a monotone utility function, then:

\underbrace{\frac{\partial X_i^W}{\partial p_\ell}(P, \bar{X})}_{\text{total effect}} = \underbrace{\frac{\partial X_i^C}{\partial p_\ell}(P, \bar{u})}_{\text{substitution effect}} - \underbrace{\frac{\partial X_i^M}{\partial I}(P, I) X_\ell^W(P, \bar{X})}_{\text{net income effect}}

and

\underbrace{\frac{\partial X_i^W}{\partial p_j}(P, \bar{X})}_{\text{total effect}} = \underbrace{\frac{\partial X_i^C}{\partial p_j}(P, \bar{u})}_{\text{substitution effect}} - \underbrace{\frac{\partial X_i^M}{\partial I}(P, I) X_j^W(P, \bar{X})}_{\text{net income effect}}

for all $i \neq j$ .

walrasian decision parameters

propositionnet demands

(i) if $X_i^N < 0$ , then the consumer sells $x_i$ .

(ii) if $X_i^N = 0$ , then the consumer consumes their endowment of $x_i$ .

(iii) if $X_i^N > 0$ , then the consumer buys $x_i$ .

propositionendowments

(i) if $\text{MRS}(\bar{x}_i, \bar{x}_j) < p_i/p_j$ , then the consumer sells $x_i$ and buys $x_j$ .

(ii) if $\text{MRS}(\bar{x}_i, \bar{x}_j) > p_i/p_j$ , then the consumer buys $x_i$ and sells $x_j$ .

propositionchanges in utility

(i) if $\frac{\partial V}{\partial p_i} < 0$ , then the consumer sells $x_i$ .

(ii) if $\frac{\partial V}{\partial p_i} > 0$ , then the consumer buys $x_i$ .

leisure-consumption model

example

\begin{align*} \max_{\{h, c\}} \quad & u(h, c) \\ \text{subject to} \quad & c \leq l^{NL} + w(T - h), \\ & h \leq T, \\ & 0 \leq h, c. \end{align*}

\begin{array}{ll} h & \text{leisure derived from leisure} \\ T & \text{total time} \\ w & \text{salary} \\ l^{NL} & \text{non-labor income} \end{array}

leisure-consumption solution

propositionoptimality conditions

given the utility function $u(h, c)$ and suppose that, at optimum, $c = l^{NL} - hw$ , then:

if $h^*, c^* > 0$ , at optimum: $\text{MRS}(h^*, c^*) = w$ .

definitionconsumption demand

the consumption demand is one of the solutions to the leisure-consumption model and is denoted:

c^W(w, l^{NL}).

definitionleisure demand

the leisure demand is one of the solutions to the leisure-consumption model and is denoted:

h^W(w, l^{NL}).

definitionlabor supply

the labor supply of the consumer is determined by:

l(w, l^{NL}) = T - h^W(w, l^{NL}).

definitionreservation salary

the reservation salary of the consumer is given by:

w^R = \text{MRS}(T, l^{NL}).

leisure-consumption duality

propositionduality relations

in the leisure-consumption model the following duality relations hold:

(i) $h^W(w, l^{NL}) = h^M(w, l^{NL} + wT)$ .

(ii) $c^W(w, l^{NL}) = c^M(w, l^{NL} + wT)$ .

theoremslutsky equation

let $u$ be a monotone utility function, then:

\underbrace{\frac{\partial h^W}{\partial w}(w, l^{NL})}_{\text{total effect}} = \underbrace{\frac{\partial h^C}{\partial w}(w, \bar{u})}_{\text{substitution effect}} + \underbrace{\frac{\partial h^M}{\partial I}(w, l^{NL} + wT) l(w, l^{NL})}_{\text{net income effect}}

and

\underbrace{\frac{\partial c^W}{\partial w}(w, l^{NL})}_{\text{total effect}} = \underbrace{\frac{\partial c^C}{\partial w}(w, \bar{u})}_{\text{substitution effect}} + \underbrace{\frac{\partial c^M}{\partial I}(w, l^{NL} + wT) l(w, l^{NL})}_{\text{net income effect}}

leisure-consumption decision parameters

propositionreservation salary

(i) if $w < w^R$ , then the consumer will decide to work.

(ii) if $w > w^R$ , then the consumer will demand $T$ hours of leisure.

propositionchanges in utility

(i) if $\frac{\partial V}{\partial w} = \mu_R l < 0$ , then the consumer will have greater affinity for work.

(ii) if $\frac{\partial V}{\partial w} = \mu_R l > 0$ , then the consumer will have greater affinity for leisure.

intertemporal consumption model

example

\begin{align*} \max_{\{c_0, \dots, c_n\}} \quad & u(c_0, \dots, c_n) \\ \text{subject to} \quad & c_0 \leq \left(\sum_{t=0}^{n-1} (1 + r) (I_t - c_t)\right) + I_n, \\ & 0 \leq c_0, \\ & \quad \vdots \\ & 0 \leq c_n. \end{align*}

\begin{array}{ll} c_t & \text{consumption in period } t \\ I_t & \text{income in period } t \\ r_t & \text{interest rate in period } t \end{array}

intertemporal solution

propositionoptimality conditions

given the utility function $u : \mathbb{R}^2 \to \mathbb{R}$ and suppose that, at optimum, $c_1 = I_1 + (1 + r_0)(I_0 - c_0)$ , then:

if $c_0^*, c_1^* > 0$ , at optimum: $\text{MRS}(c_0^*, c_1^*) = 1 + r_0$ .

definitionconsumption demand in period $t$

consumption demands are the solutions to the intertemporal consumption model and are denoted:

c_t(r_0, \dots, r_{n-1}; I_0, \dots, I_n).

definitionsavings

the savings of the consumer in period $t$ is defined as:

s_t = I_t - c_t.

definition$\bar{r}$ definition

we define $\bar{r}$ as the interest rate at which the consumer does not wish to save nor borrow in period $t$ :

\bar{r}_t = \text{MRS}(I_t, I_{t+1}).

theoremslutsky equation

let $u$ be a monotone utility function, then:

\underbrace{\frac{\partial c_t}{\partial r_\ell}(R, I)}_{\text{total effect}} = \underbrace{\frac{\partial c_t^C}{\partial r_\ell}(R, \bar{u})}_{\text{substitution effect}} + \underbrace{\frac{\partial c_t^M}{\partial r_\ell}(R, I) s_\ell(R, I)}_{\text{net income effect}}

and

\underbrace{\frac{\partial c_{t+1}}{\partial r_t}(R, I)}_{\text{total effect}} = \underbrace{\frac{\partial c_{t+1}^C}{\partial r_t}(R, \bar{u})}_{\text{substitution effect}} + \underbrace{\frac{\partial c_{t+1}^M}{\partial r_t}(R, I) s_t(R, I)}_{\text{net income effect}}

intertemporal decision parameters

propositionsavings

(i) if $s_t < 0$ , then the consumer is a debtor in period $t$ .

(ii) if $s_t = 0$ , then the consumer neither saves nor borrows in period $t$ .

(iii) if $s_t > 0$ , then the consumer is a saver in period $t$ .

propositioninterest rate

(i) if $r < \bar{r}$ , then the consumer will decide to borrow in period $t$ .

(ii) if $r > \bar{r}$ , then the consumer will decide to save in period $t$ .

propositionchanges in utility

(i) if $\frac{\partial V}{\partial r_t} = \mu_R s_t < 0$ , then the consumer will have greater affinity for borrowing.

(ii) if $\frac{\partial V}{\partial r_t} = \mu_R s_t > 0$ , then the consumer will have greater affinity for saving.

firm decisions

definitionproduction function

the production function describes the relation between the production of goods and the quantity of inputs (capital and labor) required to produce the same, in this case:

f : \mathbb{R}^2 \to \mathbb{R} \quad \text{given by some} \quad f(l, k).

definitionisoquant

an isocuant curve level $k$ is defined as those baskets that, with given prices, produce the same quantity $\bar{q}$ :

\text{IC}_k = \{(l, k) \in \mathbb{R}^2_+ \mid f(l, k) = \bar{q}\}.

definitionmarginal product

the marginal product of an input measures the change in firm production given a marginal change in said input.

i.e. $\text{PM}_{k,L} = \frac{\partial f}{\partial l}(l, k)$ and $\text{PM}_{l,K} = \frac{\partial f}{\partial k}(l, k)$ .

definitionmarginal rate of technical substitution

the marginal rate of technical substitution $\text{MRTS}(l, k)$ measures the number of units of capital that are willing to sacrifice to obtain an additional unit of labor maintaining production constant.

i.e. $\text{MRTS}(l, k) = \frac{\text{PM}_{k,L}(l, k)}{\text{PM}_{l,K}(l, k)}$ .

definitionaverage product

the average product of an input measures the average quantity produced by each unit of said input.

i.e. $\text{PMe}_L = \frac{f(l, k)}{l}$ and $\text{PMe}_K = \frac{f(l, k)}{k}$ .

theoremdecreasing returns

we say that $f(l, k)$ has decreasing returns to scale if $f(\lambda l, \lambda k) < \lambda f(l, k)$ , $\forall l, k \geq 0$ and $\forall \lambda > 1$ .

theoremconstant returns

we say that $f(l, k)$ has constant returns to scale if $f(\lambda l, \lambda k) = \lambda f(l, k)$ , $\forall l, k \geq 0$ and $\forall \lambda > 1$ .

theoremincreasing returns

we say that $f(l, k)$ has increasing returns to scale if $f(\lambda l, \lambda k) > \lambda f(l, k)$ , $\forall l, k \geq 0$ and $\forall \lambda > 1$ .

proposition

we say that $f(l, k)$ has decreasing returns to scale if and only if the average productivity of inputs is decreasing.

proposition

we say that $f(l, k)$ has constant returns to scale if and only if the average productivity of inputs is constant.

proposition

we say that $f(l, k)$ has increasing returns to scale if and only if the average productivity of inputs is increasing.

cost minimization

example

\begin{align*} \min_{\{l, k\}} \quad & wl + rk \\ \text{subject to} \quad & 0 \leq l, \\ & 0 \leq k, \\ & \bar{q} \leq f(l, k). \end{align*}

\begin{array}{ll} f(l, k) & \text{production function} \\ k & \text{capital} \\ l & \text{labor} \\ \bar{q} & \text{minimum supply} \\ r & \text{interest rate (capital price)} \\ w & \text{salary (labor price)} \end{array}

cost minimization solutions

propositionoptimality conditions

given a production function $f(l, k)$ and suppose that, at optimum, $f(l^*, k^*) = \bar{q}$ , then:

if $l^*, k^* > 0$ , at optimum: $\text{MRTS}(l^*, k^*) = \frac{w}{r}$ .
if $l^* = 0$ , at optimum: $\text{MRTS}(0, k^*) \leq \frac{w}{r}$ .
if $k^* = 0$ , at optimum: $\text{MRTS}(l^*, 0) \geq \frac{w}{r}$ .

definitioncontingent demands

if the optimal solution of the cost minimization problem $(l^*, k^*)$ is also known as contingent input demands and is usually denoted as:

l^* = l^c(w, r, \bar{q}) \quad k^* = k^c(w, r, \bar{q})

definitionminimum cost function

the minimum cost function, denoted $C(w, r, \bar{q})$ , is the value function of the cost minimization problem:

C(w, r, \bar{q}) = wl^c(w, r, \bar{q}) + rk^c(w, r, \bar{q}).

theoremcontingent demand law

let $l^c(w, r, \bar{q})$ and $k^c(w, r, \bar{q})$ , then each input is non-increasing in its own price and non-decreasing in cross price.

i.e. $\frac{\partial l^c}{\partial w} \leq 0$ ; $\frac{\partial k^c}{\partial r} \leq 0$ ; $\frac{\partial l^c}{\partial r} \geq 0$ ; $\frac{\partial k^c}{\partial w} \geq 0$ .

lemmashephard's lemma

l^c(w, r, \bar{q}) = \frac{\partial C(w, r, \bar{q})}{\partial w} \quad k^c(w, r, \bar{q}) = \frac{\partial C(w, r, \bar{q})}{\partial r}

properties of the minimum cost function

proposition

(i) homogeneous of degree one in prices, i.e. $C(\lambda w, \lambda r, \bar{q}) = \lambda C(w, r, \bar{q})$ .

(ii) increasing in quantity, i.e. $\frac{\partial C}{\partial \bar{q}} > 0$ .

(iii) non-decreasing in prices, i.e. $\frac{\partial C}{\partial w} \geq 0$ and $\frac{\partial C}{\partial r} \geq 0$ .

(iv) concave in prices, i.e. $\frac{\partial^2 C}{\partial w^2} \leq 0$ and $\frac{\partial^2 C}{\partial r^2} \leq 0$ .

properties of contingent input demands

proposition

(i) homogeneous of degree zero in prices, i.e. $l^c(\lambda w, \lambda r, \bar{q}) = l^c(w, r, \bar{q})$ and $k^c(\lambda w, \lambda r, \bar{q}) = k^c(w, r, \bar{q})$ .

(ii) contingent demand law holds.

(iii) shephard's lemma holds.

(iv) there exists symmetry in cross effects, i.e. $\frac{\partial l^c}{\partial r} = \frac{\partial k^c}{\partial w}$ .

definitionmarginal cost

the marginal cost measures the change in costs given a marginal change in production.

i.e. $\text{CMg}(w, r, \bar{q}) = \frac{\partial C(w, r, \bar{q})}{\partial \bar{q}}$ .

definitionaverage cost

the average cost measures the average costs to produce each unit.

i.e. $\text{CMe}(w, r, \bar{q}) = \frac{C(w, r, \bar{q})}{\bar{q}}$ .

proposition

we say that $f(l, k)$ has decreasing returns to scale if and only if average cost is increasing.

proposition

we say that $f(l, k)$ has constant returns to scale if and only if average cost is constant.

proposition

we say that $f(l, k)$ has increasing returns to scale if and only if average cost is decreasing.

propositionrelation between marginal cost and average cost

(i) if $\text{CMg} < \text{CMe}$ , then $\frac{\partial \text{CMg}}{\partial \bar{q}}(w, r, \bar{q}) < 0$ .

(ii) if $\text{CMg} = \text{CMe}$ , then $\frac{\partial \text{CMg}}{\partial \bar{q}}(w, r, \bar{q}) = 0$ .

(iii) if $\text{CMg} > \text{CMe}$ , then $\frac{\partial \text{CMg}}{\partial \bar{q}}(w, r, \bar{q}) > 0$ .

corollary

let $f$ be a production function and $\bar{q} = f(l, k)$ , then:

\text{CMe}(w, r, \bar{q}) = \frac{w}{\text{PM}_{k,L}(l^c, k^c)} + \frac{r}{\text{PM}_{l,K}(l^c, k^c)}.

benefit maximization

example

\begin{align*} \max_{\{q\}} \quad & pq - C(w, r, \bar{q}) \\ \text{subject to} \quad & 0 \leq q. \end{align*}

\begin{array}{ll} C(w, r, \bar{q}) & \text{minimum cost function} \\ p & \text{market price} \\ \bar{q} & \text{supply} \end{array}

benefit maximization solution

propositionoptimality conditions

given a minimum cost function $C(w, r, \bar{q})$ , then:

if $q^* = 0$ , at optimum $p \leq \text{CMg}(w, r, 0)$ .
if $q^* > 0$ , at optimum $p = \text{CMg}(w, r, q^*)$ .

definitionsupply

if the optimal solution of the benefit maximization problem $(q^*)$ is also known as supply and is usually denoted as:

q^* = q(w, r, p).

definitionmaximum benefit function

the maximum benefit function, denoted $\pi(w, r, p)$ , is the value function of the benefit maximization problem:

\pi(w, r, p) = pq(w, r, p) - C(w, r, q(w, r, p)).

propositiondecision parameters

so that $q^*$ produces a maximum, it must be fulfilled that $\frac{\partial \text{CMg}(w, r, \bar{q})}{\partial \bar{q}} \geq 0$ . additionally:

(i) if $p < \text{CMeV}(w, r, q^*)$ , then the firm's benefits of NOT producing are greater than producing.

(ii) if $p = \text{CMeV}(w, r, q^*)$ , then it is the same for the firm to produce or not.

(iii) if $p > \text{CMeV}(w, r, q^*)$ , then the firm's benefits of producing are greater than not producing.

note: $\text{CMeV} = \frac{C(w, r, q^*) - C(w, r, 0)}{q^*}$ .

proposition

let $q^*$ be the efficient production scale such that $0 < q^* < q^*$ , where $q^*$ is a solution of the maximization problem. then, $q^*$ is not optimal.

proposition

if the production function has increasing returns to scale, then $q^* = 0$ or $q^* \to \infty$ .

theoremsupply law

let $q(w, r, p)$ be derivable, then supply is non-decreasing in selling price.

i.e. $\frac{\partial q}{\partial p}(w, r, p) \geq 0$ .

properties of the maximum benefit function

proposition

(i) homogeneous of degree one in prices, i.e. $\pi(\lambda w, \lambda r, \lambda p) = \lambda \pi(w, r, p)$ .

(ii) non-decreasing in selling price, i.e. $\frac{\partial \pi}{\partial p} \geq 0$ .

(iii) non-increasing in input prices, i.e. $\frac{\partial \pi}{\partial w} \leq 0$ and $\frac{\partial \pi}{\partial r} \leq 0$ .

properties of supply

proposition

(i) homogeneous of degree zero in prices, i.e. $q(\lambda w, \lambda r, \lambda p) = q(w, r, p)$ .

(ii) supply law holds, by obvious reasons.

(iii) the sign of the impact on input prices of supply depends on the contingent demands in the following form:

\frac{\partial l}{\partial p} = \frac{\partial l^c}{\partial \bar{q}} \frac{\partial q}{\partial p} \quad \text{and} \quad \frac{\partial k}{\partial p} = \frac{\partial k^c}{\partial \bar{q}} \frac{\partial q}{\partial p}.

lemmahotelling's lemma

\frac{\partial \pi}{\partial p} = q^* \geq 0.

corollary

\frac{\partial \pi}{\partial w} = -l(w, r, p) \quad \frac{\partial \pi}{\partial r} = -k(w, r, p)

definitionproducer surplus

the producer surplus represents the difference between the price at which the producer sells and the disposition to sell a determined quantity and is defined as follows:

\begin{align*} \text{EP} &= \pi(w_1, r_1, p_1) - \pi(w_1, r_1, p_0) \\ &= \int_{p_0}^{p_1} q(w, r, p) dp \\ &= \pi(w_1, r_1, p_1) - \pi(w_0, r_1, p_1) \\ &= -\int_{w_0}^{w_1} l(w, r, p) dw \\ &= \pi(w_1, r_1, p_1) - \pi(w_1, r_0, p_1) \\ &= -\int_{r_0}^{r_1} k(w, r, p) dr \end{align*}

definitioninput-induced demands

input-induced demands determine the quantity of capital and labor that the firm demands for each level of $w$ , $r$ and $p$ . it is denoted:

l(w, r, p) = l^c(w, r, q(w, r, p));

k(w, r, p) = k^c(w, r, q(w, r, p)).

properties of input-induced demands

proposition

(i) homogeneous of degree zero in prices, i.e. $l(\lambda w, \lambda r, \lambda p) = l(w, r, p)$ and $k(\lambda w, \lambda r, \lambda p) = k(w, r, p)$ .

(ii) non-increasing in input prices, i.e. $\frac{\partial l}{\partial w} \leq 0$ and $\frac{\partial k}{\partial r} \leq 0$ .

(iii) the sign of the impact on selling price of input demands depends on the contingent demands in the following form:

\frac{\partial l}{\partial p} = \frac{\partial l^c}{\partial \bar{q}} \frac{\partial q}{\partial p} \quad \text{and} \quad \frac{\partial r}{\partial p} = -\frac{\partial k^c}{\partial \bar{q}} \frac{\partial q}{\partial p}.

theoremslutsky equation

let $u$ be a monotone utility function, then:

\underbrace{\frac{\partial l}{\partial w}(w, r, p)}_{\text{total effect}} = \underbrace{\frac{\partial l^c}{\partial w}(w, r, q)}_{\text{substitution effect}} - \left(\frac{\partial l^c}{\partial \bar{q}}(w, r, q)\right) \frac{\partial q}{\partial p}(w, r, p),

\underbrace{\frac{\partial k}{\partial r}(w, r, p)}_{\text{total effect}} = \underbrace{\frac{\partial k^c}{\partial r}(w, r, q)}_{\text{substitution effect}} - \left(\frac{\partial k^c}{\partial \bar{q}}(w, r, q)\right) \frac{\partial q}{\partial p}(w, r, p),

\underbrace{\frac{\partial l}{\partial r}(w, r, p)}_{\text{total effect}} = \underbrace{\frac{\partial l^c}{\partial r}(w, r, q)}_{\text{substitution effect}} - \frac{\partial l^c}{\partial \bar{q}}(w, r, q) \frac{\partial q}{\partial p}(w, r, p),

and

\underbrace{\frac{\partial k}{\partial w}(w, r, p)}_{\text{total effect}} = \underbrace{\frac{\partial k^c}{\partial w}(w, r, q)}_{\text{substitution effect}} - \frac{\partial k^c}{\partial \bar{q}}(w, r, q) \frac{\partial q}{\partial p}(w, r, p).

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