Graphical

Budget constraints

Budget curve moves outward

Slope of budget curve changes

Optimization of utility

Utility maximization involves the choice of bundles under a resource constraint. For example, individuals select the amount of goods, services and transportation by comparing the utility increase with an increase in consumption against the utility loss associated with the giving up of resources (or equivalently forgoing the consumption which those resources command).

Often one price is taken to be 1, and one good is taken to be money. An income increase can be represented by the outward movement of the budget line.

An increase in the price of good ${\displaystyle X_{1}}$ can be represented by a change in the slope of the budget line (still anchored at one end).

In graphic terms the process of optimization is accomplished by equating the rate at which an individual is willing to trade off one good for another to the rate at which the market allows him/her to trade them off. This can be represented in the following graph

The individual maximizes utility by moving down the budget constraint to that point at which the slope of the budget line (${\displaystyle -P_{1}/P_{2}}$) which is the rate of exchange dictated by the market is just equal to the rate at which the individual is willing to trade the two goods off. This is the slope of the indifference curve or the marginal rate of transformation (MRT). A point such as 'e' is an equilibrium point at which utility is being maximized.

Equilibrium is the tangency between the indifference curve/utility and the budget constraint.

Optimization

As an optimization problem, this can be written:

${\displaystyle {\text{Maximize}}U(X)\,\!}$

subject to:

${\displaystyle px\leq m\,\!}$

${\displaystyle x}$ is in ${\displaystyle X}$

where:

• ${\displaystyle p}$ = price vector,
• ${\displaystyle x}$ = goods vector,
• ${\displaystyle m}$ = income

(Because of non-satiation, the constraint can be written as px=m.) This kind of problem can be solved with the use of the Lagrangian:

${\displaystyle \Lambda =U(X)-\lambda (px-m)\,\!}$ where

• ${\displaystyle \lambda }$ is the Lagrange multiplier

Take derivatives with respect to ${\displaystyle x}$, and set the first order conditions to 0

${\displaystyle {\frac {\partial \Lambda }{\partial x_{i}}}={\frac {\partial U(X)}{\partial x_{i}}}-\lambda p_{i}=0\,\!}$

Divide to get the Marginal rate of substitution and Economic Rate of Substitution

${\displaystyle MRS={\frac {\frac {\partial U(X)}{\partial x_{i}}}{\frac {\partial U(X)}{\partial x_{j}}}}={\frac {p_{i}}{p_{j}}}=ERS\,\!}$

Example: Optimizing Utility

Indirect Utility

The Marshallian Demand relates price and income to the demanded bundle. This is given as ${\displaystyle x(p,m)}$. This function is homogenous of degree 0, so if we double both ${\displaystyle p}$ and ${\displaystyle m}$, ${\displaystyle x}$ remains constant. We can develop an indirect utility function:

${\displaystyle v(p,m)=maxU(X)\,\!}$

subject to: ${\displaystyle px=m\,\!}$

where X that solves this is the demanded bundle

Expenditure Function

The inverse of the indirect utility is the expenditure function ${\displaystyle e(p,u)={\text{min}}px\,\!}$

subject to: ${\displaystyle u(x)\geq u\,\!}$

Properties of the expenditure function ${\displaystyle e(p,u)}$:

• is non-decreasing in ${\displaystyle p}$
• homogenous of degree 1 in ${\displaystyle p}$
• concave in ${\displaystyle p}$
• continuous in ${\displaystyle p}$ for ${\displaystyle p>>0}$

Roy's Identity

The Hicksian Demand or compensated demand is denoted h(p,u).

${\displaystyle h_{i}(p,u)={\frac {\partial e(p,u)}{\partial p_{i}}}\,\!}$

vary price and income to keep consumer at fixed utility level vs. Marshallian demand. Roy's Identity allows going back and forth between observed demand and utility

${\displaystyle x_{i}(p,m)=-{\frac {\frac {\partial v(p,m)}{\partial p_{i}}}{\frac {\partial v(p,m)}{\partial m}}}\,\!}$

Equivalencies

${\displaystyle e(p,v(p,m))=m\,\!}$ the minimum expenditure to reach ${\displaystyle v(p,m)}$ is ${\displaystyle m}$

${\displaystyle v(p,e(p,u))=u\,\!}$ the maximum utility from income ${\displaystyle e(p,u)}$ is ${\displaystyle u}$

${\displaystyle x_{i}(p,m)=h_{i}(p,v(p,m))\,\!}$ Marshallian demand at ${\displaystyle m}$ is Hicksian demand at ${\displaystyle v(p,m)}$

${\displaystyle h_{i}(p,u)=x_{i}(p,e(p,u))\,\!}$ Hicksian demand at ${\displaystyle u}$ is Marshallian demand at ${\displaystyle e(p,u)}$

Money Metric Indirect Utility Function

The Money Metric Indirect Utility Function tells how much money at price p is required to be as well off as at price level q and income m. Define it as

${\displaystyle \mu (p;q,m)=e(p,v(q,m))\,\!}$

Equivalent Variation

${\displaystyle EV=\mu (p^{0};p^{1},m^{1})-\mu (p^{0};p^{0},m^{0})\,\!}$

note 1 indicates after, 0 indicates before

Current prices are the base, what income change will give equivalent utility

Compensating Variation

${\displaystyle CV=\mu (p^{1};p^{1},m^{1})-\mu (p^{1};p^{0},m^{0})\,\!}$ New prices are the base, what income change will compensate for price change

Consumer's Surplus

Utility and Consumer's Surplus

${\displaystyle \Delta CS=\int \limits _{p^{0}}^{p^{1}}{x(t)dt}\,\!}$

Generally

${\displaystyle EV\geq CS\geq CV\,\!}$

When utility is quasilinear (${\displaystyle U=U(X1)+X0)\,\!}$, then:

${\displaystyle EV=CS=CV\,\!}$

Consumption Bundles

Define a consumption set X, e.g. {house, car, computer},

'x', 'y', 'z', are bundles of goods, such as x{house,car}, y{car, computer}, z{house, computer}.

Goods are not consumed for themselves but for their attributes relative to other goods

We want to find preferences that order the bundles. Utility is ordinal, so we only care about which is greater, not by how much.

Conditions

There are several Conditions on preferences to produce a continuous (well-behaved) utility function.

• Completeness: Either ${\displaystyle x{\underline {\succ }}y}$ (Read x is preferred to y) or ${\displaystyle y{\underline {\succ }}x}$ or both
• Reflexive: ${\displaystyle x{\underline {\succ }}x}$
• Transitive: if ${\displaystyle x{\underline {\succ }}y}$ and ${\displaystyle y{\underline {\succ }}z}$ then ${\displaystyle x{\underline {\succ }}z}$ (This poses a problem for social welfare functions)
• Monotonicity: If ${\displaystyle x\geq y}$then ${\displaystyle x{\underline {\succ }}y}$
• Local Non-satiation: More is better than less
• Convexity: if ${\displaystyle x{\underline {\succ }}y}$ and ${\displaystyle y{\underline {\succ }}z}$ then ${\displaystyle tx+(1-t)y{\underline {\succ }}z}$
• Continuity: small changes in input beget small changes in output. The preference relation ${\displaystyle {\underline {\succ }}}$ in X is continuous if it is preserved under the limit operation

The function f is continuous at the point a in its domain if:

• ${\displaystyle {\underset {x\to a}{\mathop {\lim } }}\,f\left(x\right)\,\!}$ exists

• ${\displaystyle {\underset {x\to a}{\mathop {\lim } }}\,f\left(x\right)=f\left(a\right)\,\!}$

If 'f' is not continuous at 'a', we say that 'f' is discontinuous at 'a'.