Consumer choice in one period

Utility functions recap

Utility functions

We have \(U=f(\mathbf x)\).

Throughout this we will be using a Cobb-Douglas utility function, and discuss the properties of other utility functions at the end.

We have \(U=\prod_i x_i^{\alpha_i}\).

Marginal utility

The marginal utility of product \(x_1\) is:

\(\dfrac{\delta }{\delta x_1}f(\mathbf x)\)

For Cobb-Douglas:

\(U=\sum_i x_i^{\alpha_i})\).

\(\dfrac{\delta }{\delta x_1}f(\mathbf x) = \dfrac{\delta}{\delta x_1} \prod_i x_i^{\alpha_i}\)

\(\dfrac{\delta }{\delta x_1}f(\mathbf x) = \dfrac{1}{x_1}\alpha_1\prod_i x_i^{\alpha_i}\)

Indifference curves

We have \(U=f(x,y)\)

An indifference curve is a curve where a consumer is indifferent to all points on it.

\(f(x,y)=c\)

Marginal rate of substitution

The marginal rate of substitution is the amount of one good that a customer is willing to give up for another.

This is the gradient of the indifference curve.

\(MRS(x_1, x_2)=\dfrac{MU(x_1)}{MU(x_2)}\)

\(MU(x_1) = \dfrac{1}{x_1}\alpha_1\prod_i x_i^{\alpha_i}\) \(MU(x_2) = \dfrac{1}{x_2}\alpha_2\prod_i x_i^{\alpha_i}\)

\(MRS(x_1, x_2)=\dfrac{\dfrac{1}{x_1}\alpha_1\prod_i x_i^{\alpha_i}}{\dfrac{1}{x_2}\alpha_2\prod_i x_i^{\alpha_i}}\)

\(MRS(x_1, x_2)=\dfrac{\dfrac{1}{x_1}\alpha_1}{\dfrac{1}{x_2}\alpha_2}\)

\(MRS(x_1, x_2)=\dfrac{\dfrac{\alpha_1}{x_1}}{\dfrac{\alpha_2}{x_2}}\)

Elasticity of substitution

The utility maximisation problem

First-order conditions

We have a utility function:

\(U=f(\mathbf x)\)

And the constraint:

\(\sum_i (x_i-c_i)p_i\le 0\)

This gives us the constrained optimisation problem:

\(L=f(\mathbf x)-\lambda \sum_i (x_i-c_i)p_i\)

The first-order conditions are:

\(L_{x_i}=\dfrac{\delta }{\delta x_i}f(\mathbf x)-\lambda p_i=0\)

Or:

\(\lambda = \dfrac{MU(x_i)}{p_i}\)

This means for any pair we have:

\(\dfrac{MU(x_i)}{p_i}=\dfrac{MU(x_j)}{p_j}\)

For the Cobb-Douglas utility function the first-order conditions are:

\(\dfrac{\delta }{\delta x_1}f(\mathbf x) = \dfrac{1}{x_1}\alpha_1\prod_i x_i^{\alpha_i}\)

\(L_{x_i}=\dfrac{\delta }{\delta x_i}f(\mathbf x)-\lambda p_i=0\)

Marshallian demand

We can write a demand function:

\(x_i=f(I, \mathbf p)\)

We can derive this from the first-order conditions of a specific utility function.

Own-price elasticity of demand

We have our Marshallian demand function:

\(x_i=x_{di}(I, \mathbf p)\)

The derivative of this with respect to price is the additional amount consumed after prices increase.

\(\dfrac{\delta }{p_i}x_{di}(I, \mathbf p)\)

For the Cobb-Douglas utility function, this is:

\(\dfrac{\delta }{p_i}x_{di}(I, \mathbf p)\)

In addition to the derivative, we may be interested in the elasticity. That is, the proportional change in output after a change in price.

\(\xi_i =\dfrac{\dfrac{\Delta x_i}{x_i}}{\dfrac{\Delta p_i}{p_i}}\)

\(\xi_i =\dfrac{\Delta x_i}{\Delta p_i}\dfrac{p_i}{x_i}\)

For the point-price elasticity of demand we evaluate infintesimal movements.

\(\xi_i =\dfrac{\delta x_i}{\delta p_i}\dfrac{p_i}{x_i}\)

Constant price elasticity of demand

If the point-price elasticity of demand is constant we have:

\(\xi_i =\dfrac{\delta x_i}{\delta p_i}\dfrac{p_i}{x_i}=c\)

This means that small changes in the price at low level cause large changes in quantity.

Arc-price elasticity of demand

We may have price changes which are non-infintesimal.

\(E_d=\dfrac{\Delta Q/\bar Q}{\Delta P/\bar P}\)

Where \(\bar Q\) and \(\bar P\) are the mid-points between the start and end.

Super elasticty of demand

If elasticity is constant super elasticity is 0.

Veblen goods

The demand curve is sloping up.

Cross-price elasticity of demand, complements and substitutes

Cross-price elasticity of demand

We have our Marshallian demand function:

\(x_i=x_{di}(I, \mathbf p)\)

The derivative of this with respect to price is the additional amount consumed after prices increase.

\(\dfrac{\delta }{p_i}x_{di}(I, \mathbf p)\)

Complements

Substitutes

Diversion ratio

Income effects

Engel curves and income elasticity of demand

The Engel curve shows demand for a good as a function of income.

Derivative \(x_{di}=x_{di}(I, \mathbf p)\)

\(\dfrac{\delta }{\delta I}x_{di}(I, \mathbf p)\)

Income elasticity of demand \(\xi_i =\dfrac{\delta x_{di}}{\delta I}\dfrac{I}{x_{di}}\)

Normal goods

As income rises, demand also rises.

This is the same as saying the income elasticity of demand is above \(0\).

\(\xi_i =\dfrac{\delta x_{di}}{\delta I}\dfrac{I}{x_{di}}\)

Inferior goods

An inferior good is one where the demand falls as income increases.

This is the same as the income elasticity of demand being less than \(0\).

Necessities

Necessities are goods which increase in demand as income rises, but by a smaller proportion.

The income elasticity of demand is between \(0\) and \(1\).

Luxuries

Luxuries are goods which increase in demand as income rises, by a larger proportion.

The income elasticity of demand is above \(1\).

Ordinary goods

As price rises, demand goes down.

Giffen goods

As price rises, demand goes up. Not because of the slope of the demand curve, but because the income effect and inferior effect are strong.

Substitution means you buy less.

Income means you buy less generally, but move towards inferior goods.

Income effect

Slutsky equation

Indirect utility functions

Indirect utility functions

The normal utility function is:

\(U=f(\mathbf x)\)

We have our demand:

\(x_{di}=x_{di}(I, \mathbf p)\)

The indirect utility function

We can plut this in to get:

\(U=g(I, \mathbf p)\)

The expenditure minimisation problem

The expenditure function

Shephard’s lemma

Hick’s demand

Roy’s identity

Specific utility functions

Cobb-Douglas utility function

\(U=A\sum_i X_i^{\alpha_i} \)

Leontief utility function

\(U=\sum_i X_i^{\alpha_i} \)

Constant Elasticity of Substitution (CES) utility function

For some constant \(r\).

\(U=A[\alpha_i X_i^r]^{\dfrac{1}{r}}\)

Almost Ideal Demand System

Representative consumer

Other

Consumer surplus

Aggregating to demand curves

Aggregating individual preferences to the demand curve for a product

Move Representative consumers here.