_____$$**p _{x}x + p_{y}y = I _______________ (1) **

A graph of the budget constraint is shown below:

$$**z > 0 => U(x + z,y) > U(x,y) ** and $$**z > 0 => U(x,y + z) > U(x,y) **.

That is more is better than less. An indifference curve of a utility function is the line representing all combinations of $$**x** and $$**y** which result in the same value of the utility function. In the example above the points $$**(3,4)** and $$**(4,3)** lie on the same indifference curve. To keep the calculus as simple as possible we shall use concave utility functions. Concave utility functions have indifference curves of the following form:

_____max $$**U(x,y)**

_____subject to $$**p _{x}x + p_{y}y = I _____________ (2) **

In words, the consumer compares alternatives bundles and chooses the one which has the highest utility. We do not ask how the consumer performs this operation, but rather we examine the consequences of such action which we will soon see results in demand functions for $$**x**

To solve this problem by substitution let us consider a specific concave utility function: $$**U(x,y) = x ^{1/2}y^{1/2}**

**Example I:**

The first example is:

_____max $$**x ^{1/2}y^{1/2}**

_____subject to $$**p _{x}x + p_{y}y = I ____________ (3) **

We use the budget constraint to determine:

_____$$**y = (1/p _{y})[I - p_{x}x ]________________ (4) **

We substitute $$**(4)** into the utility function to obtain:

_____$$**f(x) = x ^{1/2}[(1/p_{y})(I - p_{x}x)]^{1/2}________ (5) **

The optimization problem has been reduced to one variable $$**x**:

_____max$$**f(x) = x ^{1/2}[(1/p_{y})(I - p_{x}x)]^{1/2}_____ (6) **

We take the first derivative using the product rule:

_____$$**f'(x) = (1/2)x ^{-1/2}[(1/p_{y})(I - p_{x}x)]^{1/2} + (1/2)x^{1/2}[(1/p_{y})**

__________$$

We set $$**f'(x) = 0 ** and obtain:

_____$$** (1/2)[(1/p _{y})(I - p_{x}x)]^{1/2}/[x^{1/2}] = (p_{x}/2p_{y})x^{1/2}/[(1/p_{y})**

_______________________________$$

Multiplying both sides by the denominators we obtain:

_____$$** (I - p _{x}x) = p_{x}x_____________________ (9) **

Solving for $$**x**, we obtain:

_____$$** x = I/2p _{x}_________________________ (10) **

From the budget constraint, we obtain the demand curve for $$**y**:

_____$$** y = I/2p _{y}_________________________ (11) **

We have obtained the demand curve for $$**x** and $$**y**as functions of the income, $$**I** and price of $$**x**, $$**p _{x}**. Now when we draw a graph, we do so in two dimensions. But we have three variables. The two dimensional demand curve is drawn showing $$

Again the two dimensional demand curve for $$**y** shows the relationship between $$**y** and $$**p _{y}**. The shift parameter is $$

**Example II:**

The second example is:

_____max $$**x ^{2/5}y^{3/7}**

_____subject to $$**p _{x}x + p_{y}y = I _______________ (12) **

We use the budget constraint to determine:

_____$$**y = (1/p _{y})[I - p_{x}x ]___________________ (13) **

We substitute $$**(13)** into the utility function to obtain the following optimization problem:

_____max$$**f(x) = x ^{2/5}[(1/p_{y})(I - p_{x}x)]^{3/7}________ (14) **

We take the first derivative using the product rule:

_____$$**f'(x) = (2/5)x ^{-3/5}[(1/p_{y})
(I - p_{x}x)]^{3/7} + (3/7)x^{2/5}[(1/p_{y})**

___________$$

We set $$**f'(x) = 0 ** and obtain:

_____$$** (2/5)[(1/p _{y})(I - p_{x}x)]^{3/7}/[x^{3/5}] = (3p_{x}/7p_{y})x^{2/5}/[(1/p_{y})**

_______________________________$$

Multiplying both sides by the denominators we obtain:

_____$$** (2/5)(I - p _{x}x)] = (3p_{x}/7)x_____________ (17) **

Solving for $$**x**, we obtain the demand curve for $$**x**:

_____$$** x = 14I/29p _{x}_______________________ (18) **

From the budget constraint, we obtain the demand curve for $$**y**:

_____$$** y = 15I/29p _{y}_______________________ (19) **

The graphs of the families of demand curves for the two variables are similar to those shown above for example I. **The degrees of difficulty of this example corresponds to a B level of performance.**

**Example III:**

The third example is:

_____max $$**x ^{1/2} + y^{1/2}**

_____subject to $$**p _{x}x + p_{y}y = I ________________ (20) **

We use the budget constraint to determine:

_____$$**y = (1/p _{y})[I - p_{x}x ]____________________ (21) **

We substitute $$**(21)** into the utility function to obtain the following optimization problem:

_____max$$**f(x) = x ^{1/2} + [(1/p_{y})(I - p_{x}x)]^{1/2}_______ (22) **

We take the first derivative using the product rule:

_____$$**f'(x) = 1/(2x ^{1/2}) - p_{x}/2p_{y}[(1/p_{y})(I - p_{x}x)]^{1/2}_ (23) **

We set $$**f'(x) = 0 ** and multiplying both sides by the denominators we obtain:

_____$$** [(I - p _{x}x)(py)] = p_{x}^{2}x___________________ (25) **

Solving for $$**x**, we obtain the demand curve for $$**x**:

_____$$** x = Ip _{y}/[p_{x}p_{y} + p_{x}2^{}]_____ (16) **

From the budget constraint, we obtain the demand curve for $$**y**:

_____$$** y = Ip _{x}/[p_{x}p_{y} + p_{y}2^{}_____ (19) **

The graphs of the families of demand curves for the two variables have two shift parameters: Income and the price of the other good. In a model with many goods the prices of a subset of the goods would be shift parameters.