This applet illustrates utility maximization subject to a budget constraint.   Utility is described by the Cobb-Douglas function
U(x, z) = x1/3z2/3

where x and z are quantities of different goods such as "food and clothing" or "national park usage versus all other goods."  In the graph, the x axis is the horizontal and the z axis is the vertical.  The yellow lines show the different combinations of x and z that obtain a fixed level of utility, with utility levels increasing to the north east.  The blue line is the budget constraint.  It is described by the equation

pxx + pzz = 100

where the income level is 100 and px and pz are the prices of x and z, respectively.  The line can by drawn on the graph by rearranging the equation to be the following function of z:

z = 100/pz - (px/pz)x

where (px/pz) is the price ratio.  The price ratio is the slope of the budget line.  The maximum affordable amount of z, obtained by setting x equal to zero, is 100/pz.  The maximum affordable amount of x is obtained in a similar fashion.  These maximums give us the two intercepts of the line on the graph.

The red line is the  highest indifference curve achievable under the budget constraint.  This curve "just touches" the budget constraint at a single point.  The amount of x and z consumed at this point (the optimum) are displayed to the northeast of that point.


Click here to see how a demand curve can be obtained from the utility maximizing decision.

Geoffrey Gerdes
Last Update: October 27, 1997