Suppose that a firm uses both labour (X) and capital (Y) to produce a certain good. An isoquant is a curve made up of the set of points [math] that gives the same level of output. For example, if 10 units of labour and 6 units of capital produce the same output as 11 units of labour and 5 units of capital, than (10,6) and (11,5) are on the same isoquant. The production function is the set of all isoquants given any level of output.

The Cobb-Douglas production function is a simple example:
[math] , where [math] .

The cost function gives the cost of using a given combination of labour and capital. If [math] is the cost of using labour and [math] is the cost of using capital then: [math]. A set of points with a constant cost is known as an isocost.

To minimize cost given a certain level of output, an isocost will be tangent to an an isoquant. This is not the same situation as with utility maximization where the total income was fixed. Here, you are minimizing cost for a given level of output.

Let [math], [math] and [math].
Suppose the firm wants to produce [math] units. Find the cost minimizing amount of labour and capital to use as a function of [math].
[math]
[math]

Now, express the cost of production as a function of output (Q). Note that this will be the long run cost function.
[math]

(you will lose 25% of your points if you do)