Lecture #7

• Review• Go over Exam #1• Continue production economic theory

Using the previous example, let’s now solve for the input levels which maximize profits.Maximizing profit

1st order conditions:

For the first partial derivative:

For the second partial derivative:

2nd order conditions:

and Profits are maximized when

Summary:Least cost combinations and profit maximization.

Summary:Least cost combinations and profit maximization.Maximizing output subject to a cost constraint:

Minimizing cost subject to an output constraint:

Profit maximization:

An expansion path is a locus of points which represent the tangencies between isoquant and isocost lines assuming that input prices are constant. Note that these tangencies are least cost combinations.Since the tangencies represent points where slopes of isoquant and isocost lines are equal, we can solve for the equation of the expansion path:Slope of isoquant:

Slope of isocost line: Using our previous production example:

equation of the expansion path

Points of the expansion path are leastcost combinations of inputs.

(so the least cost solutions lie on the expansion path)

So the profit max solution is a least cost solution which lies on the firm’s expansion path.

What about the profit max solution?

The two constrained optimization methods, constrained output maximization and constrained cost minimization, yield a unique relationship called duality. It implies the possibility of deriving cost functions from production functions and vice versa.Given the same cost and production constraints, we see that both problems (constrained output maximization and constrained cost minimization) can yield the same optimal combinations of inputs

However, the other bit of information – the Lagrange multiplier is different yet related. For the constrained output maximization problem,

In the constrained minimization problem,

The following production function, based on actual Iowa experimental data, has been altered slightly to simplify computations:

Given the above production function and input prices, what is the least cost combination to produce a 240 pound hog?

This is a problem of constrained cost minimization. Why? Because costs are in cwt and 240 lbs = 2.4cwt

1st order conditions:

Solve for λ:From the first equation:

From the second equation:

Plug this relationship between into the third equation.

or

Least cost combination: Feed 448 lbs of corn and 80 lbs of soybean meal to produce a 240 lb hog (2.4 cwt)

The function is said to be homogeneous of degree k if each independent variable (the x’s for this function) is multiplied by a scalar λ, and the dependent variable (in this case y) changes by λk.So the function is homogeneous of degree k if:

Find the degree of homogeneity for this function.If the function is homogeneous then we can write:

the function is homogeneous of degree 2 (HD2)

What does this mean?If inputs x1 and x2 are increased by a factor λ, then output (y) will increase by λ2.

Another example: case of Cobb- Douglas formFind the degree of homogeneity for this function.

Linearly homogeneous production functions are HD1 or HD = 1General Cobb-Douglas production function: is homogeneous of degree

Another example:

Is this function homogeneous and of what degree?

However, we can not get back to λky. So this function is not homogeneous.

The demand for final products is derived from constrained utility maximization (objective function of consumers). The demand for inputs is derived from profit maximization (objective function of producers).Take the case of production with one variable input:

Recall first order conditions for profit maximization:

We can write the factor demand for input x1 as:

Changes in r1 can be represented as movements along the input demand function. Changes in p can be represented as shifts of the input demand function. Generally, output price is a positive shifter of the input demand function.

We can graph the input demand function for x1 by assuming p is held constant. We call the input demand function the derived demand function since it is a result of (or derived from) profit maximization.So the input demand function is often called the derived demand function since it is derived from profit maximization. For the one variable input case (factor-product case), economic theory states that the factor demand function is that portion of the VMP or MVP curve in Stage II.

What is the law of demand? Does it hold for inputs?Yes. The law of demand underlies the negative relationship between own price and quantity demanded for final products as well as inputs.

What about input demands for the factor-factor case?Given the following production function:

Assume also that p is output price, r1 is the per unit cost of x1 and r2 is the per unit cost of x2.Derive the factor demand function for x1.Input or factor demands are derived from profit maximization.

1st order conditions:Solving for x2:

In a similar manner, one can derive the factor demand for x2.Is the factor demand function for x1 downward sloping?

Is output price, p, a positive shifter of the input demand function?

So if output price increases demand for x1 increases

What effect does changes in r2 have on the input demand for x1?

Own price elasticity of demand: