Production FunctionsQ = F(K, L | given Tech)

OrOutput = F(Inputs | Chosen Tech)

Q = F(K, L |T)◦ But K = K0 (Fixed at level K0 and can’t be changed)

Short-run -> size of the plant; machinery can’t be increased/decreased immediately -> fixed in the SR

long-run it can be increased or decreased -> variable in the LR

Look at the short-run for the time being

◦ However L (labor) can be changed very quickly Layoffs/hiring So it is variable in both the SR and LR

Production FunctionWhere Only 1 Input Is Variable

Short-run Production FunctionQ = F(L |K=Ko,T)

production function or total product function A numerical or mathematical expression of a relationship between inputs and outputs. It shows units of total product as a function of units of inputs.

TABLE 7.2 Production Function

(1) Labor Units(Employees)

(2) Total Product

(Sandwiches per Hour)

(3) Marginal Product

of Labor

(4) Average Product of Labor

(Total Product ÷ Labor Units)

0123456

0102535404242

-101510 5 2 0

-10.012.511.710.0 8.4 7.0

Production Functions: Total Product, Marginal Product, andAverage Product

FIGURE 7.3 Production Function for Sandwiches

production function is of the relationship between inputs and outputs.

marginal product of labor is the additional output that one additional unit of labor produces.

Buy if We Could Increase K? K0 -> K1

Able to increase the amount of output each laborer can produce

• Increases Total Product at all levels of employment

How about if we could increase both inputs at the same time?

Let’s Do This in 2D (instead of 3D)

Let’s Do This in 2D (instead of 3D)

Finally in 2D (really)Isoquants (q1, q2, q3) represent lines of equal (iso) production – or of the same height (product output) from the 3D graph

2 Input Production Function◦ Q = F(K, L | T)◦ Cost

Depend on the quantity of the inputs used K = # of units of K L = # of employees (units of L) Pk , PL = per unit price of Kapital and Labor (wage rate)

Costs of production for a given level of output C(Q) = PkK + PLL

How About Costs?

TABLE 7.3 Inputs Required to Produce 100 Diapers Using Alternative Technologies

Technology Units of Capital (K) Units of Labor (L)

ABCDE

2346

10

106432

TABLE 7.4 Cost-Minimizing Choice among Alternative Technologies (100 Diapers)

(1)Technology

(2)Units of Capital (K)

(3)Units of Labor (L)

Cost = (L X PL) + (K X PK)

(4) (5)PL = $1PK = $1

PL = $5PK = $1

ABCDE

2346

10

106432

$12989

12

$52 33 24 21 20

Choice of Technology

Two things determine the cost of production: (1) technologies that are available and (2) input prices. Profit-maximizing firms will choose the technology that minimizes the cost of production given current market input prices.

isocost line A graph that shows all the combinations of capital and labor available for a given total cost.

FIGURE 7A.3 Isocost Lines Showing the Combinations of Capital and Labor Available for $5, $6, and $7An isocost line shows all the combinations of capital and labor that are available for a given total cost.

Factor Prices and Input Combinations: Isocosts

(PK K) + (PL L) = TC

Substituting our data for the lowest isocost line into this general equation, we get

 

FIGURE 7A.4 Isocost Line Showing All Combinations of Capital and Labor Available for $25

Slope of isocost line:

One way to draw an isocost line is to determine the endpoints of that line and draw a line connecting them.

Factor Prices and Input Combinations: Isocosts

 

Finding the Least-Cost Combination of Capital and Labor to Produce 50 Units of Output

2. Profit-maximizing firms will minimize costs by producing their chosen level of output where the isoquant is tangent to an isocost line. 3. Here the cost-minimizing technology—3 units of capital and 3 units of labor—is represented by point C.

Note: Could produce 50 units at TC = $7; but it would cost more. Could not produce 50 units if your “cost budget” is $5

Finding the Least-Cost Technology with Isoquants and Isocosts

1. Find the least cost “iso-cost” line that will produce the desired level of output. That is, the least cost (IC) line that touches the 50 output Production isoquant

FIGURE 7A.6 Minimizing Cost of Production for qX = 50, qX = 100, and qX = 150Plotting a series of cost-minimizing combinations of inputs—shown in this graph as points A, B, and C— on a separate graph results in a cost curve like the one shown in Figure 7A.7.

FIGURE 7A.7 A Cost Curve Shows the Minimum Cost of Producing Each Level of Output

Finding the minimum cost inputs (quantities) for different levels of output

K

L

K

L

P

P

MP

MP isocost of slope isoquant of slope

At the point where a line is just tangent to a curve, the two have the same slope. At each point of tangency, the following must be true:

Thus,

K

L

K

L

P

P

MP

MP

Dividing both sides by PL and multiplying both sides by MPK, we get

K

K

L

L

P

MP

P

MP

The Cost-Minimizing Equilibrium Condition

isocost line

isoquant

marginal rate of technical substitution

1. Slope of isoquant:

2. Slope of isocost line:

K

L

MP

MP

L

K

A P P E N D I X R E V I E W T E R M S A N D C O N C E P T S