2. Production Function Multiple Variable Input  The 3D production function can be compressed into a set of isoquants plotted in a 2D (K,L) space.  An isoquant shows all the input mixes of (L,K) that produce the same amount of Q.  One isoquant for each level of Q  Isoquants have negative slope: more output from one input can offset less output from the other input

4. Isoquant  The shape of the isoquant reveals the degree of substitution between the inputs.  If the technology allows for perfect substitution between labor and capital, the isoquants are straight lines.  The inputs can substitute each other at the same rate to produce the same output. e.g. each unit of K can always be replaced by 2 units of L.

5. Isoquant  In reality, inputs are imperfect substitutes and the isoquants are convex to the origin.  Check: as you move up an isoquant in Fig 7.4, you need to use more and more capital to replace each unit of labor in order to stay on the same isoquant.  The magnitude of the slope of the isoquant is greater as you move “up” an isoquant.

7. Isoquant  Slope of an isoquant dK/dL is called the marginal rate of technical substitution.  MRTS is the amount of capital input (dK) needed to replace an amount of labor input (dL) to keep the same level of output: it measures the degree of substitution.  Q = Q(K, L).  As you move “up” along an isoquant, the total differential of Q is zero

8. Isoquant  dQ = ( ∂ Q/ ∂ L) (dL) + ( ∂ Q/ ∂ K) (dK) = 0 (MP L ) (dL) + (MP K ) (dK) = 0 (dK)/dL = - MP L /MP K < 0  MRTS at different points on the isoquant is different since MP L and MP K are changing.  As you move “up” an isoquant, MRTS becomes larger “in magnitude” (more negative).

9. Isoquant  This is the result of the law of diminishing marginal productivity (LDMP).  LDMP states that “as you increase the amount of one input while keeping other inputs fixed, the marginal product of this input declines with output.  LDMP is a physical constraint – we can’t escape from it.

10. Isoquant  If LDMP is true, so is a weaker version of it: “as you increase the amount of one input while reducing other inputs, the marginal product of this input declines with output.  As we use less labor and more capital inputs along an isoquant, MP K , MP L   MRTS = (dK)/dL = - MP L /MP K  “magnitude” of MRTS  (more negative)

11. Optimal level of Input Mix to Employ Multiple variable input  For a profit max output Q, there are now many input mixes of L and K (along the isoquant for this output Q) that can be used to produce Q  Which input mix is cheapest? You need to take into account inputs prices

12. Isocost line  The amount of input you can employed depends on (1) their prices and (2) budget.  Define: an isocost line consists of all the input mixes (L, K) that can be employed with an expenditure budget M and a set of inputs prices of P L and P K.

13. Isocost line  Isocost line are all the (L, K) such that  P L L + P K K = M  K = M/P K – (P L /P K ) L  Plot this eqt in the K-L space  Slope of isocost line = – (P L /P K )  X-axis intercept = M/P L  Y-axis intercept = M/P K

15. Isocost line  If M double, the line shifts out parallelly and the intercepts doubled (same effect if inputs prices are halved)  If both M and input prices are doubled, the isocost line is unchanged  If the price of labor inputs , the isocost line rotate: steeper and the x-axis intercept decreases.

16. Optimal level of Input (Mix) to Employ Multiple variable input  Suppose we know the profit max output Q.  Plot the isoquant of Q  Given input prices P L and P K, plot several isocost lines, each corresponds to a different expenditure budget M  The optimal input mix is one such that the isocost line just “touch” the isoquant of Q