Cost Concepts for Decision Making and Cost-Minimization Problem

Cost Concepts for Decision Making:

Costs:

a) Explicit costs which involve a direct monetary outlay.
Indirect costs that don’t involve outlays of cash.

b) Accounting Costs (total sum of explicit costs)
Economic Costs (sum of explicit and implicit costs)

c) Opportunity Costs

d) Sunk (unavoidable) costs v/s Non-sunk (avoidable) costs, Initial set-up costs.

Cost-Minimization Problem:

a) iso-cost curve:

wL + rK = TC = C,  K =[-(w/r) L]+ C/r

1338_cost minimization problem.jpg

b) Cost minimization:

At point F, (slope of iso - cost curve) = (slope of isoquant curve). So, w/r = MRTSLK
And we already know that MRTSLK = MPL/MPk= w/r or MPL/w= MPk/r

The last expression implies that the additional output per dollar spent on labor services equals the additional output per dollar spent on capital services (“equal bang for the buck”).

You are able to check the condition at points E and G.

How in regards the corner point solutions? Does the above equation grapple at the corner solution?

In general if we have m units of inputs, the cost minimization condition (FOC) is:
  
c) Expansion Path:

Q0 < Q1< Q2

d) Comparative Statics Analysis of Changes in Input Prices: Demand for inputs is Derived Demand depending upon change in quantities of final products.

So, we can’t use the method in consumption theory to derive input demand curves.

To analyze the effect of change in input prices without considering change in quantities, we need to confine our analysis to change in input combinations that can produce same amount of output (analogous to consumption theory of only substitution effect!).

940_comparative statistics.jpg

e) Cost Minimization in SR: When the firm’s capital is fixed at K, the short-run cost-minimizing input combination is at point F. If the firm were free to adjust each of its inputs the cost-minimizing combination would be at point A.

1827_cost minimization in SR.jpg

min(L,K)wL + rK
subject to: Q = f (L, K)

We proceed by defining a Lagrangean function

Λ(L, K, λ ) = wL + rK +λ[ f (L, K) − Q], where λ is a Lagrange multiplier.

The conditions for an interior optimal solution (L > 0, K > 0) to this problem are:

1394_optimal solution for langrange multiplier.jpg

From (1) and (2), we can get MPL/MPK = w/r ............(4)

Equations (3) and (4) are two equations with two unknowns, L and K. They are alike to the conditions that we derived for an interior solution to the cost-minimization problem using graphical arguments. The solutions to these conditions are the long-run input demand functions:

L*(w, r, Q) and K*(w, r, Q) .

Example: Production function is Q = 50 √LK . What are the demand curves for labour as well as capital?

f) Duality: “Backing Out”

The above analysis shows how we can start with a production function and derive the input demand function. However we can also reverse directions: If we begin with input demand functions we can characterize the properties of a production function and sometimes even write down the equation of the production function. This is because of duality, which refers to the correspondence between the production function and the input demand function.

Example- Suppose we are given respective labour demand function and capital demand function:

L = Q/50(√r/w) and K = Q/50 (√w/r) Solving for w in terms of Q, r, and L: w = (Q/50L)2r

Plugging the last expression in capital demand function:

K = (Q/50) [(Q/50L)2r]1/2 = Q2/2500L Finally, Q2 = 2500LK, Q = 50 √LK = 50L0.5 K0.5

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