Managerial economics guide 8

Firm choice over production plans and technologies

Concepts

Technology set ℎ This is a possible production plan or technology set. All Ys which are feasible lie within this set. Only the production plan within the production set can be carried out. We want to simplify now to essential parts. We only consider one goods as output and others are input = ( , , … ) ⏟ 1 2 −1 Y=output produced. Because output is positive and inputs are negative, we can write the production plan as: (− , − , … , − , ) 1 2 −1.

Efficient Plan

An efficient plan produces the given output with the least amount of input and produces maximum output from the given input. We use the production function to represent the production plan: = (). It gives us the maximum output with a given input (X), defining the efficient frontier of your technology. Taking a random point of X’, Y’. This is a feasible production plan but as you can see, you can use a small amount of input (X*<X’) to produce Y’ and vice versa (Y*>Y’). So, the efficient production plans are captured by the production function.

Production Functions

Just as in the consumer part, we have families of production functions, these functions are also empirically fit models:

• Cobb-Douglas
• Leontief
• Linear = min( ; 1 2 = + 1 21 21−, > 0 = Represents convex technology, perfect complements productive functions, perfect substitute productive functions

Marginal Productivity

The marginal product of factor “i” tells us by how much the production increases when we increase a little that factor.

Marginal productivity in Physical term: ∗
Marginal productivity in Value term: Law of diminishing marginal utility.

We often assume that the production factor is characterized by decreasing marginal productivity meaning that as you increase the quantity of the input, the marginal productivity will fall. The idea is you have constraints in your tech, once you fixed all input but once, you are getting an increase in output but the increase marginal is getting less and less because you can’t use that input in isolation.

Isoquant and the Convexity of Technology

The isoquant of production is the set of vectors which produce the same output. This represents convex technology, convexity in technology implies that when your bundle falls out of the curve you will get something less than efficient. To identify convex combination of two points A and B a segment and an average point. If we move the isoquant upward we have more input inside, which represents monotonic technology, implying that more input you use the more output you get.

Production Function Graphs

Cobb-Douglas, Leontief, Linear
Production min( ; = + = = 1 2 1 21 2 Function ̅ Efficient Because values are cardinal, we cannot = → = − 2 1 2 1 combination convert them using monotonic transformation, instead we have to work with what we have.

Implicit expression: ( , =1 2 21 = + = 0 → = − → − = | 1 2 1 2 2 2 1 1 1 ̅− 2. This is ultimately the derivative slope. The slope of the isoquant at the tangent point is called the marginal rate of technical substitution.

Return of Scale

Returns scale tell you what happens when you multiply both and 1 2 = ( , ) 1 2. Constants returns of scale means that if you multiply the output by kappa then all inputs have to be multiplied by kappa () = ().

Increasing returns means that () > () > 1.
Decreasing returns will be () < () > 1.

Models

Profit maximization
Cost minimization: max (, , ) = − − min( , = + 1 2 1 1 2 2 1 2 1 1 2 2) ). = ( , . ( , ≥ 1 2 1 2ℎ , = ( , )1 2( = , , )1 1 1 2( = , , )(, );→ , (, , ) 2 2 1 21 1 2 2 1 2

Conditional Demand and Market Demand

Market demand → ( , , ) = )=→ (, , market supply function 1 21 2

Profit maximization: max (, , ) = − − 1 2 1 1 2 2 s.t. ) = ( , 1 2. In that: = ; = ; = 1; = 11 1. Plugging the constraint function to the objective function we get: ) )max ( , = ( , − − 1 2 1 2 1 1 2 2

If the second derivative condition is satisfied, which basically means assuming the profit function is concave, we have to look at the first condition: =∗ − = 0 → ∗ = 1 1 1 1 1 max ↔ =∗ − = 0 → ∗ = 2 2{ 1 1 2.

Marginal Product

Indicate how much output goes up when the quantity of each factor goes up by a small amount in physical terms. If we multiply the increases in output by price, these become value terms or revenue, so how much revenue goes up or how much more value we get by increasing a small amount of the factors.

Efficient Point and Optimal Mix of Inputs

What is the efficient point? We employ those two factors at to a point where what we get more in terms of revenue is exactly balanced by what we pay more by producing a bit more of that product. If MR is higher than MC of that factor, I will use more of that factor. When the congestion effect kicks in and MR starts to decrease as I increase the factor, I will decrease the quantity until MC=MR again. The marginal condition is basically MR=MC. By dividing MR and MC by p we are expressing this condition in a physical term.

To obtain the optimal mix of inputs and gain optimal return, the MRTS which is the ratio of marginal productivity must be equal to the relative factor prices. At the optimum return, we have MRTS as above 11 = 22.

Market Demand for Input Factor

Market demand for input factor (, )→ , 1 1 2→ (, , )2 1 2. Market supply function → :)(, , 1 2. So, you want to produce supply based on the price and demand of input on the market.

Cost Minimization

Cost minimization )min( , = + 1 2 1 1 2 2. The problem here is to minimize cost, subjecting to a constraint )( , ≥ 1 2. The production needs to get at least the given output. Tech is monotonic so if you use more input you will not get less input. If we have to minimize x1 x2, we can minimize this as )( , = 1 2. Minimizing cost using the Lagrange function ))( , , ) = + + ( − ( , 1 2 1 1 2 2 1 2 = − ∗ =01 1 1 = − ∗ =02 2 2 11→ = 22. This is the same as in profit maximization so when we are maximizing profit, we are also minimizing cost but we had more conditions for maximizing profit. It is always possible to minimize cost because you already have a fixed y (output is decided in advance) and therefore this becomes a compact set. By the bias criterion, compact continuous functions always have a maximum and