M. Muniagurria Econ 464 Microeconomics Handout (Part 1) I. TECHNOLOGY : Production Function, Marginal Productivity of Inputs, Isoquants (1) Case of One Input: L (Labor): q = f (L) • Let q equal output so the production function relates L to q. (How much output can be produced with a given amount of labor?) • Marginal productivity of labor = MPL is defined as q = Slope of prod. Function L Small changes i.e. The change in output if we change the amount of labor used by a very small amount. • How to find total output (q) if we only have information about the MPL: “In general” q is equal to the area under the MPL curve when there is only one input. Examples: (a) Linear production functions. Possible forms: q = 10 L| MPL = 10 q = ½ L| MPL = ½ q = 4 L| MPL = 4 The production function q = 4L is graphed below. -1- Notice that if we only have diagram 2, we can calculate output for different amounts of labor as the area under MPL: If L = 2 | q = Area below MPL for L Less or equal to 2 = = in Diagram 2 8 Remark: In all the examples in (a) MPL is constant. (b) Production Functions With Decreasing MPL. Remark: Often this is thought as the case of one variable input (Labor = L) and a fixed factor (land or entrepreneurial ability) (2) Case of Two Variable Inputs: q = f (L, K) L (Labor), K (Capital) • Production function relates L & K to q (total output) • Isoquant: Combinations of L & K that can achieve the same q -2- • Marginal Productivities )q MPL ' Small changes )L K constant )q MPK ' Small changes )K L constant )K • MRTS = - Slope of Isoquant = Absolute value of Along Isoquant )L Examples (a) Linear (L & K are perfect substitutes) Possible forms: q = 10 L + 5 K Y MPL = 10 MPK = 5 q = L + K Y MPL = 1 MPK = 1 q = 2L + K Y MPL = 2 MPK = 1 • The production function q = 2 L + K is graphed below. MRTS = 2 -3- Marginal Productivities are constant in all these examples, then MRTS = constant (b) Fixed coefficients (i.e. no factor substitution) Possible Forms • Example (i) Unit Labor requirement = aL = 2 Unit Capital requirement = aK = 1 aL= Amount of L needed to produce one unit of output aK = Amount of K needed to produce one unit of output -4- q = L Average Productivity of Labor = Average Productivity of Capital = Example (ii) Average Productivity of Labor = APL = q/L = 1 Average Productivity of Capital = APK = q/K = ½ Notice that the APL = 1/aL and APK = 1/aK MRTS not easy to calculate. -5- (c) “Nice Isoquants” Possible Forms: Slope of Isoquants in absolute value 9 as L8 ( i.e. 9MRTS as L8, as L8it becomes more and more costly to replace capital.) II. COST MINIMIZATION How to choose inputs to minimize cost: (1) One input: Trivial: Price of labor Total Cost = w. L , where w is the price of labor (2) Two variable inputs (a) Fixed coefficients: No substitution (still easy) Let r be the price of capital, then the cost of producing one unit of output (Average cost: AC) is: ' % AC aL . w aK . r Notice that this does not change for different levels of output: The AC is constant. -6- If either w or r change, the AC changes but the firm still uses aL units of L and aK units of K to produce one unit of output. (b) Linear Isoquants (perfect substitutes) (Hard - we will skip this) (c) “Nice Isoquants” - Define Isocost lines - Find tangency between Isoquant and Isocost lines Isocost = c = w.L + r.k (Combinations of L & K that cost same amount of money) Example: w = 5 , r = 2 Find the Isocost for $10 5 Slope of Isocost ' 2 In General: cw w Slope of Isocost ' r Example: Cost minimizing combination of K & L to produce q = 10 w slope ' r -7- Effect of changes in prices: •8w, r constant (i.e. 8w/r) Y 9 L* &8 K* New Combination: K$,L$ K$ •8 r w constant (i.e. 9 w/r) Y 8 L* & 9 K* III. PROFIT MAXIMIZING FACTOR DEMANDS We assume that each firm maximizes profits - taking the product price as given (P) and - taking input prices as given (w, r) How many units of L & K should the firm hire? (1) Case of One Variable Input: Labor (L) A profit maximizing firm will hire L up to the point at which: Marginal product x output price = input price In our case: MPL . p = w Marginal Marginal cost benefit for of hiring an extra the firm unit of labor Y MPL = w/p -8- Example: Production function with decreasing MPL (Think about a fixed factor). (2) Case of Two Variable Inputs: L & K “Nice Isoquants” IV. DISTRIBUTION OF INCOME AMONG PRODUCTIVE FACTORS (FIXED & VARIABLES) (1) Case of one variable input with decreasing marginal productivity due to the existence of a fixed factor. Example: Think of the fixed factor as land or entrepreneurial ability. Labor is the variable factor. L* is the profit maximizing amount of labor. Remember: Area below MPL to the left of L* is total output produced with L* (Let’s call it q*) -9- Total Revenue p . q* = w . L* + Income to the Fixed Factor 99 Labor Residual Income OR q* = w/p . L* + Income to Fixed Factor/P 99 Real Labor Real Income of Income The Fixed Factor (yL) (yF) Then yF = q* - yL Notice in Diagram that: yF = •What happens if real wage (w/p) goes up? Y 9 Profit max. amount of labor AND The surplus that the fixed factor goes down (i.e. yF 9 ) -10- • What happens if p8? Y w/p 9Y More labor is hired Y Surplus to the fixed factor 8 i.e. yF 8 “The owners of the fixed factor, say land, benefit when the price of corn 8”. 464part1.wpd -11-.