Theory of the Firm Firms want to maximize profit This implies minimizing cost Need to identify underlying technological relationships between inputs and outputs.

Factors of Production Inputs Broadly – labor, land, raw materials, capital Motion picture studio: –producers, directors, actors, lots, sound stages, equipment, film. Electricity Generator –Managers, technicians, coal, generation equipment, (clean air)

Specification of Technology – one output Production function: f(x) = {y such that y is the maximum output associated with –x} –Cobb Douglas f(x1,x2) = x1 a x2 (1-a) –Leontiff f(x1,x2) = min(ax1,bx2) –CES f(x1,x2) = [ax1 r + bx2 r ] 1/r –Linear f(x1,x2) = x1 + x2 Sketch the Cobb Douglas, Leontiff, and Linear Isoquants

Specification of Technology – one output Cobb Douglas Isoquants f(x1,x2) = x1 a x2 (1-a)

Specification of Technology – one output input 1 input 2 Q(y1) Q(y2) slope a/b Leontiff isoquant f(x1,x2) = min(ax1,bx2)

Technical Rate of Substitution The technical rate of substitution is the amount that you need to adjust one input in order to keep output constant for a small change in another output. This is equivalent to the slope of the isoquant. If the production function is f(x1,x2) what is the technical rate of substitution?

Elasticity of substitution The elasticity of substitution measures the curvature of the isoquant:  = d(x1/x2)/(x1/x2) d(TRS)/TRS The higher , the less curved the isoquant is, and easier it is to substitute between inputs.

Elasticity of substitution Input 1 Input 2 Constant Elasticity of Substitution f(x1,x2) = [ax1  + bx2  ]   )  = ∞  = 1 (Cobb Douglas)  = - ∞

Example Let f(x1,x2) = Find the TRS Then, try to find the elasticity of substitution, using the ln formulation.

Elasticity of substitution is a good measure of flexibility.

Returns to Scale Constant returns to scale: a doubling of inputs will result in a doubling of output. –f(tx) = tf(x) the production function is homogeneous of degree 1. Increasing returns to scale –f(tx) > tf(x) –example: fixed costs Decreasing returns to scale –f(tx) < tf(x) –example: a fixed input

Returns to Scale: Examples Bausch and Lomb. A 12-oz bottle of saline solution costs $2.79. A 1-oz bottle of eye drops costs $5.65. People’s Express. Very successful, attributed partly to management practices (minimal hierachy, training, profit sharing, performance pay). Grew from 300 to 5000 employees. Active involvement became difficult, more hierarch necessary, output increased less than input. Oil shippers: unlimited liability in case of a spill. Therefore a small firm with only one ship is preferable.

Short run vs. Long run In the short run some inputs may be fixed. In the long run, we generally consider all inputs to be variable. Example: Capacity

Profit Maximization Basic assumption of Economics. Is it right? The firm takes actions that will maximize the total revenues – costs. Max R(a) – C(a) F.O.C. R’(a) = C’(a) MR = MC

Profit Maximization - examples The level of output should be chosen so that the cost of producing the last unit of output is equal to the revenue from that unit. In 1962 Continental Airlines filled only 50% of certain flights. It considered dropping some of these flights, but each flight had a (marginal) cost of $2000 and revenue of $3100.

Profit Maximization Revenue = the price of what is sold X the amount that is sold Cost = the price of the inputs X the amount of input used. Technological Constraints Market Constraints Assume that the firm is a price-taker (competitive firm)

Profit Maximization- price taker Max pf(x) –wx foc characterizes profit-maximizing behavior. “value marginal product = its price” What is the requirement on f(x) for a maximum to exist? What is the SOC?

Profit, supply, and demand functions demand function x(p,w) = argmax pf(x) – wx Supply function y(p,w) = f(x(p,w)) Profit Function  (p,w) = py(p,w) – wx(p,w)

Profit, supply, and demand functions A function is homogeneous of degree n if f(tx)=t n x If a function is homogneous of degree 0, then doubling all its inputs doesn’t change the output. If a function is homogenous of degree 1, then doubling all inputs doubles the outputs.

Profit, supply, and demand functions Consider the demand function x(p,w). What happens if you double all prices? Consider the profit function,  (p,w). What happens if you double all prices? What about the supply function? What does this say about behavior under inflation?

Properties of the Profit Function  (p,w) = py(p,w) – wx(p,w) Increasing in output prices; decreasing in input prices Homogeneous of degree 1 in prices. Convex in p (!) These properties follow only from the assumption of profit maximization.

The profit function is convex  (p,w) = Max pf(x) –wx Prove it using the envelope theorem

Alternative proof that the profit function is convex Let y max profits at p, y’ at p’, and y’’ at p’’, where p’’ = tp + (1-t)p’ Then  (p’’) = p’’y’’ = (tp + (1-t)p’)y’’ = tpy’’ + (1-t)p’y’’ < tpy + (1-t)p’y’ = t  (p) + (1-t)  (p’) Why is this true?

The profit function is convex output price profits p*  (p*) py*-w*x*  (p)

Profit function Suppose the price of output is randomly fluctuating. Is it desirable to stabilize this price? Jensen’s inequality:  (E[p]) < E[  (p)] for any convex . Thus, firms do better when price fluctuates! Because they change their production plan accordingly. When might this not be true?

Hotelling’s Lemma Use the envelope theorem to find

Hotelling’s Lemma Use the envelope theorem to find

Example Consider a firm with 3 inputs – x, e c,e nc with prices w,p c,p nc Carbon emissions are equal to e c Carbon is taxed at level t Thus total price of e c is p c +t Now consider technical change that reduces the carbon intensity. Let technical change  reduce the carbon intesity from 1 to (1-  )

Example-continued After technical change the total price for e c would be p c +(1-  )t The profit function is  (w, p c +(1-  )t,p nc )

Cost Minimization In order to maximize profit, a firm must be minimizing the cost of producing its output. Cost minimization is an alternate characterization of price-taking firms. Max py – c(w,y) Where c(w,y) = min wx s.t. y = f(x) Cost minimization is equally valid for other types of firms

Cost Minimization Min wx s.t. f(x) = y Lagrangian L(,x) = wx – (f(x) – y) F.O.C. w i - df(x*)/dx i = 0 f(x*) = y

Cost Minimization Divide the ith constraint by the jth df(x*)/dx i = w i df(x*)/dx j w j Economic rate of substitution = TRS

Cost Minimization Input 1 Input 2 f(x1,x2) = y isocost line. slope = -w1/w2

Cost Minimization Input 1 Input 2 f(x1,x2) = y isocost line. slope = -w1/w2 How does this differ from the utility maximization problem?

Example Graph the demand for factor x1 as a function of its price w1 for the 3 production functions below. x1 x2 x1 x2 x1 x2

Kuhn-Tucker Theorem In order to solve constrained optimization problems when there might be a corner solution, we need to use the Kuhn-Tucker Theorem.

Cost Minimization df(x*)/dx i = df(x*)/dx j w i w j Marginal product per $ One dollar invested in x i increases output by df(x*)/dx i w i To minimize cost you must equalize the rates of return on each input.

Cost minimization - examples parking lots: tall versus expansive

Properties of the Cost Function Increasing or decreasing in w? Homogeneous of degree ? in w? Convex or concave?

Elasticity of substitution as a measure of flexibilty Input 1 Input 2  = ∞  = 1 (Cobb Douglas)  = - ∞