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Review Class Seven. Producer theory. Key sentence: A representative , or say, typical firm will maximize his profit under the restriction of technology and market structure. Hey, guys! What we are going to do is only to translate this key sentence. Translation is an easy job for you, Ok?.
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Producer theory • Key sentence: A representative , or say, typical firm will maximize his profit under the restriction of technology and market structure. • Hey, guys! What we are going to do is only to translate this key sentence. • Translation is an easy job for you, Ok?
Market structures • Perfect competition • Monopoly • Oligopoly (Duopoly) • Monopolistic competition • This term we will introduce above market structures, and now we focus on the simplest case of Perfect Competition.
Ch 18:Technological constraints • We can consider the firm as a BLACK BOX, which means we do not think about the Production Process ,(That is the matter of the operating managers and workers.) but we care the capability that transforms the inputs to outputs facing the typical firm――Technological constraint.
Technological constraints • Nature imposes the constraint that there are only certain feasible ways to produce outputs from inputs: There are only certain kinds of technological choices that are possible. • Precisely speaking, only certain combinations of inputs are feasible ways to produce a given amount of output, and the firm must limit itself to technologically feasible production plans. • So we can describe the technological constraints by listing the feasiblebundle of inputs and outputs.
Technological constraints • Def.1 of the production set: The set of all combinations of inputs and outputs that comprise a technologically feasible way to produce is called a production set. • Def.2 of the production function: The function describing the boundary of this set is known as the production function. It measures the maximum possible output that you can get from the given amount of input. • Warning: production function is the cardinal function.
Technological constraints • We often consider the two-input and one-output case. (Two inputs are often enough.) • Arithmetically, • Geometrically, we can use the definition of isoquants to describe the technology. An isoquant is the set of all possible combinations of inputs 1 and 2 that are just sufficient to produce a given amount of output.
Examples of technology (isoquants analysis): • Fixed proportions, • Perfect substitutes, • Cobb-Douglas.
Fixed proportion x2 Isoquants x1
Perfect subsitutes x2 Isoquants x1
Well-behaved isoquants • Monotonic (free disposal) To avoid the positive slope of any Isoquant • Convex: Given any two possible input bundles to produce the certain level of production, the weighted average of these two bundles can produce a higher level of output. To avoid the concavity case
Three Indicators of Technology • Marginal Product (MP) • Technical Rate of Substitution (TRS) • Returns to Scale (RS) • Relationship?
The Law of Diminishing MP The Law of Diminishing MP：It’s the assumption we usually apply. It’s a short-run concept.
Technical Rate of Substitution • Def 3: It measures the rate at which the firm will have to substitute one input for another in order to keep output constant. • Geometrically, TRS is just the slope of the given isoquant.
Technical Rate of Substitution • Roughly speaking, the assumption of diminishing TRS means that the slope of an isoquant must decrease in absolutevalue as we move along the isoquant in the direction of increasing x1, and it must increase as we move in the direction of increasing x2. Do you know how to prove it? • It is based on the assumption of well-behaved isoquant.
Ch 19Short-run and Long-run • In Microeconomics, Short-run and Long-run are based on the capability of adjustment of factors of production. • So when we analyze the problem of firms, we should think about both short-run and long-run. • In Macroeconomics, Short-run and Long-run are based on the capability of adjustment of price level.
Profit-maximization(short-run) • A representative , or say, typical firm will maximize his profit under the restriction of technology and market structure. • The value of the marginal product of a factor should equal its price. • Geometrically, find highest isoprofit line in the production set!
Profit-maximization(short-run) • Comparative statics: change w and p and see how x1 , y and respond?
产品价格 f(x1) High p Low p Low w1 x1 Comparative statics: • Increasing p increases x1 and then y.
Profit-maximization(long-run) • Profit-maximization, together with perfect competition and constant returns to scale implies 0 economic profit in the long run! • How to prove it ?
Deeper sight • Profit maximization problem facing a typical firm can be divided into two parts: • Part 1: Given any production level of ,how to choose the optimal inputs bundle to minimize cost? (Ch 20 and Ch 21) • Part 2: How to choose the proper output level of y, such that the profit can be maximized? (Ch 22 and Ch 23 are referred to perfect competition, while Ch 24 and Ch 25 are referred to Monopoly.)
Deeper sight • For part 1, you should remember that cost minimization problem is not the objective or the end, the purpose for us is to obtain the cost function, which measures the minimum cost of producing y units of output when factor prices are (w1,w2). • For part 2, having got the cost function, we will return to the profit maximization problem, and analyze the behavior of firm supply. Especially in the perfectly competitive market, we will see how the supply curve is proved. • In chapter 20 and 21, we will learn the cost analysis.
Cost minimization(long-run) • Geometric solution: Find the lowest isocost line along the given isoquant! • Some examples: • y = ax1 + bx2; Perfect substitutes • y = min{ax1 , bx2};Fix proportion • y = x1a x2b. Cobb-Douglas
The relationship between the objective function and the cost function (Take long-run for example) • When , the minimum of the objective function at point is , where , . • Please distinguish the objective function and the cost function seriously.
Three definitions of fixed costs • (Common) Fixed costs are the costs associated with the fixed factor: they are independent of the level of output, and in particular, they must be paid whether or not the firm produces output. • Quiz: Are there any fix costs in the long run? • Quasi-fixed costs are costs that are also independent of the level of output, but only need to be paid if the firm produces a positive amount of output. • Quiz: Can there be any quasi-fixed cost in the long run? • Sunk costs are the costs that can not be recovered. • Of cause, generally we only consider the variable cost and the common fixed cost.
AC MC AC AVC MC . AVC . y
Two examples • Specific cost function: • How to obtain the AC, MC and AVC?
Two examples • Marginal cost curves for two plants: • Suppose that you have two plants that have two different cost functions, and . • You want to produce units of output in the cheapest way. • How much should you produce in each plant?
Two examples • Arithmetically, • Geometrically, seek for the horizontal sum of MC curves, and solve for the value of MC at the given output level of y. Then setting each MC equal to that value yields to the desired output of each plant.
Two examples • The above arithmetic and geometric solution methods are the common methods for the case of any plants. • In the case of two-plant case, one shortcut can be used.
Approach 1:Long-run and short-run(average) cost curves (Please refer to the Dissertation.) • The long-run cost curve is the lower envelope of the short-run cost curves. • The long-run average cost curve is the lower envelope of the short-run average cost curves. • Here I will prove the first conclusion, and you should prove the second according to my method.
Approach 1:Long-run and short-run(average) cost curves • 1)Any one short-run cost curve is tangent to the long-run cost curve at some y. • 2)short-run cost curves are above the long-run cost curve.
Approach 2:Relationship between returns to scale and LAC (Please refer to the Dissertation.) • Constant R.T.S ______constant LAC • Increasing R.T.S ______downward –sloping LAC • Decreasing R.T.S _____upward-sloping LAC • How to prove?
