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Today Returns to Scale

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Today Returns to Scale 3/31/2016 1 Intermediate Microeconomics W3211 Lecture 14: Introduction Cost Functions and Optimal Output Columbia University, Spring 2016 Mark Dean: [email protected] 2 3 4 The Story So Far…. Today • We have set up the firm’s problem • Describe the relationship between returns to scale and • Converts inputs to outputs in order to maximize profit cost functions • Thought about how to solve the firm’s problem when there is • Solve the second part of the firm’s problem: only one input • E.g. economists who only need labor to produce paper • Choose the output level that maximizes profit given costs • Thought about how to solve the problem when the firm needs • i.e. the profit maximization problem to use more than one input to produce output • Think about comparative statics • Suggested that we can solve this problem by splitting it into two • i.e. what happens when we change the prices of inputs and outputs 1. Construct the cost function for the firm, by finding the lowest cost way of producing each output (the cost minimization problem) 2. Choose the output level that maximizes profit given these costs (the profit maximization problem) • Figured out how to solve the firm’s cost minimization problem 6 Returns to Scale and Cost Functions • In the last lecture we defined returns to scale for production functions • Decreasing returns to scale: doubling the inputs less than doubles the output Returns to Scale and Cost ,,,…,,,… • Constant returns to scale: doubling the inputs exactly doubles Functions the output ,,,…,,,… • Increasing returns to scale: doubling the inputs more than doubles the output ,,,…,,,… 5 1 3/31/2016 7 8 Returns-to-Scale Returns-to-Scale One input, one output One input, one output Output Level Output Level 2f(x’) y = f(x) y = f(x) 2y’ f(2x’) Decreasing Constant returns-to-scale returns-to-scale f(x’) y’ x’ 2x’ x x’ 2x’ x Input Level Input Level 9 10 Returns-to-Scale Returns to Scale and Cost Functions One input, one output Output Level • We showed that, a Cobb Douglas production function ,,,… … Increasing returns-to-scale y = f(x) Will exhibit • Decreasing returns to scale if …1 f(2x’) • Constant returns to scale if …1 • Increasing returns to scale if …1 2f(x’) f(x’) x’ 2x’ x Input Level 11 12 Returns to Scale and Cost Functions Returns to Scale and Cost Functions • Ultimately, what we are interested in is the firm’s costs • What are the marginal costs for a firm with Cobb Douglas production function? • In the worked example, we showed that for technology • Specifically, their marginal costs, which will determine optimal output , • We showed that, in the case of one input, that there was • Costs were given by a relationship between returns to scale and marginal costs , , • Because of the inverse function theorem It turns out (and you should check) that for technology , 1 Costs are given by ,, • E.g., decreasing returns to scale means increasing marginal costs Where A is a constant which depends on and 2 3/31/2016 13 14 Returns to Scale and Cost Functions Returns to Scale and Cost Functions ,, = 1 ,, What does this mean for marginal costs? Is this positive? Take the first derivative with respect to y Only if is greater than 1 ,, i.e. only if <1 = i.e. Only if the production function exhibits decreasing returns to Now take the second derivative scale ,, = 1 Is this positive or negative? i.e. are marginal costs increasing or decreasing? 15 16 Returns to Scale and Cost Functions Returns-to-Scale and Marginal Cost Marginal Cost • So, if there is only one input, or technology is Cobb Douglas decreasing returns to scale • Decreasing returns to scale if and only if marginal costs increase as increases • Constant returns to scale if and only if marginal cost unchanged constant returns to scale as increases • Increasing returns to scale: marginal cost fall as increases increasing returns to scale. y 17 18 Returns to Scale and Cost Functions Returns-to-Scale and Total Costs • What about Average Cost? What does this imply for the shapes of total cost functions? • Say marginal cost is always increasing Lets start by thinking about decreasing returns to scale • The cost of the making the second car is higher than the cost of making the first car And therefore increasing average cost • The average cost of producing two cars is higher than that of producing one car • Following the same logic • Decreasing returns to scale → increasing marginal cost → increasing average cost • Constant returns to scale → constant marginal cost → constant average cost • Increasing returns to scale → decreasing marginal cost → decreasing average cost 3 3/31/2016 19 20 Returns-to-Scale and Total Costs Returns-to-Scale and Total Costs $ $ c(2y’) Slope = c(2y’)/2y’ = AC(2y’). Slope = c(y’)/y’ Slope = c(y’)/y’ = AC(y’). = AC(y’). c(y’) c(y’) y’ y y’ 2y’ y 21 + 22 Returns-to-Scale and Total Costs Returns-to-Scale and Total Costs $ What does this imply for the shapes of total cost functions? c(y) Now increasing returns to scale c(2y’) Slope = c(2y’)/2y’ And therefore decreasing average cost = AC(2y’). Slope = c(y’)/y’ = AC(y’). c(y’) y’ 2y’ y 23 24 Returns-to-Scale and Total Costs Returns-to-Scale and Total Costs Av. cost decreases with y if the firm’s Av. cost decreases with y if the firm’s technology exhibits increasing r.t.s. technology exhibits increasing r.t.s. $ $ c(y) c(2y’) c(2y’) Slope = c(2y’)/2y’ Slope = c(2y’)/2y’ c(y’) = AC(2y’). c(y’) = AC(2y’). Slope = c(y’)/y’ Slope = c(y’)/y’ = AC(y’). = AC(y’). y’ 2y’ y y’ 2y’ y 4 3/31/2016 + 25 26 Returns-to-Scale and Total Costs Returns-to-Scale and Total Costs Av. cost is constant when the firm’s technology exhibits constant r.t.s. What does this imply for the shapes of total cost functions? $ c(2y’) c(y) Finally constant returns to scale =2c(y’) Slope = c(2y’)/2y’ And therefore constant average cost = 2c(y’)/2y’ = c(y’)/y’ c(y’) so AC(y’) = AC(2y’). y’ 2y’ y 28 Profit Maximization Remember that the firm’s original problem was 1. CHOOSE inputs and output - , ,,… 2. IN ORDER TO MAXIMIZE profit - ⋯ Solving The Firm’s Profit 3. SUBJECT TO technological constraints - y ,,,….) Maximization Problem 27 29 30 Profit Maximization Profit Maximization Which we split into two: First the cost minimization problem Let’s think a bit more about the profit maximization problem 1. CHOOSE inputs ,,… Using the Cobb Douglas case as an example. 2. IN ORDER TO MINIMIZE costs ⋯ Remember I claimed that for Cobb Douglas technology , 3. SUBJECT TO achieving output y ,,,….) This allowed us to identify the cost function ,,… The cost function is given by Which we can then use to solve the profit maximization , , problem 1. CHOOSE output And so profits by 2. IN ORDER TO MAXIMIZE profits ,,… 5 3/31/2016 31 32 Profit Maximization Profit Maximization Is this going to work? Depends on whether we have increasing, decreasing or How can we find the profit maximizing output? constant returns to scale Well, we could take derivatives with respect to y If we have decreasing returns to scale (i.e. <1) then marginal And use the first order conditions costs are increasing , 1 Which implies 33 34 Marginal Revenue and Marginal Cost Profit Maximization $ Marginal Cost In this case, setting marginal cost equal to price is optimal Marginal The cost of producing one more unit of output would be higher Revenue than the revenue of doing so This would lower profits We can also see this from the second order conditions y* y 35 36 Profit Maximization Profit Maximization Profits are given by 1 1 , ,… 1 Remember, from calculus, for us to have found a maximum, we need this to be negative First derivative is Which it will be if 1 is positive 1 i.e. <1 Second derivative is i.e. decreasing returns to scale 1 1 1 6 3/31/2016 37 38 Profit Maximization Marginal Revenue and Marginal Cost $ Marginal What about if we have increasing returns to scale? Cost Then marginal costs are decreasing Marginal Revenue y* y 39 40 Profit Maximization Profit Maximization ∗ Imagine we are producing at 1 1 1 What would happen if we produced one more unit of output? If 1 (i.e. increasing returns to scale) then this expression is Now the costs of additional output are below revenue positive ∗ If we wanted to produce at then we would certainly want to Means we have found a profit minimizing level of output produce more So, if the ‘tangency point’ is not the solution, what is? The more we produce, the lower the costs and the higher the profits A corner solution! ∗ is the profit minimizing output Produce infinite output (assuming marginal costs at some point get below price) You can see this from the second order conditions 41 42 Profit Maximization Marginal Revenue and Marginal Cost $ Marginal What about constant returns to scale? Cost 1 Marginal 1 Revenue = So in the constant returns to scale case marginal cost doesn’t depend on output y This leaves us with three possibilities 7 3/31/2016 43 44 Marginal Revenue and Marginal Cost Marginal Revenue and Marginal Cost $ $ Marginal Marginal Cost Cost Marginal Marginal Revenue Revenue y y 45 46 Profit Maximization Profit Maximization Putting this all together 1. Marginal costs above price Lose money on each item produced Diminishing returns to scale: MC increasing Optimal to produce zero Optimal output is an interior solution as long as there is a y such that price equals marginal cost What happens otherwise? 2.
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