ajaz_eco_204_2012_2013_chapter_15_Competition

Then 17 eco 204 chapter 15 competitive firms and

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Unformatted text preview: with identical Cobb-Douglas production functions (note how capital is fixed). We can solve each firm’s Cost Minimization Problem (CMP) to derive the optimal labor required to produce an arbitrary target output . From this we can get the short run cost function: You should solve this CMP and show that: ( ) ⏟ ⏟ For example, suppose there are firms (A, B) producing output according to the decreasing returns Cobb-Douglas production function where ( and ) ⏟ ⏟ ⏟ . Then: () ⏟ 17 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. Noting that we have: { The market supply curve is: { As another example, suppose there are firms (A, B) producing output according to the constant returns Cobb-Douglas production function where ( Noting that and ) ⏟ ⏟ ⏟ . Then: () ⏟ we have: { The market supply curve is: { __________________________________________________________________________________________________ 5. Competitive Market Equilibrium We have seen how to derive the market demand curve by aggregating individual rational consumers’ demand curves, and the market supply curve by aggregating individual rational firms’ (with decreasing/constant returns) supply curves. The equilibrium “market clearing” price is where aggregate demand equals aggregate supply: From the equilibrium price we can calculate aggregate demand and aggregate supply, as well as the individual quantities demanded by consumers and supplied by producers. Let’s do a “super example”. 18 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. __________________________________________________________________________________________________ Example: A competitive market consists of: ● 3 price taking consumers with identical Cobb-Douglas utility functions and individual-specific incomes and where (i.e. good 2 is the “base” good). As shown earlier, we can solve each consumer’s UMP (do it!), derive their individual demand functions for good 1, and from this we can derive the market demand function for good 1 as: Dropping the subscript for good 1 we have the individual demand function (recall all consumers have the utility parameters and incomes): From this we have the total quantity demanded: ∑ ∑ From this we have the market demand curve: ● 2 price taking firms with identical Cobb-Douglas production functions (note how this production function has decreasing returns and constant returns to scale) with capacity and . As shown earlier, we can solve each firm’s Cost Minimization Problem (CMP) to derive the optimal labor required to produce an arbitrary target output from which we can get their short run cost functions, from which we can get their individual supply curves, and from which we can get the market supply curve: ( Noting that ) ⏟ () ⏟ we have: 19 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. { The market supply function is: { The market supply curve is: { ● The market demand and supply curves are shown below: notice how the supply curve spikes up at industry capacity): Demand Curve (total Supply Curve Industry Capacity The equilibrium price “clears” the market: From the graphs we see that demand and supply curves “cross” at an output . Put another way, the curves cross at a price (how do we know this from the equations?) and we can solve for the price by equating market demand to the market supply corresponding to : 20 ECO 204 Chapter 15: Competitive Firms and Markets (this version 2012-2013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. √ From this: √ √ Notice how this is below industry capacity (had we gotten a price we would’ve said and ). From the equilibrium price we can solve for quantity demanded by each consumer and quantity supplied by each producer. First, the quantity demanded by each individual consumer (recall we assumed all consumers have the same utility parameters and incomes): √ Check to make sure that ( √ ) √ √ (yay!). Next, the quantity supplied by each individual firm (recall we assumed all producers have the same production function parameters and face identical input prices): √ Check to make sure that (yay!). I hope you appreciate the depth of this example: we have derived individual demand functions of price taking consumers from their preferences and budget constraints; we have derived individual supply functions of price taking firms from their “production processes” and input prices; we have combined consumers and firms in a competitive market and derived the “price” from which we can back out individual quantities demanded and supplie...
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This document was uploaded on 01/19/2014.

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