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Unformatted text preview: with identical CobbDouglas 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 CobbDouglas
production function where
( and
) ⏟ ⏟ ⏟ . Then: ()
⏟ 17
ECO 204 Chapter 15: Competitive Firms and Markets (this version 20122013) 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 CobbDouglas
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 20122013) 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 CobbDouglas utility functions
and individualspecific 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 CobbDouglas 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 20122013) 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 20122013) 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.
 Fall '14

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