5.1 THE ROBINSON CRUSOE ECONOMY

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1 Essential Microeconomics -- 5 THE ROBINSON CRUSOE ECONOMY Ke ideas: Walrasian equilibrium allocation, otimal allocation, invisible hand at work A simle econom with roduction Two commodities, H consumers, one firm, an aggregate endowment vector ω = ( ω,0) Commodit can be consumed or used as an inut in the roduction of commodit The firm, with strictl convex roduction set Y utilizes commodit to roduce commodit h h h Consumers have identical convex homothetic utilit function U = U( x, x), h=,, H Given the assumtion of identical homothetic references the consumers can be reresented b a single consumer Robinson Crusoe who has an endowment of ω John Rile October 9, 0

2 Essential Microeconomics -- The Otimum x, If Robinson Crusoe, with endowment vector ω, chooses x = + ω the roduction lan, his consumtion vector is x= + ω Y Y+ { ω} U x ( ) = U( x ) The set of feasible consumtion bundles is therefore the set ω x, Y + ω This set is also deicted in Figure Figure : The Otimum Robinson Crusoe then chooses the consumtion bundle x that maximizes his utilit from the bundles in the set Y + { ω} Formall, x solves the following maximization roblem: Max{ U ( x) x Y + { ω}} x John Rile October 9, 0

3 Essential Microeconomics -3- Examle: Y = {(, ) 0, }, U( x) = ln x + ln x, ω = (,0) We substitute for x= + ω and write utilit as U( + ω) = ln( ω+ ) + ln Because utilit is increasing, the otimum must be on the boundar of the roduction set so that = / 64 Substituting for ω and, FOC: U = ln( / 64) + ln du d /3 = + = / 64 0 Hence = 3 64 Solving, = 6 and so = ( ) / 64 = 4 Hence = ( 4,6) and x = + ω = (8,6) John Rile October 9, 0

4 Essential Microeconomics -4- Suorting herlane Since x is the otimal consumtion bundle the Interior of the uer contour set X = { x U( x) U( x )} = x= x = ( + ω) x, and the set Y+ { ω} is emt Therefore x = + ω x = + ω is on the boundar of the sets Y+ { ω} and X Y Y+ { ω} U x ( ) = U( x ) The line x= x is drawn tangent to the boundaries of ω x, these two sets Since the sets are both convex, the line is a suorting line Figure : Suorting lines for both sets That is (i) x x for all x Y + { ω} and (ii) x> x for all x int X John Rile October 9, 0

5 Essential Microeconomics -5- Robinson the Manager (i) x x for all x Y + { ω} Therefore ( + ω) ( + ω) for all Y, equivalentl, for all Y, Note that with a rice vector (, ) the total revenue of the firm is and total cost is ( ) and so the rofit of the firm is π (, ) = Therefore Robinson the manager maximizes the rofit of the firm b choosing the roduction lan John Rile October 9, 0

6 Essential Microeconomics -6- Crusoe the Consumer At home Robinson becomes Crusoe the reresentative consumer Given the rice vector, the value of Crusoe s endowment is ω As the single consumer, Crusoe also receives a dividend equal to the firm s rofit Thus his total income is ω+ = ( ω+ ) = x and so his budget constraint is x x We have argued that (ii) x> x for all x int X Thus an strictl referred consumtion bundle costs strictl more Therefore x is a maximizer for Max{ U( x) x x = ( + ω)} x The otimal allocation is therefore a WE allocation John Rile October 9, 0

7 Essential Microeconomics -7- Examle (continued): Robinson the rice taking manager Robinson the manager solves the following maximization roblem: Max{ Y }, that is Max{ 0, } For a maximum the constraint must be binding Then substituting for = / 64, rofit is / 64 + Solving for the rofit-maximizing outut we obtain, = = = and maximized rofit is 6 Π ( ) = ( ) 3 Thus ( ) ( ) / 64 6( ) John Rile October 9, 0

8 Essential Microeconomics -8- Crusoe the Price-Taking Consumer Next consider the choice of Crusoe the consumer The value of his endowment is ω In addition, as the single shareholder in the econom he collects all the dividends Π ( ) = ( ) His sending on corn is therefore constrained as follows: x ω +Π ( ) He therefore solves the following maximization roblem Max{ln x + ln x x Π ( ) + ω} x Because utilit is strictl increasing, the budget constraint must be satisfied with equalit at the maximum From the FOC, and aealing to the Ratio Rule, U U x x = = = = x x x + x ω +Π( ) Π ( ) + ω Therefore x( ) = ( ) John Rile October 9, 0

9 Essential Microeconomics -9- Walrasian Equilibrium We have alread seen that 6 ( ) 3 and ( ) = Π = 6 Therefore demand for commodit is x( ) = ( ) + It follows that excess demand for commodit is ( ) = ( ) ( ) = ( + ) = 6 ( 4( ) z x = = 0 if Thus = (,) is a WE rice vector and the WE allocation is x ( ) = (6,8) Class Exercise: Wh must the other market clear as well? Exercise: Show that the WE allocation is the same if there are 4 identical firms, each with a f f f f roduction set Y = { 6 + ( ) 0} John Rile October 9, 0