; p. p y p y p y. Production Set: We have 2 constraints on production - demand for each factor of production must be less than its endowment

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1 Exercise 1. Consider an economy with produced goods - x and y;and primary factors (these goods are not consumed) of production A and. There are xedcoe±cient technologies for producing x and y:to produce one unit of x;you require units of A and one unit of ; to produce one unit of y;you require one unit of A and one unit of. The economy is endowed with 150 units of A and 100 units of : De ne mathematically the production possibility set of this economy and illustrate it graphically.. Suppose that there us a single consumer in the economy with utility function u (x; y) =logx +logy Find the set of Pareto-optimal allocations. Find the competitive equilibrium prices µ px ; p A ; p Production Set: We have constraints on production - demand for each factor of production must be less than its endowment Factor A : x + y 150 Factor : x + y 100 The shaded region where both constraints hold is the production possibility set Y Pareto Optimal Allocation: The Pareto-optimal allocation is where the highest indi erence curve is tangent to Y:The optimization problem is subject to The Lagrangian is max logx +logy x;y x + y 150 x + y 100 $ =logx +logy + 1 (150 x y)+ (100 x y)

2 y x+ y = 150 x+ y = 100 x Figure 1: The rst order Kuhn-Tucker conditions are µ x 1 0; x 0; x x 1 =0 µ y 1 0; y 0; y y 1 =0 (150 x y) 0; 1 0; 1 (150 x y) =0 (100 x y) 0; 0; (100 x y) =0 From the utility function x =0ory = 0 cannot be optimal. Therefore, the rst two order conditions become x 1 = 0 y 1 = 0 Thereare4casehere: Case (1): 1 = =0: This implies x = y =0

3 which can be ruled out. since x; y are nite Case (): 1 > 0; =0: This implies This implies that (150 x y) = 0 x 1 = 0 y 1 = 0 x =45;y =60; 1 = 1 0 ut here x + y = 105 > 100 so the second feasibility condition is violated. So we can rule out this case. Case (): 1 =0; > 0: This implies This implies that (100 x y) = 0 x = 0 y = 0 x =60;y =40; = 1 0 ut here x + y = 160 > 150 so the rst feasibility condition is violated. So we can rule out this case. Case (4): 1 > 0; > 0: This implies (150 x y) = 0 (100 x y) = 0 x 1 = 0 y 1 = 0

4 Solving the rst equations we get Substituting in the second we get x = y =50 1 = = 1 50 > 0 So all rst order conditions are satis ed and this is the unique Pareto optimal allocation. Competitive Equilibrium: This is a Robinson Crusoe economy where the \ rm" and the consumer are the same person. Here the rm's problem is to to Choose x s ;y s ;A d ; d max p x x s + y s p A A d p d s.t x s + y s A d x s + y s d The Lagrangian is $ = p x x s + y s p A A d p d + A A d x s y s + d x s y s Since the consumer consumes positive amounts of the goods, the rm must supply strictly positive amounts of the outputs. Since both inputs are required in xed proportions, the rm must use strictly positive amounts of inputs. The Kuhn- Tucker conditions are x s (p x A ) = 0;x s 0; (p x A ) 0 y s ( A ) = 0;y s 0; ( A ) 0 A d (p A + A) = 0;A d 0; (p A + A) 0 d (p + ) = 0; d 0; (p + ) 0 A A d x s y s = 0; A 0; A d x s y s 0 d x s y s = 0; 0; d x s y s 0 Since the consumer consumes positive amounts of the goods, the rm must supply strictly positive amounts of the outputs. Since both inputs are required in

5 xed proportions, the rm must use strictly positive amounts of inputs. That is x s ;y s ;A D ; D > 0:This implies that The rst order conditions become A = p A = p (p x p A p ) = 0 ( p A p ) = 0 p A A d x s y s = 0;p A 0; A d x s y s 0 p d x s y s = 0;p 0; d x s y s 0 The \consumer" has endowments (A s ; s )=(150; 100) and will choose x D ;y D to max logx d +logy d x d ;y d s.t. p x x s + y s = 150p A + 100p D Since the utility function is Cobb-Douglas and satis es local non-satiation we know that we will have an interior solution and the consumer will spend his entire income. So the Lagrangian rst order conditions are The rst conditions can be written as Competitive equilibrium requires that Againwehave4cases. x d = p x x d = p x x d + y d = 150p A +100p y d x d = p x x d = x s : y d = y s A d = 150 : d = 100

6 Case (1): p A = p =0:We can this rule this out since it implies that p x = = 0 so the problem will be unde ned. Case (): p A > 0; p =0: This implies from the rm's rst order conditions p x = p A = p A x s + y s = 150 x s + y s 100 From the consumer's rst order conditions y d x d = p x = Combining x + y = 150; y = (since demand equals supply in competitive x equilibrium) we get x =45;y =60 ut then x s + y s =105> 100 so the second feasibility condition is violated. We can rule this out. Case (): p A =0;p > 0: This implies from the rm's rst order conditions p x = p = p x s + y s 150 x s + y s = 100 From the consumer's rst order conditions Combining x + y = 100; y x equilibrium) we get y d x d = p x =1 = 1 (since demand equals supply in competitive x =60;y =40 ut then x s + y s = 160 > 150 so the rst feasibility condition is violated. We can rule this out as well.

7 Case (4): p A > 0; p > 0: This implies from the rm's rst order conditions (p x p A p ) = 0 ( p A p ) = 0 x s + y s = 150 x s + y s = 100 We can solve this to get From the consumer's problem x =50;y =50 (50) (50) = = p x which is the competitive equilibrium relative price of good x: We can rewrite the rst conditions of the rm as µ px p A p = 0 p µ y p A p = 0 p µ y 1 p A p = 0 We can solve this to obtain p A Therefore the competitive equilibrium is px = ; p A = p = 1 = p = 1 ; x = y =50;A = 150; =100