Prerequisites
Almost essential Consumer: Optimisation
Useful, but optional Firm: Optimisation
A SIMPLE ECONOMY MICROECONOMICS Principles and Analysis Frank Cowell
April 2018 Frank Cowell: Simple Economy 1 The setting
. A closed economy • Prices determined internally . A collection of natural resources • Determines incomes . A variety of techniques of production • Also determines incomes . A single economic agent • Robinson Crusoe
Needs new notation, new concepts..
April 2018 Frank Cowell: Simple Economy 2 Notation and concepts available for consumption or production
R = (R1, R2, ..., Rn) .resources
more on this soon... q = (q1, q2, ..., qn) .net outputs
just the same as before
x = (x1, x2, ..., xn) .consumption
April 2018 Frank Cowell: Simple Economy 3 Overview A Simple Economy
Structure of production Production in a multi- output, multi-process The Robinson world Crusoe problem
Decentralisation
Markets and trade
April 2018 Frank Cowell: Simple Economy 4 Net output clears up problems
. “Direction” of production • Get a more general notation . Ambiguity of some commodities • Is paper an input or an output? . Aggregation over processes • How do we add my inputs and your outputs?
. We need to reinterpret production concepts
April 2018 Frank Cowell: Simple Economy 5 Approaches to outputs and inputs
NET OUTPUTS INPUTS OUTPUTS .A standard “accounting” approach .An approach using “net outputs” q1 z1 .How the two are related q z 2 2 .A simple sign convention ......
qn-1 zm
qn q
q –z net additions to the 1 1 Outputs: + stock of a good q2 –z2 reductions in the Inputs: − ... = ... stock of a good q –z Intermediate your output and my n-1 m 0 input cancel each goods: other out qn +q
April 2018 Frank Cowell: Simple Economy 6 Multistage production
. Process 1 produces a 1 unit land 1000 consumption good / input 10 potatoes 1 potatoes . Process 2 is for a pure 10 hrs labour intermediate good
. Add process 3 to get 3 interrelated processes 22 90 potatoes 2 10 hrs labour pigs 2 pigs
1000 10 hrs labour sausages 20 pigs 3
April 2018 Frank Cowell: Simple Economy 7 Combining the three processes
Economy’s net Process 1 Process 2 Process 3 output vector sausages 0 0 +1000 +1000 potatoes +990 – 90 0 +900 pigs 0 + 20 + –20 = 0 labour –10 –10 –10 –30 land –1 0 0 –1
30 hrs labour
1 unit land 22 pigs 1000 22 pigs potatoes 100 potatoes 1000 sausages
April 2018 Frank Cowell: Simple Economy 8 More about the potato-pig-sausage story
. We have described just one technique . What if more were available? . What would be the concept of the isoquant? . What would be the marginal product? . What would be the trade-off between outputs?
April 2018 Frank Cowell: Simple Economy 9 An axiomatic approach
. Let Q be the set of all technically feasible net output vectors • The technology set . “q ∈ Q” means “q is technologically do-able” . The shape of Q describes the nature of production possibilities in the economy . We build it up using some standard production axioms
April 2018 Frank Cowell: Simple Economy 10 Standard production axioms
. Possibility of Inaction • 0 ∈ Q . No Free Lunch n • Q ∩ + = {0} . Irreversibility ℝ • Q ∩ (– Q ) = {0} . Free Disposal • If q ∈ Q and q' ≤ q then q' ∈ Q . Additivity • If q ∈ Q and q' ∈ Q then q + q' ∈ Q . Divisibility • If q ∈ Q and 0 ≤ t ≤ 1 then tq ∈ Q A graphical interpretation
April 2018 Frank Cowell: Simple Economy 11 * detail on slide can only be seen if you run the slideshow The technology set Q
. Possibility of inaction . “No free lunch”
sausages q1 . Some basic techniques q° . Irreversibility . Free disposal . Additivity No points . Divisibility here! . Implications of additivity + divisibility 2q' ½q°
.Additivity+Divisibility imply q'+q" q' 0 constant returns to scale • q" . In this case Q is a cone
All these points No points here! Let’s derive some familiar production concepts
April 2018 Frank Cowell: Simple Economy 12 A “horizontal” slice through Q
sausages q1
Q 0
April 2018 Frank Cowell: Simple Economy 13 ... to get the tradeoff in inputs
q3 pigs q4 labour
(500 sausages)
(750 sausages)
April 2018 Frank Cowell: Simple Economy 14 …flip these to give isoquants
(750 sausages)
(500 sausages) labour
q4 pigs
q3
April 2018 Frank Cowell: Simple Economy 15 A “vertical” slice through Q
sausages q1
Q 0
April 2018 Frank Cowell: Simple Economy 16 The pig-sausage relationship
(with 10 units of labour) sausages q1
(with 5 units of labour)
q3 pigs
April 2018 Frank Cowell: Simple Economy 17 The potato-sausage tradeoff
(potatoes) q 2 . Again take slices through Q . Heavy shade: low level of inputs . Light shade: level of inputs . Rays illustrate CRTS
high input
low input
q1 (sausages)
April 2018 Frank Cowell: Simple Economy 18 Now rework a concept we know
. In earlier presentations we used a simple production function • A way of characterising technological feasibility in the 1-output case . Now we have defined technological feasibility in the many- input many-output case • using the set Q . Use this to define a production function • for the general multi-output version • it includes the single-output case that we studied earlier
…let’s see.
April 2018 Frank Cowell: Simple Economy 19 Technology set and production function
. The technology set Q and the production function Φ are two ways of representing the same relationship: q ∈ Q ⇔ Φ(q) ≤ 0 . Properties of Φ inherited from properties of Q
. Φ(q1, q2,…, qn) is nondecreasing in each net output qi . If Q is a convex set then Φ is a concave function . As a convention Φ(q) = 0 for efficiency, so... . Φ(q) ≤ 0 for feasibility
April 2018 Frank Cowell: Simple Economy 20 The set Q and the production function Φ
.A view of set Q: production possibilities of two q2 outputs Φ(q) > 0 .If frontier is smooth (many basic techniques) .Feasible but inefficient points in the interior .Feasible and efficient points on the boundary .Infeasible points outside the boundary Φ(q) = 0
.Boundary is the transformation curve Φ(q) < 0 .Slope: marginal rate of transformation .MRTij := Φj (q) / Φi (q)
q1
April 2018 Frank Cowell: Simple Economy 21 How the transformation curve is derived
. Do this for given stocks of resources . Position of transformation curve depends on technology and resources . Changing resources changes production possibilities of consumption goods
April 2018 Frank Cowell: Simple Economy 22 Overview A Simple Economy
Structure of production A simultaneous production-and- The Robinson consumption problem Crusoe problem
Decentralisation
Markets and trade
April 2018 Frank Cowell: Simple Economy 23 Setting
. A single isolated economic agent • No market • No trade (as yet) . Owns all the resource stocks R on an island . Acts as a rational consumer • Given utility function . Also acts as a producer of some of the consumption goods • Given production function
April 2018 Frank Cowell: Simple Economy 24 The Crusoe problem (1)
. max U(x) by choosing x and q . a joint consumption and subject to: production decision
• x ∈ X . logically feasible consumption
Φ ≤ • (q) 0 . technical feasibility:
equivalent to “q ∈ Q ” The facts of life • x ≤ q + R . materials balance: can’t consume more than is available from net output + resources
April 2018 Frank Cowell: Simple Economy 25 * detail on slide can only be seen if you run the slideshow Crusoe’s problem and solution
. Attainable set with R1= R2 = 0 x2 . Positive stock of resource 1 . More of resources 3,…,n . Crusoe’s preferences . The optimum . The FOC
. Attainable set derived from technology and materials • x* balance condition . MRS = MRT: U (x) Φ (q) ——1 = ——1 U2(x) Φ2(q)
x1 0
April 2018 Frank Cowell: Simple Economy 26 The nature of the solution
. From the FOC it seems as though we have two parts: 1. A standard consumer optimum 2. Something that looks like a firm's optimum . Can these two parts be “separated out”? . A story: • Imagine that Crusoe does some accountancy in his spare time • If there were someone else on the island he would delegate production… • …then use the proceeds from production to maximise his utility . To investigate this possibility we must look at the nature of income and profits
April 2018 Frank Cowell: Simple Economy 27 Overview A Simple Economy
Structure of production The role of prices in separating The Robinson consumption and Crusoe problem production decision- making Decentralisation
Markets and trade
April 2018 Frank Cowell: Simple Economy 28 The nature of income and profits
. The island is a closed and the single economic actor (Crusoe) has property rights over everything . Property rights consist of 1. “implicit income” from resources R 2. the surplus (profit) of the production processes . We could use • the endogenous income model of the consumer • the definition of profits of the firm . But there is no market • therefore there are no prices • we may have to “invent” the prices . Examine the application to profits
April 2018 Frank Cowell: Simple Economy 29 Profits and income at shadow prices
. We know that there is no system of prices . Invent some “shadow prices” for accounting purposes . Use these to value national income
ρ1q1 + ρρ22q2 +...+... ρnqn profits ρ R + ρ R +...+ ρ R value of 1 1 2 2 n n resource stocks
ρ [q +R ] +...+ ρ [q +R ] value of 1 1 1 n n n national income
April 2018 Frank Cowell: Simple Economy 30 National income contours
q2+R2
contours for progressively higher values of national income
ρ1[q1+ R1] + ρ2[q2+ R2 ] = const
q1+R1
April 2018 Frank Cowell: Simple Economy 31 “National income” of the Island
x2 .Attainable set . Iso-profit – income maximisaton . The Island’s “budget set” . Use this to maximise utility
U (x) ρ ——1 = —1 U2(x) ρ2 . Using shadow prices ρ we’ve broken down the • * • x Crusoe problem into a Φ (x) ρ two-step process: ——1 = —1 Φ (x) ρ 2 2 1.Profit maximisation 2.Utility maximisation
x1 0
April 2018 Frank Cowell: Simple Economy 32 A separation result
. ρ By using “shadow prices” : max U(x) subject to • a global maximisation problem x ≤ q + R • is separated into sub-problems: Φ(q) ≤ 0
n 1. An income-maximisation problem max Σ ρi [ qi+Ri] subj. to i=1 Φ(q) ≤ 0 2. A utility maximisation problem max U(x) subject to n Σ ρi xi ≤ y i=1 . Maximised income from 1 is used in problem 2
April 2018 Frank Cowell: Simple Economy 33 The separation result
. The result is central to microeconomics • although presented in a very simple model • generalises to complex economies . But it raises an important question: • can this trick always be done? • depends on the structure of the components of the problem . To see this let’s rework the Crusoe story • visualise it as a simultaneous value-maximisation and value minimisation • then try to spot why the separation result works
April 2018 Frank Cowell: Simple Economy 34 * detail on slide can only be seen if you run the slideshow Crusoe problem: another view
. The attainable set . The “Better-than-x* ” set x . The price line 2 . Decentralisation
. A = {x: x ≤ q + R, Φ(q)≤0}
. B = {x: U(x) ≥ U(x*)}
Here “Better-than-x*” is used as B shorthand for “Better-than-or- • x* just-as-good-as-x*” ρ1 . x* maximises income ρ2 over A A . x* minimises expenditure over B x1 0
April 2018 Frank Cowell: Simple Economy 35 The role of convexity
. The “separating hyperplane” theorem is useful here • Given two convex sets A and B in Rn with no points in common, you can pass a hyperplane between A and B • In R2 a hyperplane is just a straight line . In our application: . A is the Attainable set • Derived from production possibilities + resources • Convexity depends on divisibility of production . B is the “Better-than” set • Derived from preference map • Convexity depends on whether people prefer mixtures Let's look at another case . The hyperplane is the price system
April 2018 Frank Cowell: Simple Economy 36 Optimum cannot be decentralised
profit maximi- x2 sation here . A nonconvex attainable set x° . The consumer optimum • . Implied prices: MRT=MRS . Maximise profits at these prices
consumer optimum here • x* . Production responses do not A support the consumer optimum . In this case the price system “fails”
x1
April 2018 Frank Cowell: Simple Economy 37 Overview A Simple Economy
Structure of production How the market simplifies the model The Robinson Crusoe problem
Decentralisation
Markets and trade
April 2018 Frank Cowell: Simple Economy 38 Introducing the market
. Now suppose that Crusoe has contact with the world • not restricted to “home production” • can buy/sell at world prices . This development expands the range of choice . It also modifies the separation argument in an interesting way
Think again about the attainable set
April 2018 Frank Cowell: Simple Economy 39 Crusoe's island trades
x2 . Equilibrium on the island . The possibility of trade ** q2 • q** . Max national income at world prices . Trade enlarges the attainable set . Equilibrium with trade
. x* is the Autarkic equilibrium: * * * * x1 = q1 ; x2 =q2 ** • x* x2 Domestic • x* .World prices imply a revaluation prices *World of national income prices .In this equilibrium the gap between x** and q** is bridged by imports & exports
** ** x q1 x1 1
April 2018 Frank Cowell: Simple Economy 40 The nonconvex case with world trade
x2 . Equilibrium on the island . World prices ** . Again maximise income at world prices q2 • q** . The equilibrium with trade . Attainable set before and after trade
** .Trade “convexifies” the x2 • x** attainable set
After • x* opening Before opening trade trade A A′
** ** q1 x1 x1 April 2018 Frank Cowell: Simple Economy 41 “Convexification”
. There is nothing magic about this . When you write down a conventional budget set you are describing a convex set
• Σ pi xi ≤ y, xi ≥ 0 . When you “open up” the model to trade you change • from a world where Φ(·) determines the constraint • to a world where a budget set determines the constraint . In the new situation you can apply the separation theorem
April 2018 Frank Cowell: Simple Economy 42 The Robinson Crusoe economy
. The global maximum is simple . It can be split up into two separate parts: • Profit (national income) maximisation • Utility maximisation . Relies on the fundamental decentralisation result for the price system . Follows from the separating hyperplane result • “You can always separate two eggs with a single sheet of paper”
April 2018 Frank Cowell: Simple Economy 43