Prerequisites Almost essential Consumer Optimisation Useful but optional

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 = (R 1, R 2, . . . , Rn) q = (q 1, q 2, . . . , qn) §resources more on this soon. . . §net outputs just the same as before x = (x 1, x 2, . . . , xn) April 2018 §consumption Frank Cowell: Simple Economy 3

Overview A Simple Economy Structure of production Production in a multioutput, multi-process world The Robinson 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 q 1 z 1 q 2 z 2 . . . qn-1 zm qn §An approach using “net outputs” §How the two are related §A simple sign convention q q 1 –z 1 q 2 –z 2. . . =. . . qn-1 –zm qn +q April 2018 §A standard “accounting” approach Outputs: Inputs: + net additions to the stock of a good reductions in the stock of a good Intermediate 0 goods: Frank Cowell: Simple Economy your output and my input cancel each other out 6

Multistage production 1 unit land 10 potatoes 10 hrs labour 1 1000 potatoes § Process 1 produces a consumption good / input § Process 2 is for a pure intermediate good § Add process 3 to get 3 interrelated processes 90 potatoes 10 hrs labour 2 pigs 10 hrs labour 20 pigs April 2018 2 3 22 pigs 1000 sausages Frank Cowell: Simple Economy 7

Combining the three processes Process 1 Economy’s net output vector Process 3 Process 2 sausages 0 0 +1000 potatoes +990 – 90 0 +900 pigs 0 labour – 10 – 30 land – 1 0 0 – 1 + 20 + – 20 0 = 30 hrs labour 1 unit land 22 pigs 100 potatoes April 2018 22 pigs 1000 potatoes 1000 sausages 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 Î 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 § 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 sausages q° 2 q' q'+q" April 2018 No points here! ½q° q' q" All these points q 1 • 0 pigs labour § Possibility of inaction § “No free lunch” § Some basic techniques § Irreversibility § Free disposal § Additivity § Divisibility § Implications of additivity + divisibility q 3 §Additivity+Divisibility imply constant returns to scale q 4 § In this case Q is a cone No points here! Frank Cowell: Simple Economy Let’s derive some familiar production concepts 12

A “horizontal” slice through Q sausages Q q 1 0 pigs labour April 2018 q 3 q 4 Frank Cowell: Simple Economy 13

. . . to get the tradeoff in inputs q 3 pigs labour (500 sausages) q 4 (750 sausages) April 2018 Frank Cowell: Simple Economy 14

…flip these to give isoquants labour (750 sausages) q 4 (500 sausages) pigs q 3 April 2018 Frank Cowell: Simple Economy 15

A “vertical” slice through Q sausages Q q 1 0 pigs labour April 2018 q 3 q 4 Frank Cowell: Simple Economy 16

The pig-sausage relationship (with 10 units of labour) sausages q 1 (with 5 units of labour) q 3 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 q 1 (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 F are two ways of representing the same relationship: q Î Q Û F(q) 0 § Properties of F inherited from properties of Q § F(q 1, q 2, …, qn) is nondecreasing in each net output qi § If Q is a convex set then F is a concave function § As a convention F(q) = 0 for efficiency, so. . . § F(q) 0 for feasibility April 2018 Frank Cowell: Simple Economy 20

The set Q and the production function F §A view of set Q: production possibilities of two outputs q 2 F(q) > 0 §If frontier is smooth (many basic techniques) §Feasible but inefficient points in the interior §Feasible and efficient points on the boundary F(q) = 0 F(q) < 0 §Infeasible points outside the boundary §Boundary is the transformation curve §Slope: marginal rate of transformation §MRTij : = Fj (q) / Fi (q) q 1 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-andconsumption problem The Robinson 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 production decision subject to: • x Î X § logically feasible consumption • F (q) 0 § technical feasibility: equivalent to “q Î Q ” • x q + R The facts of life § 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 R 1= R 2 = 0 § Positive stock of resource 1 § More of resources 3, …, n § Crusoe’s preferences § The optimum § The FOC x 2 g sin e a e r c inc feren pre • § Attainable set derived from technology and materials balance condition x* § 0 April 2018 MRS = MRT: U 1(x) F 1(q) —— = —— U 2(x) F 2(q) x 1 Frank Cowell: Simple Economy 26

The nature of the solution § From the FOC it seems as though we have two parts: 1. 2. § § Can these two parts be “separated out”? A story: • • • § April 2018 A standard consumer optimum Something that looks like a firm's optimum 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 Frank Cowell: Simple Economy 27

Overview A Simple Economy Structure of production The role of prices in separating consumption and production decisionmaking The Robinson Crusoe problem 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. 2. “implicit income” from resources R 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 April 2018 . . . rnqn r 1 q 1 + rr 22 q 2 +. . . + profits r 1 R 1 + r 2 R 2 +. . . + rn. Rn value of resource stocks r 1[q 1+R 1] +. . . + rn[qn+Rn] value of national income Frank Cowell: Simple Economy 30

National income contours q 2+R 2 contours for progressively higher values of national income r 1[q 1+ R 1] + r 2[q 2+ R 2 ] = const q 1+R 1 April 2018 Frank Cowell: Simple Economy 31

“National income” of the Island x 2 §Attainable set § Iso-profit – income maximisaton § The Island’s “budget set” § Use this to maximise utility U 1(x) r 1 —— = — U 2(x) r 2 § Using shadow prices r we’ve broken down the Crusoe problem into a two-step process: x* F 1(x) r 1 —— = — F 2(x) r 2 0 April 2018 1. Profit maximisation 2. Utility maximisation x 1 Frank Cowell: Simple Economy 32

A separation result § By using “shadow prices” r : • a global maximisation problem • is separated into sub-problems: 1. An income-maximisation problem max U(x) subject to x q + R F(q) 0 n max S ri [ qi+Ri] subj. to 2. A utility maximisation problem i=1 F(q) 0 max U(x) subject to n S ri 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 § The price line § Decentralisation x 2 § A = {x: x q + R, F(q) 0} § B = {x: U(x) U(x*)} x* B Here “Better-than-x*” is used as shorthand for “Better-than-orjust-as-good-as-x*” r 1 r 2 § x* maximises income over A A 0 April 2018 x 1 § x* minimises expenditure over B 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 R 2 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 § The hyperplane is the price system April 2018 Frank Cowell: Simple Economy Let's look at another case 36

Optimum cannot be decentralised x 2 profit maximisation here • § A nonconvex attainable set § The consumer optimum § Implied prices: MRT=MRS x° § Maximise profits at these prices consumer optimum here • x* § Production responses do not support the consumer optimum § In this case the price system “fails” A x 1 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 x 2 q 2** • § Equilibrium on the island § The possibility of trade § Max national income at world q** prices § Trade enlarges the attainable set § Equilibrium with trade • x 2** x* • x* *World Domestic prices q 1** April 2018 § x* is the Autarkic equilibrium: x 1*= q 1*; x 2*=q 2* §World prices imply a revaluation of national income §In this equilibrium the gap between x** and q** is bridged by imports & exports x 1** Frank Cowell: Simple Economy x 1 40

The nonconvex case with world trade x 2 • q 2** § Equilibrium on the island § World prices § Again maximise income at world prices q** § The equilibrium with trade § Attainable set before and after trade • x 2** Before opening trade q 1** April 2018 • §Trade “convexifies” the attainable set x** After opening trade x* A′ A x 1** x 1 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 • S pi xi ≤ y, xi ≥ 0 § When you “open up” the model to trade you change • from a world where F(·) 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