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T_CHAPTER12 - Chapter 12 From Arbitrage to Equilibrium In...

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Chapter 12. From Arbitrage to Equilibrium In chapter 9, we described outcomes of investing either on the °nancial market or the capital goods market. We had also made the hypothesis that market forces would be at play to establish an equilibrium between those outcomes. Our aim in this chapter is to describe those forces. We begin with the case of a risk-free world in the sense that the existence of forward markets enables some investors to protect themselves against uncertainty (section 1). We then introduce uncertainty (section 2). 1 The case of risk-free transactions We have considered four prices in the Fisher-Solow equation of interest: p ( t ) is the price of the capital good at time t ; p ( t + h ) is the forward price for time t + h , decided upon at time t ; q ( t + h ) is the forward rental rate of the capital good, to be received at time t + h ; and i ( t ) , the interest rate, is the price of loanable funds at time t . Until now, we have just supposed that an equilibrium would exist between these four prices, stemming from an equality between the available returns on the °nancial market and on the capital goods market. We will now describe the forces that come into play to establish this equilibrium. Two types of forces should be distinguished. The °rst is arbitrage; we de°ne arbitrage as the action of individuals who will earn bene°ts without committing their own resources. The second is investing, i.e. the action of buying capital goods (or assets) with own resources. In order to simplify the exposition, we suppose that the capital good may be borrowed, and also that we are able to sell it on a forward market. Suppose that at time 0 we observe the four following prices: p 0 ; p 1 ; q 1 , and i 0 , resulting from supply and demand curves as described in °gure 1. The question now is: can these four prices be considered in equilibrium in the sense that, if nothing else is disturbed in the economy, these numbers will remain constant at least for a short time? To answer this question, let us retrace the steps of Solow±s °rst method, supposing that h is one year. An investor of $1 on the risk-free °nancial market will earn 1 + i 0 after one year. 1
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If he invests in the capital good, he will buy 1 =p 0 units of the capital good; in order to protect himself against any uncertainty, he does two things: he sells his capital good on the forward market at price p 1 , and rents it for one year at rate q 1 , with such guarantees that the contract is riskless. 1.1 The case of an undervalued asset Suppose now that the proceeds of investing in the capital good are higher than those on the °nancial market; this translates as the inequality 1 + i 0 < q 1 p 0 + p 1 p 0 : (1) Two sets of forces will come into play to establish equilibrium. The °rst is due to the appearance of arbitrageurs. The second results from the behaviour of investors who were operating initially on these markets.
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