[TIME] The Fisher Separation Theorem The function of capital markets in intertemporal decisions

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1 [TIME] The Fisher Separation Theorem The function of capital markets in intertemporal decisions Prof. Dr. Heinz Zimmermann University of Basel Theory of Finance I Spring 2015

2 Objective How do capital markets (without uncertainty) affect consumption and production decisions in a decentralized economy? 1 Production and (real) investment without capital markets the intertemporal production possibilities frontier (ppf) 2 Production and investment with capital markets the net present value (NPV) rule 3 Production and consumption with capital markets Fisher separation of production, investment and consumption decisions 2 / 15

3 Intertemporal production possibilities frontier (ppf) The intertemporal ppf (also called "transformation curve") for 2 commodities reveals the combinations of maximum quantities which can be produced with given factor inputs. From the dynamic resource constraint we get the intertemporal ppf: k t+1 k t = f (k t ) c t c t+1 = f (k t+1 ) + k t+1 k t+2 = f ( f ( k t ) + k t c t ) + ( f ( kt ) + k t c t ) kt+2 (where k clarifies that the input of the production function is exogenously given) 3 / 15

4 Intertemporal production possibilities frontier (ppf) The marginal rate of transformation (MRT): dc t+1 dc t = f (k t+1 ) dk t+1 + ( 1) k=const. dc t = f (k t+1 )( 1) + ( 1) [ ] = 1 + f (k t+1 ) Show that the frontier is concave. MRT t,t+1 < 0 (1) 4 / 15

5 Optimal production, investment and consumption Optimal production and consumption: no capital market u (c t ) = βu (c t+1) ( 1 + f (k t ) ) consumption must be equal to production in each period. Or expressed in terms of MRS: MRS t,t+1 = u (c t ) βu (c t+1 ) = 1 + f (k t ) = MRT t,t+1 Optimal real investment is: i t = c t c t 5 / 15

6 Optimal production and investment without capital market Figure : Fisher Separation Theorem I: No capital market 6 / 15

7 Optimal production, investment and consumption Optimal production and (real) investment: with capital market the production decision is not governed by households preferences, but by profit maximization, which leads to the optimal stock of capital: f (k t+1) = R, ct+1 ) is determined The optimal production of commodities (ct from: dc [ ] = 1 + f (kt+1) = (1 + R) k=const. The optimum: t+1 dc t MRT t,t+1 = (1 + R) The market interest rate - not the preferences of hh - determines the optimal production-investment decision. 7 / 15

8 Optimal production and investment with capital market Figure : Fisher Separation Theorem II: Production with capital market 8 / 15

9 The net present value rule (NPV rule) The investment decision which results from the optimum production decision is known as NPV-rule If a capital market exists, each allocation (c t, c t+1 ) on the intertemporal ppf can be re-allocated, at cost respectively return R, to current or future consumption along the capital market budget schedule. Put it differently: The firm can sell its entire production forward and cash-in the proceeds immediately by to get maximum possible current consumption The present value of the production PV t = c t + c t R 9 / 15

10 Optimal consumption with capital market Optimal consumption with capital market Optimality is characterized by the Euler equation: MRS t,t+1 = u (c t ) βu (c t+1 ) = 1 + R Graphically, this is the tangency point of the indifference curve and the negatively sloped capital market budget schedule. 10 / 15

11 Optimal consumption with capital market Figure : Fisher Separation Theorem III: Consumption with capital market 11 / 15

12 Fisher separation If a capital markets exist, then production-investment decision of firms: c, resp. i = c c consumption-savings decisions of households: c, resp. s = y c can be separated, i.e. regarded as separate or disconnected decisions. Fisher (1930) 12 / 15

13 Fisher separation Figure : Fisher Separation Theorem IV 13 / 15

14 Implications of the Fisher separation theorem Welfare: it shifts individuals to a higher utility function due to expansion of the consumption set of households. Decision: interest rates provide an objective, market determined real investment criterion for firms Management: It explains why shareholders and management can be separated in firms; capital markets as a fundament for modern joint-stock company 14 / 15

15 References Fisher, I. (1930): The Theory of Interest. Macmillan. 15 / 15

MICROECONOMICS AND POLICY ANALYSIS - U8213 Professor Rajeev H. Dehejia Class Notes - Spring 2001

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