Economics and Finance,

Transcription

1 Economics and Finance, Lecture 2: Consumption and Investment decisions without capital markets Luca Deidda UNISS, DiSEA October 2014 Luca Deidda (UNISS, DiSEA) EF October / 19

2 Plan Introduction The model Optimal investment decision Preferences and investment decisions Luca Deidda (UNISS, DiSEA) EF October / 19

3 Objective Introduction This is the first lecture about firm governance We will analyze individual consumption-investment decisions in an economy where there are no capital markets (financial autarky) This in order to understand how the state of development of capital markets shapes the relationship between investment decisions and subjective preferences This exercise allows us to draw some conclusions of how the absence of capital markets affects the potential conflict of interests among firm s owners Luca Deidda (UNISS, DiSEA) EF October / 19

4 The model The model Population, and time horizon Endowments, and individual preferences Optimal consumption-investment decision Luca Deidda (UNISS, DiSEA) EF October / 19

5 The model Population and time horizon To start with, we analyze the following environment: Population: a number N of individuals. We look at the decision problem of one of them. Agents are price-takers. Individuals live for two periods, 1 e 2, where 1 indicates the beginning of the first period and 2 indicates the second period Luca Deidda (UNISS, DiSEA) EF October / 19

6 The model Endowments and preferences Each agent: is endowed with y 1 unit of wealth in period 1 and y 2 unit of wealth in period 2 has preferences defined over consumption in both periods, u = u(c 1, c 2 ) (1) where c 1 is consumption in period 1 and c 2 is consumption in period 2 We further assume: u increases as c1 or c 2 increases, more specifically: Marginal utility of consumption is positive: Holding ci constant, (c i = c), u is increasing in c j Margin utility of consumption is decreasing Luca Deidda (UNISS, DiSEA) EF October / 19

7 The model Graphing the utility function Holding c 2 costant, the shape of u as a function of c 1 is described by this graph: Figure 1 The shape of u in three-dimensions, allowing both c t and c t+1 to vary is described by this graph: Figure 2 Holding u constant, indifference curves are desecribed by this graph: Figure 3 Luca Deidda (UNISS, DiSEA) EF October / 19

8 The model Subjective intertemporal discount rate The slope of the indifference curve U c 1 U measures the subjective intertemporal discount rate Its absolute value, dc 2 = (2) dc 1 c 2 U dc 2 dc 1 = c 1 U tells us how many units of consumption in period 2 the individual is willing to give up to get an extra unit of consumption in the current period The greater its value, the more the individual is willing to pay, in terms of lower future consumption, to consume one unit more in the current period c 2 Luca Deidda (UNISS, DiSEA) EF October / 19

9 The model Production technology Each individual has access to a production technology such that an amount of real investment I generates an overall return R(I) We assume decreasing marginal returns to investment Marginal product of an investment I is, dr di = R (I) > 0 (3) Marginal product of an investment I is decreasing, d 2 R di 2 = R (I) < 0 (4) Esempio: R(I) = A I α (5) Question: Which restrictions should satisfy α and A in order to have positive and decreasing marginal returns to investment? Luca Deidda (UNISS, DiSEA) EF October / 19

10 The model Individual behavior under financial autarky (FA) Under FA, with all individuals being identical, there are no transactions. Each individual chooses optimally how much to invest in period 1, I, subject to the constraint that I cannot be negative and more over, cannot exceed y 1, that is: I y 1 Moreover, individuals will take into account that the choice of I affects her consumption profile, since: c 1 y 1 I (6) c 2 y 2 + R(I) (7) Non satiation implies that, for any given value of I, the above constraints will always be satisfied as equalities c 1 = y 1 I (8) c 2 = y 2 + R(I) (9) Luca Deidda (UNISS, DiSEA) EF October / 19

11 The model Budget constraint and production possibilities Imposing strict equality and rearranging perod-1 constraint, I = y 1 c 1 (10) Which confirms that investment decisions determine consumption in the first period or, viceversa, consumption decision in the first period determines the level of investment. Substituting for I into the second constraint we find: c 2 y 2 + R(y 1 c 1 ) (11) Which describes the set of production possibilities, in terms of levels of wealth to be consumed in the first or second period, associated with the possible levels of investment I The above constraint taken as strict equality describes the production possibilities frontier The production possibilities frontier is described by Figure 4 Its slope is: dc 2 dc 1 = R (y 1 c 1 ) (12) Luca Deidda (UNISS, DiSEA) EF October / 19

12 Optimal investment behavior Optimal choice under FA The individual will choose how much to consume in the to periods, and hence how much to invest in the first period, so to maximize u(c 1, c 2 ) subject to: c 2 = y 2 + R(y 1 c 1 ) The individual will invest up to the point that SMS c 1,c 2 = SMT c 1,c 2 R (y 1 c 1 ) = The optimal decision is represented in Figure 5 U c 2 1 U c 2 (13) Luca Deidda (UNISS, DiSEA) EF October / 19

13 Preferences and investment choices Preferences, opportunity cost of capital and optimal level of investment It can be easily verified that the optimal choice, and hence optimal level of investment depends on individual preferences More impatience leads to lower levels of investments Hence individuals preference affect the value created by means of investment activity For an example of how different preferences result in different levels of investment see Figure 6 Luca Deidda (UNISS, DiSEA) EF October / 19

14 Preferences and investment choices Figure 1: Utility u as a function of c 1 for given c 2 u(c 1,c ) 0 c 1 go back Luca Deidda (UNISS, DiSEA) EF October / 19

15 Preferences and investment choices Figure 2: Utility u as a function of c 1 and c 2 u(c 1,c 2 ) u(c,c 2 ) c 2 u(c 1,c ) c 1 go back Luca Deidda (UNISS, DiSEA) EF October / 19

16 Preferences and investment choices Figure 3: Indifference curves c 2 0 c 1 go back Luca Deidda (UNISS, DiSEA) EF October / 19

17 Preferences and investment choices Figure 4: Production possibilities frontier c 2 0 c 1 go back Luca Deidda (UNISS, DiSEA) EF October / 19

18 Preferences and investment choices Figure 5: Optimal investment decision under financial autarky c 2 c 2 * R'(I) 0 c 1 * I t y 1 c 1 1 go back Luca Deidda (UNISS, DiSEA) EF October / 19

19 Preferences and investment choices Figure 6: Preferences and investment decisions c 2 u A u B 0 c 1 go back Luca Deidda (UNISS, DiSEA) EF October / 19