Foundations of Financial Economics GE under uncertainty

Transcription

1 Foundations of Financial Economics GE under uncertainty Paulo Brito 1 pbrito@iseg.ulisboa.pt University of Lisbon March 16, 2018

2 Topics Stochastic Robinson-Crusoe (representative agent) economy Gains from trade under risk aversion

3 A Robinson-Crusoe economy 1. an agent receives a contingent endowment Y = (y s ) N. 2. from which he can consume the uncertain amount X = (x s ) N 3. he evaluates his contingent consumption with the utility function N U(X) = E[u(X)] = π s u(x s ) where 0 < π s < 1 and N and u(.) is continuously differentiable, increasing and concave 4. he faces the constraint x s y s, s = 1,..., N 5. he wants to maximize expected utility

4 A Robinson-Crusoe economy The problem: maximize the value extracted from the resource Y { N } V(Y) = max π s u(x s ) : x 1 y 1,..., x N y N {x s} N Generalized Lagrangean N L = π s u(x s ) + λ s (y s x s ) First order conditions π s u (x s) = λ s, s = 1,..., N λ s (y s x s) = 0, s = 1,..., N with x s = y s and λ s > 0 or x s < y s and λ s = 0 (Note: as we have n inequality constraints, for every state of nature, therefore we need n Lagrange multipliers and n KKT conditions)

5 A Robinson-Crusoe economy Case 1: u (x) > 0 for all x > 0 and π s > 0 for all s then π s u (x s ) > 0 for all x s Solution x s = y s, s = 1,..., N Case 2: if the utility function has a satiation point, i.e there is a point x such that u ( x) = 0 and π s > 0 Solution then x s = { y s, x if x = (u ) 1 (0) y s if x = (u ) 1 (0) < y s Question: what is the solution to the problem if there is a state, r, such that π r = 0?

6 Gains from trade under risk aversion Assumptions: 1. an agent receives a contingent endowment Y = (y s ) N and let π s > 0 for all s; 2. from which he can consume the uncertain amount X = (x s ) N ; 3. he evaluates his contingent consumption with the utility function (assumption: no satiation) U(X) = E[ln (X)] = N π s ln (x s ) 4. one of two alternative institutional settings are possible: setting 1- autarky : he cannot trade: then x s = y s setting 2 - insurance: there is a market for contingent goods in which he can buy x a y a in the state of nature a, at price p a, or sell y b x b if state of nature b occurs at prices p b, given to him. In which setting the consumer will be better off?

7 The consumer problem in the two environments In order to answer the question: We have to determine in which case he can obtain more value out of the endowment, Y(ω), in the two alternative settings and compare. 1. in autarky his utility is V a (Y) = max X = max {x s} N {E[ln (Y)] : X Y} { N } π s ln (x s ) : x s y s 2. if there is an insurance market he has to solve the problem { N } N V m (Y) = max π s ln (x s ) : p s (y s x s ) 0 X we write h N p sy s the market value of its endowment.

8 Utility in autarky Because there is no satiation, the solution of the consumer problem is x s = y s for s = 1,..., N Note: the solution is state-dependent (no insurance) The utility at the optimum is V a (Y) = E[ln (Y)] = ln (GE[Y]) is the log of the geometric mean (GM). This measures the the value of the resource Y in autarky

9 Utility with trade in the market economy the solution to the agent s problem is(prove this) x s = π sh p s, s = 1,..., n. Then his utility is V m (Y) = N ( ) πs π s ln h = ln p s [ N ( ) ] πs πs h p s If p s = π s (the market and the nature probabilities are the same) then consumption is state-independent (complete insurance) x s = h, for all s = 1,..., n. Then the value of the resource Y with trade is V m (Y ) = ln (h) = ln (E[Y]) the log of the arithmetic mean (AM).

10 Comparing autarky with trade Because then E[Y] GE[Y] V m (Y) = ln (E[Y]) V a (Y) = ln [GE[Y]] Conclusion: with the assumptions regarding the utility function (non-satiation and concavity) and the market prices (equal to the probabilities) there are two implications: 1. The consumer will be better off if an insurance market exists: risk aversion implies risk sharing is utility enhancing 2. There is complete insurance: the consumer is able to have access to a deterministic consumption profile although he has stochastic endowments X a = (y 1,... y N ) with y 1... y N versus X m = (h, h,..., h) Are those results always true?

11 Application: the cost of volatility for Portugal Histogram of growth.rate growth.rate Frequency Time growth.rate Figure: Real per capita growth rates: Portugal An R-file importing PWT data and with the computations can be found here

12 Application: the cost of volatility Figure: Portugal : average growth rate and certainty equivalent growth rate for a CRRA utility function with parameter values between 1 and 10

13 References (LeRoy and Werner, 2014, Part III), (Lengwiler, 2004, ch. 2), (Altug and Labadie, 2008, ch. 3) Sumru Altug and Pamela Labadie. Asset pricing for dynamic economies. Cambridge University Press, Yvan Lengwiler. Microfoundations of Financial Economics. Princeton Series in Finance. Princeton University Press, Stephen F. LeRoy and Jan Werner. Principles of Financial Economics. Cambridge University Press, Cambridge and New York, second edition, 2014.