Problem Set (1 p) (1) 1 (100)

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1 University of British Columbia Department of Economics, Macroeconomics (Econ 0) Prof. Amartya Lahiri Problem Set Risk Aversion Suppose your preferences are given by u(c) = c ; > 0 Suppose you face the following options: you can either take $0 for sure or you could accept a gamble in which you get $00 with probability p and $with probability p. Clearly with p = 0 you would always take the $0. For values of = f0:; 0:; :00; :; g compute the corresponding p such that for all p > p you would take the gamble rather than the certain payo of $0. What does this exercise tell you about the value of that one should use when one calibrates a model? Speci cally, should one use values of in excess of or? Why or why not? Payo from taking $0 for sure is: u s = 0 Payo from taking the gamble is: p u g = (00) + ( p) () Expected utiltities get equated at the following combinations of and p: p EU The thing to not about these numbers is that the p above which one would take the gamble gets close to its natural upper bound of one at pretty small levels of. Hence, just from a lottery perspective, it is hard to defend terribly high numbers for the risk aversion parameter.

2 Computing asset prices and returns Suppose preferences of a representative agent in a two-period lived economy are given by V = u (c ) + E fu (c )g where u(c) = c ; > 0 Agents receive an endowment every period. The endowment is stochastic and can take on two values in each period: or. The joint density function of the four possibilities in this two-state, two-period world is Outcome Probability, 0., 0., 0., 0. Suppose there are three assets on o er in this economy. One is a risk free bond which costs q and pays in every state next period. The second is a state-contingent bond that costs q L and pays in the low state when y = and zero in the good state when y =. The third asset is also a state-contingent claim that costs q H and pays in the good state when y = and zero in the low state. Let = 0:0. (a) Compute q L ; q H ; q and the associated rates of return r H ; r L and r: (b) What is the e ect of a rise in on r H ; r L and r? What is the e ect of on the return premium on each state-contingent asset, i.e., on r i r, i = H; L? (c) Can you explain your results in (b) above? Answers: Part (a) First compute conditional probabilities: p(y = jy = ) = p(y = jy = ) = Second, obtain an euler equation: Now let us look at two cases in turn. p(y = jy = ) = p(y = jy = ) = c = E ( + r) c

3 Case : y = Riskless bond The price of the riskless bond is given by: c q = E c The payo of the riskless bond is one in each possible state for y. probabilities we have calculated above: and thus and the returns are given by: q = 0 q = Thus using the r = q Low-state-contingent bond The payo of q L are one in the low state and zero otherwise. Thus, which reduces to: q L = 0 () + (0) The returns if y = are: q L = 8 And if y = : r y = = 8 = 7 8 r y = = 0 8 = So that the expected rate of return is: E r L = ( ) =

4 High-state-contingent bond The payo of q H are one in the high state and zero otherwise. Thus, which reduces to: q H = 0 (0) + () The returns if y = are: And if y = : q H = 0 r y = = 0 q H = So that the expected rate of return is: Case : y = r y = = = 0 q H E r H = 0 + ( ) Redo the same step but using y =, we get: Riskless bond and the returns are given by: q = 0 + () r = q Low-state-contingent bond The returns if y = are: q L = () 0 r y = = q L = 0 () ()

5 And if y = : So that the expected rate of return is: r y = = 0 q L = E r L = 0 () () + ( ) High-state-contingent bond The returns if y = are: And if y = : q H = 8 r y = = 0 q H = r y = = = So that the expected rate of return is: Part (b) Take Case : y = We know that: Hence, E r L = 7 8 r = dr d = ( ) = ln > 0 where we have used the fact that dax dx = ax ln a and ln < 0. The rate of return on the riskless bond increases with an increase in risk aversion. This is because agents being in state y = know that their situation will either stay the same or improve. Now for the asset which pays o in the low state, we have:

6 Er L = One sees immediately that the rate of return in this case doesnt change with the risk aversion parameter. Note that this is due to the fact that, for y = y =, the term involving drops out of the equation in this example. And nally, for the asset which pays o in the high state, we have: and thus, Er H = der H d =! 0 + ( ) 0! 0 ln > 0 where ln < 0:The rate of return on this asset increases with an increase in risk aversion. This is because the high-state contingent bond pays o when one also gets the highest endowment in period. The bond increases the volatility of consumption whereas risk-averse agents want to smooth their consumption. Hence they want to be compensated for that increased volatility in consumption. Let us now examine the e ect of a change in on the return premiums. For the asset which pays o in the low state: Hence, L = Er L r = d L d = dr d < 0 In the case of the asset which pays o in the high state: Hence, H = Er H r = which reduces to:! 0 + ( ) H = 0 " + 6 # 0 + +

7 " d H d = 0 ( ) ( ) ln + ln # Examining the two terms in brackets one can show that the rst term will have a greater magnitude in absolute value than the second one for > 0. This implies: Part (c): d H d > 0 The e ect of the risk aversion parameter on rates of return is as expected. Indeed, agents want to insure themselves against bad times. Thus the higher their risk aversion parameter, the lower [higher] the rate of return these agents will accept for a bond that pays o in bad [good] times. 3 Productivity Shocks Consider a two-period lived closed economy inhabited by a continuum of identical agents of measure one. Lifetime welfare of the representative agent is V = ln [c ] + ln [c ] There is a storage technology in that setting aside k units of the good in period produces ( ) k + f (k) of the good in period. Set the depreciation rate =, i.e., assume full depreciation. Further, assume that f (k) = k ; (0; ) where is an exogenous term that re ects total factor productivity (TFP). Suppose that the rst period endowment of the good is one and the second period endowment is zero. (a) De ne a competitive equilibrium for this economy. (b) Solve for a competitive equilibrium. (c) Suppose there are two economies which are identical in every respect except for the fact that is greater in one relative to the other economy. Which economy will have a higher interest rate? Which economy will have a higher saving rate? (a) fc ; c ; kg and fp =p g such that: households maximize utility taking prices as given; rms maximize pro ts given prices and the production function; markets clear so that supply equals demand for each good. 7

8 (b) Substitute the constraints in the objective function: max because = and y = 0. FOC: ln( k) + ln(k ); k + k k = 0 Thus, Because = ; f 0 (k) = r. So k = + c = k = + c = k = + + = r = p p Or, equivalently p p = c c = + Notice that substituting the constraints in the objective function, just as in the notes, is the simplest method. (c) Notice that the economy with higher will have a higher interest rate but that the savings rate will be una ected. This suggests that exogenous productivity changes (at least the way we have thought of them here and with log utility) cannot explain variation over time in savings rates. So there must be another explanation for changes in North American savings rates, at least in this simple framework. 8