Exercises for Chapter 7

Exercises for Chapter 7 Exercise 7.1 The following simple macroeconomic model has four equations relating government policy variables to aggregate income and investment. Aggregate income equals consumption plus investment and government expenditure: y= c+ i+ g. Consumption is positively related to the real rate of interest: c=α+βy, α,β>0. Investment is negatively related to the rate of interest, r: i= γ δr, γ,δ>0. The money supply, m s, equals the money demand, m d, and this in turn is positively related to the level of income (ρ> 0) and negatively related to the rate of interest (ϕ>0): m s = m d = ρy ϕr, ϕ,k>0. Find the impact on y of (a) a small change in g, g, and (b) a small change in m, m. 29

CHAPTER 7. SYSTEMS OF EQUATIONS, DIFFERENTIALS AND DERIVATIVES 30 Exercise 7.2 curves. Consider a market defined by the following supply and demand q d = a+ bp d + cȳ q s = d+ep s, where a, c, d, e> 0, and b<0, a>d. (a) (i) Compute the equilibrium price P and quantity q. (ii) Sign the comparative statics partial derivatives P Ȳ, q Ȳ. (b) Suppose a tax t>0 is levied on producers (i) Compute the new equilibrium price and quantity. (ii) Sign the derivatives q t, P d t, P s t. Exercise 7.3 Consider the following macroeconomic model of national income: C= α+βy α>0, 0<β<1 Y = C+I+G I = γy, 0<γ<1 What are the exogenous and endogenous variables? Write this system in matrix form and solve for the reduced form (i.e., write the system in the form Ax = b, where x is the vector of endogenous variables and b the vector of exogenous variables and solve as x = A 1 b). Using the reduced form expressions, find the following comparative statics derivatives: c G, Y G, I G. Exercise 7.4 Consider the following macroeconomic model: y= c(y,r)+ i(r)+g m=l(r, y). (a) Identify the exogenous and endogenous variables. (b) (i) What is the effect of an increase in government expenditure G, on output y and the interest rate r? (ii) What is the effect of an increase in the money supply m, on output and the interest rate? (c) Consider what happens if we replace the above money market equation with the following: m=µy. Repeat part (b) with this assumption.

31 EXERCISES FOR CHAPTER 7 (d) Represent the two different systems in (y,r) space and show the above effect diagrammatically. Exercise 7.5 Consider the following macroeconomic model: S(Y,r)+ T(Y )+ IM = I(r)+G+x(e) L(Y,r)=M where Y is national income, S is savings, which is a function of income and the rate of interest, IM is imports, x is exports, I is investment, G is government expenditure, M is money supply and e is the exchange rate. The first equation is the equilibrium condition for the goods market and the second equation is the equilibrium condition for the money market. Using this model, determine how the equilibrium values of Y and r change with a change in the exchange rate e and with a change in the exogenous level of imports (here imports is an exogenous variable, just like G). Make the usual assumptions: S y > 0,S r > 0,T y > 0, I r < 0,x e < 0,L y > 0,L r < 0. Exercise 7.6 Consider a production process using input x to produce output y. The relationship between inputs and outputs is somewhat odd: for technological reasons the input output levels must satisfy two conditions: x 2 + y 2 = α, 0<α<1 βy= x(1 x), 0<β. This pair of equations has a unique non-negative solution (x, y 0). Plot these functions in (x, y) space. At this solution, calculate dx dα, d y dα, dx dβ and d y. In relation to you are graph, interpret the signs of the derivatives (they are positive dβ or negative). Exercise 7.7 Repeat the previous question, with input and output required to satisfy the following two conditions: y γln(x+1)= 0 x α y β = γ. Exercise 7.8 Consider the following IS LM model with an import sector: y= c(y)+ I(i)+G m=l(y, i) x(π)= z(y,π).

CHAPTER 7. SYSTEMS OF EQUATIONS, DIFFERENTIALS AND DERIVATIVES 32 The first two equations are the IS and LM curves, respectively. The third equation asserts that exports (x) depend only on the exchange rate, π, while imports (z) depend on the exchange rate and the level of domestic income. (a) Calculate the slopes of the IS and LM curves. (b) Find d y dg, di dg dπ and dg. (Differentiate each of the three equations to give a 3 3 equation system in (d y, di, dπ) and use Cramer s rule.) Exercise 7.9 An individual has utility function u(x, y) = x α y β, where 0 < α < 1,0<β<1 and 0<α+β<1. The prices of x and y, respectively, are p and q, and the individual has income I. The marginal utility of x is denoted u x, and here you see that u x (x, y)=αx α 1 y β = αxα y β x = α u(x,y) x. Similarly, u y (x, y)=βx α y β 1 = βx α y β y = βu(x,y) y. The utility maximizing levels of x and y are determined by the tangency of the indifference curve to the budget constraint: px + qy = I. The tangency condition is given by the condition that the marginal rate of substitution equals the price ratio MRS = u x u x u y = p q u y = p q. Thus we have two equations in (x, y): (1) and (2) px+ qy= I. Note that equation (1) has a very simple form: it can be written as a linear equation in x and y after appropriate multiplication and cancellation. (a) Write the pair of equations in the form F 1 (x, y;α,β, p, q, I) = 0 F 2 (x, y;α,β, p, q, I) = 0 and calculate x I, x p, x q, x α, x β. (b) Solve (1) and (2) directly for x and y (as functions of (α,β, p, q, I)), then take the partial derivatives of x with respect to the parameters, to confirm your answer in (a). (c) Illustrate with a graph how x I, x p, x q, x α, x are calculated. β Exercise 7.10 A production possibilities (transformation) curve in two goods, x and y, is given by T(x, y)= k, where T(x, y)= x 2 + y. The social utility function is given by u(x, y) = αln(x) +βln( y). The efficient choices for x and y are given at the point where the marginal rate of substitution equals the marginal rate of transformation (MRS=MRT or u x u y = T x T y ). Thus, there are two equations determining the solution: (1) u x u y = T x T y and (2) x 2 + y= k.

33 EXERCISES FOR CHAPTER 7 (a) Rearrange these to get two equations in the form F 1 (x, y;α,β,k)= 0 F 2 (x, y;α,β,k)= 0. (b) Use the implicit function theorem to determine x α, y α, x k, y k. Exercise 7.11 Consider the following model C= 5+0.85Y Y = C+I+G I = 0.03Y Solve for the effects of an increase in government spending on: consumption, income, investment. Exercise 7.12 Consider the IS LM model ( ) M a+ br C = d er+ f Y (savings = investment) P M P = AY α r β (money demand = money supply). (a) Totally differentiate the pair of equations. (b) Use Cramer s rule to show what effect an increase in M has on r, P (for simplicity let dy = 0). Exercise 7.13 A firm is required to produce output Q using inputs x and y. The production technology is x α y β. In addition, the firm faces the technology constraint y= ke x, k>0. Find y Q, x Q, y k, x y k. Also, try finding α, x α. Exercise 7.14 Consider the following two equations: x 2 + y 2 = r 2 and y= kln(x+ 1), with r> 0 and k>0. Plot both equations and indicate on your graph the impact of an increase in r and k on the solution. Find d y dr, dx d y dx dr dk and dk. Exercise 7.15 Consider the following two equations: xy = k and y = 1 e γx, k> 0 and γ>0. Plot both equations and indicate on your graph the impact of an increase in k and γ on the solution. Find d y dk, dx d y dx dk dγ and dγ. Exercise 7.16 An output technology is given by x+ky 2 = a, where k and a are positive constants. Government regulations require that outputs of x and y satisfy the relation e βy e αx = c, where α, β and c are positive constants. (We assume that ln(c+1) β < a k to ensure there is a solution to the pair of equations.)

CHAPTER 7. SYSTEMS OF EQUATIONS, DIFFERENTIALS AND DERIVATIVES 34 (a) The government has decided that it is desirable to raise x and reduce y, by a policy of increasing c. Will the policy have the desired effect? Calculate the impact on x and y of an increase in c. (b) An economic consultant recommends instead that the government raise β. Would this policy have the desired effect of increasing x and reducing y? Exercise 7.17 An individual has income or wealth, y, to allocate between present, c, and future, f, consumption. Suppose that the individual has utility function u(c, f ) = c f. Income not consumed in the current period is invested at interest rate r to yield the amount available for future consumption: f = (y c)(1+ r). Since y is given, the choice of c determines f. This is a budget constraint relating current and future consumption. Find the impact of a change in income, y, on consumption c. Exercise 7.18 An individual has preferences given by indifference curves of the form xy=k: if the agent is consuming x units of one good, the consumption of k y units of the other good leads to utility level k. The problem facing the individual is that the goods x and y are bundled so that they cannot be bought in any combination. Specifically, x units of one good come in a package with y=1 e αx units of the other good. Suppose that the individual currently is at utility level k, consuming at levels x and y according to the packaging constraint. Find the impact of an increase in utility on the consumption levels x and y: x y k and k. Exercise 7.19 According to a macroeconomic model, the aggregate level of income Y equals consumption C, investment I and government expenditures G (Y = C+I + G). Consumption depends on income C = C(Y ) while investment depends on the rate of interest r, so I = I(r). Thus, Y = C(Y )+ I(r)+ G. The money supply M equals money demand, which is a function of the level of income and the rate of interest L=L(Y,r). Thus, M = L(Y,r). Finally, the exchange rate π is assumed to depend on income and the interest rate π=π(y,r). Together, this gives three equations. The level of government expenditure (G) and the money supply (M) are assumed to be controlled by the government. Given values for G and M, the three equations determine the values of Y, r and π. Suppose that these functions can be written in the form: Y = cy α γr+g M = Y r π=y β r δ.

35 EXERCISES FOR CHAPTER 7 In this model, there are two government policy parameters, government expenditure, G, and money supply M. The impact of changes in these policy variables on output, interest rates and the exchange rate are given by such expressions as Y G, r M β= δ. π π, and G. Determine these policy multipliers and confirm that G = 0 if Exercise 7.20 Suppose that the supply function for a particular good is given by the equation p=α+δq β, α,δ,β> 0. The demand function for the good is given by ln(pq)= k. Shifts in the supply function are caused by changes in α,δ and β. Plot the supply and demand equations and calculate the impact on (equilibrium) price and quantity of supply shifts.