Review

Key Concepts convergence general solution homogeneous form initial-value problem integrating factor particular solution steady-state value

Review Questions

Review Exercises

1. Describe the two-step procedure for obtaining ihe general solution to Ihe complete, autonomous, linear, first-order differential equation.

2. Explain what is meant by the steady state of a linear, first-order differential equation.

3. Under what conditions is the particular solution equal to the steady-state solution?

4. State and explain the necessary and sufficient condition for convergence in an autonomous, linear, first-order differential equation.

5. Explain how to find ihe integrating factor for a first-order differential equation.

6. Explain how to use the integrating factor to help solve a linear, first-order differential equation.

1. Suppose that energy consumption E grows at the rate of 2% and was equal to 2 units at time /<). Solve for energy consumption as a function of lime.

2. Suppose that national income Y grows at a rate of g and national population P grows at a rate of a. Define income per capita as y = Y/P Solve for income per capita as a function of time.

3. Let Kit) be the quantity of capital available in an industry at time t. If K(0) = 500 and if the depreciation rate is 5% and ihe investment rate is a constant 100 units, solve for the expression showing the quantity of capital available as a function of time. Find the steady-state capital stock, and show that Kit) converges to the steady state.

4. A perfectly competitive firm maximizes profits by producing the quantity of output at which marginal cost equals price. Assuming that il takes time for the firm to change Ihe quantity of output it produces, let it adjust its output level in proportion to the gap between price and marginal cost. Thai is. assume that

where q is the quantity of output, p is the price of output, and MC(c/) is the marginal-cost function. Let the marginal-cost function be MCk/) = aqt where a is a positive constant. Solve this differential equation for i/(/), find the steady state for q, and determine whether q(i) converges to the steady state.

5. Modify the IS-LM model of section 21.1 by letting investment demand depend on the interest rate. That is, assume that

and assume that ()</?< I. Derive the differential equation for Y. Solve for Y(t) and determine the condition the parameters must satisfy for the equilibrium to be stable.

6. Let K' be a firm's desired capital stock. Although K' will depend in general on a number of factors, such as the desired output level of the linn, the price of output, and the price of capital, assume these factors are constant so that K' is also constant. Because it is costly for the firm to adjust its capital stock (e.g.. workers have to be diverted from producing output to installing additional machinery), the firm adjusts gradually towards K*. Specifically, suppose that the charge in the capital stock K is proportional to the gap between the desired and the cuneni capital stock, where u is the factor of proportionality thaL determines the speed of adjustment. Assuming that there is no depreciation, and assuming that the initial capital stock at / =0 is K0, write out the differential equation for the capital slock and solve it. What is the steady-state stock size? Does K(i) converge to the steady-state stock size ?

7. Suppose that the government runs a deficit, gross of interest payments on the debt, which is a fixed proportion b of national income. Then debt increases by the amount of the deficit plus the interest payments on the debt. If r denotes the constant interest rate paid on the debt, then ihe differential equation for debt is

Assuming that Y = f>Y, DO)) = Z>0, and >'<0) = Yq, solve for the ratio rD(t)/Y(t). Show that this ratio converges to a finite limit if and only if the growth raie of income exceeds the interest rate.

8. Suppose that a government always runs a budgetary deficit equal to 12% of national income including interest payments. If national income is growing at a rate of 2%. will this government always be able to meet its interest payments on the debt? If the inierest rate is a constant 10%. what share of national income will go toward servicing (paying interest on) the national debt in the limit ?

Use the information in review exercise 3. but now let investment grow ai the rate <; from an initial level of /<i ai time i = 0. Solve for the expression that shows capital as a function of time. Dews Kit) converge to a finite limit''

878 CHAPTER 21 LINEAR, FIRST-ORDER DIFFERENTIAL EQUATIONS

10. In the growth model of section 21.2, assume that a = 0 and solve the new differential equation without using the integrating factor.

11. Suppose that the demand for wheat is given by q n — A + Bp hut the supply is given by qs = F + Gp + H(l p > 0

where the last term reflects productivity growth over time. That is, the supply curve tor wheat shifts (smoothly ) out over time because of continuous technological improvements in production. Assume that price adjusts if there is excess demand or supply according to p = a(ql> - qs), Solve this differential equation for p(t) given p(0) = p().

Chapter 22

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