Calculus of Variations: Maximizing Utility and Excess Burden of a Tax, Study notes of Agricultural engineering

Download Calculus of Variations: Maximizing Utility and Excess Burden of a Tax and more Study notes Agricultural engineering in PDF only on Docsity! Lecture XX Applications of Calculus of Variations I I. Example 7: Maximize Utility over Time A. An individual seeks the consumption rate over some time T. The utility function is assumed to obey the typical conditions U’(x)>0, U’’(x)<0: max ( ( ))e U C t dtrt T −∫ 0 subject to a cash flow constraint. B. The cash flow constraint. 1. Assume that the individual receives current income v(t) and a return on capital ik(t) where k(t) is the level of capital invested. 2. The uses of income are consumption c(t) and the change in investment k’(t). 3. Thus, the cash flow equation becomes v t ik t c t k t( ) ( ) ( ) ’( )+ = + with boundary conditions k(0)=k0 and k(T)=kT. C. Solution: c t v t i k t k t e U v t i k t k t dtrt T ( ) ( ) ( ) ’( ) max ( ( ) ( ) ’( ))= + − ⇒ + −−∫ 0 The Euler equation is then formed as: f t k t k t e U v t i k t k t f ie U C t f e U C t rt k rt k rt( , ( ), ’( )) ( ( ) ( ) ’( )) ’( ( )) ’( ( ))’ = + − ⇒ = = −    − − − Using the integral approach [ ] i e U C s ds d e U C t d s ds i e U C s ds e U C t e U C t e U C t i e U C s ds e U C t rs t t rt t t rs t t r t rt rt rs t t r t − + −+ − + − + − − − + − + ∫ ∫ ∫ ∫ = − = − + + = + + ’( ( )) ’( ( )) ’( ( )) ’( ( )) ’( ( )) ’( ( )) ’( ( )) ’( ( )) ( ) ( ) ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ 1. Interpretation e U C trt− ’( ( )) is the marginal utility of consuming an additional dollar at time t. While i e U C s ds e U C trs t t r t− + − +∫ +’( ( )) ’( ( ))( ) ∆ ∆ is the marginal value of the postponed dollar. Implicitly, i e dsrs t t − + ∫ ∆ is the small change in dollars resulting from investing the dollar for consumption later rather than consuming it in the current period. Thus, i e U C s dsrs t t − + ∫ ’( ( )) ∆ is the change in utility from investing the dollar of consumption and receiving the consumption value in the next period. Similarly, U C t e r t’( ( )) ( )+ − +∆ ∆ is the marginal value of consumption at the next increment in time. 2. Back to the Euler equation: [ ] ( ) d d t e U C t r e U C t e U C t C t i e U C t r e U C t e U C t C t i r U C t U C t C t i r U C t C t U C t rt rt rt rt rt rt − = − ∴ = − − = − − = − − − − − − ’( ( )) ’( ( )) ’’( ( )) ’( ) ’( ( )) ’( ( )) ’’( ( )) ’( ) ’( ( )) ’’( ( )) ’( ) ’’( ( )) ’( ) ’( ( )) Given that -U’’/U’>0, C’(t)>0 if I>r. Remember that I is the return on capital and r is the impatience. Thus, you accumulate capital if the interest rate is greater than the impatience rate. Letting U(C(t))=ln(C(t)): U C t C t U C t C t U C t C t U C t C t C t i r C t C e i r t ’( ( )) ( ) ’’( ( )) ( ) ’’( ( )) ’( ) ’( ( )) ’( ) ( ) ( ) ( ) ( ) = = − − = = − ⇒ = − 1 1 0 2 II. Levhari, D. and E. Sheskinski. “Lifetime Excess Burden of a Tax.” Journal of Political Economy 80(1972): 139-47. A. Concept of Excessive Burden. 1. For any individual, the excess burden of a tax is his loss of consumer’s surplus due to the inefficiency imposed by the tax. 2. It is the purpose of this paper to introduce a new, dynamic idea of excess burden based on intertemporal maximization of utility. a. Thus, we propose to define the excess burden of a tax as the loss of consumer’s surplus summed, with the appropriate discounting, over the lifetime of the individual. b. In principle, the Harberger formula (1964) may also be applied to the intertemporal case, interpreting some of the different goods as the same goods supplied in different time periods. There are, however, some special features of the intertemporal case which require elaboration: A. One application of calculus of variations is the design of fishing policies. B. Typically, the change in the fish population is modeled as some function of current fish stock. For example, using the logistic function: dx dt rx x k = −  1 where r is the intrinsic growth rate and k is the carrying capacity. Adding fishing to this equation of motion: d x d t rx x k qEx F x rx x k h t qEx = −   − ⇒ = −   =     1 1( ) ( ) natural reproduction harvest C. To integrate these into a model of fishing we construct a profit function for each unit of effort [ ]R t ph t c E t t∆ ∆= −( ) ( ) where R is the revenue for the increment in time ∆t, p is the price of the output, c is the cost of effort and E(t) is the level of effort. Rewriting the harvest variable as a function of effort so that h(t)=G(x(t)) we get [ ] [ ] R t pG x t c E t t p c x t h t t c x t c G x t G x t h t E t ∆ ∆ ∆ = − = − ≡ ≡ ( ( )) ( ) ( ( )) ( ) ( ( )) ( ( )) ( ( )) ( ) ( ) Therefore, the present value of a harvest scenario becomes [ ]PV e p c x t h t dtt= −− ∞ ∫ δ ( ( )) ( ) 0 D. To integrate the equation of motion, we solve the fish stock equation backward yielding: d x d t x t F x t h t h t F x t x t = = − = − ’( ) ( ( )) ( ) ( ) ( ( )) ’( ) Therefore the planning problem becomes [ ]( ) [ ]( ) ( ) [ ] [ ] PV e p c x t F x t x t dt f t x x e p c x t F x t x t f e c x t F x t x t p c x t F x t f e p c x t e c x t F t t x t x t t = − − = − − ⇒ = − + −         = − −      − − ∞ − − − − ∫ δ δ δ δ δ ( ( )) ( ( )) ’( ) ( , , ’) ( ( )) ( ( )) ’( ) ’( ( )) ( ( )) ’( ) ( ( )) ’( ( )) ( ( )) ’( ( )) ’ 0 ( ) [ ]{ } [ ]{ }( ( )) ’( ) ( ( )) ’( ( )) ( ( )) ’( ) ’( )x t x t p c x t F x t e p c x t c x x tt− + − = − +−δ δ Solving the Euler equation: [ ]− + + − = − + − − + = c x t F x t c x t x t p c x t F x t p c x t c x t x t c x t F x t p c x t F x t ’( ( )) ( ( )) ’( ( )) ’( ) ( ( ( ))) ’( ( )) ( ( )) ’( ( )) ’( ) ’( ( )) ( ( )) ( ( )) ’( ( )) δ δ