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where Y is the total amount of output, A is the total amount of capital, and N is the total labor input, including the adjustment for labor-augmenting techno- logical change. That is, if n is the labor input supplied by the representative household and g is the rate of labor-augmenting technological change, then N= ne". With these conventions, we can write the production function as y = f(k, n). (7) where y = 1/e" is output per efficiency unit and k = Kest is capital per efficiency unit. Note that we no longer assume that the production function f (k, n) is Cobb-Douglas. We will let a denote the capital share and { denote the elasticity of substitution between capital and labor. Given competitive markets, firms earn zero profits and capital earns a before- tax rate of return r equal to its marginal product: r = f*( k,n). (8) Each efficiency unit of labor is paid a wage w equal to its marginal product, w = fn( k, n). (9) Below, in Section 4, we consider generalizations to non-competitive production settings. 2.2 Households We use a conventional, infinitely-lived representative household. The house- hold's instantaneous utility function takes the isoelastic form with curvature parameter y. To incorporate elastic labor supply, we add labor n to the house- hold's utility function. This labor variable should be intrepreted broadly to include both time and effort. The household's utility function is U= e-m (ce)1-7 (1-)(m) - 1 1-7 where v(n) is a differentiable function of labor supply and all other variables are defined as before. This functional form was introduced by King, Ploaser, and Rebelo (1988) and has been more recently explored by Kimball and Shapiro (2003). We can write the household's dynamic budget constraint in per efficiency unit terms: k= (1 - T)wn+ (1 -Tx)rk - c- gk+I, lim kertalt = 0.2 A More General Ramsey Model In this and the next three sections, we extend the basic Ramsey model along a number of dimensions. In this section we include elastic labor supply and a more general production technology. We also present a more detailed derivation of our results. To allow for elastic labor supply, we use a form of preferences over consump- tion and labor proposed by King, Plosser, and Rebelo (1988). King-Plosser- Rebelo preferences have the property that the uncompensated elasticity of labor supply is zero. This feature has the appealing implication that long-run growth caused by technological progress does not lead to a trend in hours worked. The compensated (constant-consumption) elasticity of labor supply need not be zero, however. This parameter, which we will call o, will have a significant role in some of our results. 2.1 Firms We begin with production. Assume there are many identical firms in competi- tive input and output markets, producing output with constant returns to scale technology according to the production function Y = F(K, N), The feedback depends critically on the tax rate. If the capital tax rate were 0.40 instead of 0.15, and all other parameter values are the some, the feedback from a capital tax cut would be 75 percent rether than 50 percent. The literature on taxation in the United States suggests that our choice of 7% = 7 = 0.25 is within the range of plausible estimates, although perhaps a bit conservative. Mendoza, Romin, and Tesar (1994) estimate a 40.7 percent capital tax rate (applied to corporate and non-corporate capital) for the United States in 1986, the last year of their series. This is above the estimate given by Gravelle (2004), who reports a rate of 37 percent for all capital in that year. Gravelle extends her estimates through 2003, by which point the capital tax rate had fallen to 17 percent. Mendoza et al. estimate a labor tax rate of 28.5 percent in 1980, together with a consumption tax of about 5 percent; these tax rates would combine to be equivalent to a tax on labor of about 31 percent.A. factory produces racing lakes. The rst stage is to assemble the bile: on the factoryr oor, which tak an exponentially distributed length of time with mean 1|} hours. Once amenibled, the bike is then immediately hrspected. o If the bike passes inspection {which happens at a rate of 1 per hour}. then the bike is shipped to its new owner and does not return to the factory oor or for inspection. a If it Idem not pass inspection [which happens at a rate of Ill per hour] then it is start baclt to the factory oor to be reassembled. The distribution of time it takes to reassernbie is the same as for the original assembly. This transfer between assembly and inspection wili continue indenitely until the bike passes inspection. Let I; be the status of a bike at time t1 and assume that {list :3 } is a continuous time Markov chain with state space, S, given by: S = {'being assembled or reassembled', 'being inspected', Ishipped"'}. [a] Draw a statesoace diagram for the process {Int 2 ill}. [b] Reiahel the states for notational convenience: 1 is \"being assembled or reassembled', 2 is 'being inspectod', and 3 is 'shipped'. Using ofh] notation where necessary, compute the following for VERY small It: ii] Pixa - 2Win: - 1}; [ii] P{X;. = SIXD = 1} [the answer to this is not zero]; [iii] P'IIXHH = III: = 2}: (iv) PUD. = 3|.Xn = 3). {c} 1Write down the transition matrix of the jump chain of the process {Int 3 D}. {d} Does the jump chain of the process {Xi 3 D} have an equilibrium distribution? Justify your answer. If it does have an equilibrium distribution, nd it. If it doesn't, find all invariant distributions instead. {e} What is the probability that a bilte currently being inspected will pass [and so be shipped to its new miter)? {f} How many times, on average, will the biloe be inspected before eventually being shipped? (a) Biden pic's finance director is evaluating a new four-year recovery project for Uncle Sam. You are a member of Biden pic' finance team and have been asked to find out the appropriate cost of capital figure for the company's project appraisal. Assume that the current date is 31 March 2021. the company's financial year end, and the corporation tax rate is 21%. The Balance Sheet of the company provides the following information: Ordinary shares (nominal value $1 each) E20,500,000 6% preference shares (nominal value $1 each) E4,300.000 4% irredeemable debentures (nominal value (100 each) $5,200,000 3% redeemable debentures" (nominal value [100 each) E6.500.000 *Redeemable with a 2% premium (i.e., 102% of nominal value) on 31 March 2024. The market values of the company's capital as of 31 March 2021 are as follows: Ordinary shares E3.11 per share (cum-dividend) 6% preference shares E1.20 per share (ex-dividend) 4% irredeemable debentures (105% (ex-interest) 3% redeemable debentures $101% (cum-interest) Biden pic's ordinary dividend for the year to 31 March 2021 totals $5,125,000 and this is due to be paid on 14 April 2021. Biden pic's ordinary dividends have been increasing at a steady annual rate since the year ended 31 March 2016 when they totalled $4,421,000. Biden pic's issued ordinary share capital has not changed since 2015. Required: Calculate Biden pic's existing Weighted Average Cost of Capital at 31 March 2021. (18 marks) (b) Convertible bonds had been reported as a popular type of financing in year 2020. particularly for firms badly hit by the coronavirus pandemic. Discuss the pros and cons of convertible bonds as a source of finance, especially in the face of greater uncertainty. (12 marks) (c) The directors of Joe pic are seeking to raise $5 million via a rights issue. The company currently has $5 million of issued share capital (nominal value $0.50 each), and the market price per share is currently $3 cum-rights. The directors are considering a rights issue subscription price of either $1 per share or $2 per share for new shares. Required: Calculate the theoretical ex-rights price per share if the rights issue subscription price is (i) 61 per share. (ii) E2 per share. (4 marks) (d) Explain why a rights issue generally results in a fall in the market price of shares