21 +22= 1. There is a world stock of capital, equal to 1, to be allocated between the two countries. The production function is f(ki) = k₁, k, standing for per capita capital invested in country i = 1,2. Production factors are paid their marginal produc- tivity. Capital moves freely between the two countries. Denote its net price by p. The representative consumer in country i = 1,2 owns k, units of capital. Country i, i = 1,2, taxes capital at rate 1₁. Tax revenue is paid out as a transfer to the representative con- sumer. Country i chooses t, so as to maximize the net income of its representative con- sumer: y = f(ki) - f'(k;)k; + pk, + tik; a. What are the first-best tax levels 1₁ and 12 in this setting? b. Express the demand for capital in country i as a function of p and t,. Using the equi- librium condition of the international capital market, Aiki + Azk2=1, solve for p as a function of tax rates and population shares. If increases, how will p react? How do you interpret this result? c. Show that each country's objective function can also be written as f(ki) + p[ki - ki]. d. Find the first-order condition for country i's choice of the tax rate, and show how the sign of the tax rate depends on the net exporting position of the country. e. Suppose now that ₁ = ₂ =1, k₁ = 1 and ₂ =2. Draw the tax response functions, and find the equilibrium taxes and the corresponding values of p, k₁, and k₂. Is this equilibrium outcome efficient? How can you measure the inefficiency of tax competi- tion?
21 +22= 1. There is a world stock of capital, equal to 1, to be allocated between the two countries. The production function is f(ki) = k₁, k, standing for per capita capital invested in country i = 1,2. Production factors are paid their marginal produc- tivity. Capital moves freely between the two countries. Denote its net price by p. The representative consumer in country i = 1,2 owns k, units of capital. Country i, i = 1,2, taxes capital at rate 1₁. Tax revenue is paid out as a transfer to the representative con- sumer. Country i chooses t, so as to maximize the net income of its representative con- sumer: y = f(ki) - f'(k;)k; + pk, + tik; a. What are the first-best tax levels 1₁ and 12 in this setting? b. Express the demand for capital in country i as a function of p and t,. Using the equi- librium condition of the international capital market, Aiki + Azk2=1, solve for p as a function of tax rates and population shares. If increases, how will p react? How do you interpret this result? c. Show that each country's objective function can also be written as f(ki) + p[ki - ki]. d. Find the first-order condition for country i's choice of the tax rate, and show how the sign of the tax rate depends on the net exporting position of the country. e. Suppose now that ₁ = ₂ =1, k₁ = 1 and ₂ =2. Draw the tax response functions, and find the equilibrium taxes and the corresponding values of p, k₁, and k₂. Is this equilibrium outcome efficient? How can you measure the inefficiency of tax competi- tion?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Consider two countries, 1 and 2, with population shares λ1 and λ2 such that
f. Express equilibrium taxes as functions of the parameters λ1, λ2, 1, and 2, and show that the equilibrium taxes must be of opposite signs.
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Step 1: Define the term Objective Function
VIEWStep 2: a. Determine the First-best tax levels (t₁ and t₂)
VIEWStep 3: b. Analyse the Demand for capital and price of capital
VIEWStep 4: Show the Alternative representation of objective function
VIEWStep 5: Show the First-order condition for tax rate and net exporting position
VIEWStep 6: Show the Symmetric case with equal population shares and tax bases
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