dr Bartłomiej Rokicki Chair of Macroeconomics and International Trade Theory Faculty of Economic Sciences, University of Warsaw

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1 Chair of Macroeconomics and International Trade Theory Faculty of Economic Sciences, University of Warsaw

2 Intertemporal trade and consumption Intertemporal trade - occurs when resources are transferable in time, i.e. when in the current period you can consume more than you have, or save current income in order to consume more in the future. In this model we consider an optimal choice of consumer that is related to the present and future consumption. Assuming that our present and future income is given by y 1 and y 2 respectively, a consumer faces a budget constraint in the form of: c 1 c r = y 1 y r where 1 + r is a market discount factor

3 Intertemporal budget constraint

4 Intertemporal utility function The intertemporal utility function takes a form of: U(c 1, c 2 ) = u(c 1 ) + β u(c 2 ) where 1 > β > 0 Parameter β is a subjective discount factor. Its value lower than 1 means that in our model we assume that an average consumer always prefers present consumption before future consumption (if he preferred identically present and future consumption β would equal 1). The marginal rate of substitution between period 1 and 2 equals:

5 Moreover, the intertemporal choice is optimal if we cannot get any profit from reallocation of consumption between different periods. This implies that the standard optimum condition for representative consumer takes form of: MRS = 1 + r Optimal intertemporal choice Therefore, in equilibrium there must be: u (c 1 ) =(1 + r) β u (c 2 ) intertemporal Euler equation This can be solved applying Lagrange multipliers: where the First Order Conditions (FOC) are:

6 Closed economy intertemporal equilibrium Let s consider a small economy where the sum of income of all consumers in a given period equals production. The production depends on intertemporal production possibility frontier and the slope of budget constraint. In case of closed economy there is no possibility of trade. So consumption in each period must equal production in each period. This is shown on the diagram below. c 2 y 1 (1+r) + y 2 U A y 2 = c 2 y 1 = c 1 y 1 + y 2 /(1+r) c 1

7 Determinants of an autarkic interest rate The country has a comparative advantage in the current consumption when its relative price (i.e. 1 + r) is lower than abroad. This means that a country with a low autarkic interest rate will have a comparative advantage in current consumption and will be the net exporter during the current period. Parameter describing time preferences (discount factor) - interest rate rises with impatience of the consumers (small β). Relation between future and current production (income) higher future production (income), higher autarchic interest rate. Therefore a country with a comparative advantage in the current consumption will be characterized by a high value of β (patient consumers) or low relation of the future production to the current production.

8 Open economy intertemporal equilibrium When we open our economy then we get the possibility of trade and we may achieve higher level of utility. Still, we assume that we are small open economy so our country takes the world s interest rate as given. Since there is more competition on international capital markets than autarky s capital markets it seems reasonable to believe that world s interest rate should be lower than autarky s interest rate. This leads to a change in the slope of budget constraint and a change in production level. Due to the change in the slope of budget constraint we may move to higher indifference curve (hence, get higher utility).

9 Open economy intertemporal equilibrium (2)

10 Question 1. Consider a small country inhabited by identical citizens with a twoperiod lifetime and representative household s preferences over consumption given by: U(c 1, c 2 ) = log(c 1 ) + β log(c 2 ) with 1 > β > 0 Solve the households' maximization problem. Interpret the first order condition and determine the consumption function in period 1 and period 2. Suppose that y 1 and y 2 are such that the economy runs a current account deficit in the first period. Draw a graph of the equilibrium in this case. Show graphically the gains from trade (assuming that the economy was acting under autarky in the first period). Define the autarky interest rate. Explain the underlying reason for which the country is running a current account deficit. (i.e. what is the relationship between autarky and world interest rate?). You can show it graphically or more algebraically.

11 Question 2. We analyze the model of intertemporal choice without production and investment (a pure exchange model). We suppose that the utility function is given by: U ( C 1, C 2 1 1/ σ 1 1/ σ C1 C2 ) = + β 1 1/ σ 1 1/ σ where β is a subjective discount factor. Write down the intertemporal Euler equation for the utility function and calculate the autarkic interest rate. What factors and how influence the autarkic interest rate? Determine what factors conduce a country to be a net borrower in present. Explain, why in accordance to this model, developing economies usually face current account deficits.

12 Question 3. Imagine that there are only two countries, Canibalia and Vegania whit exactly the same intertemporal utility function, given by: maxu CC 1 2 = lnc1 +β lnc Now, lets denote Y 1, Y 2 as a Canibalia present and future income and, as a Vegania present and future income. Moreover, we assume that the following condition is accomplished: Y 2 > Y 1 Y Y * 2 * 1 We analyze the pure exchange model. 2 Write down the budget constraint of both countries. Determine the level of present and future consumption if there is a possibility of international trade. Determine the level of world s interest rate in equilibrium. How would the increase in parameter β influence present consumption in both countries? Justify your answer. * Y 1 * Y 2

13 The intertemporal production possibilities curve In more realistic terms, the future income is equal to future production which in turn is a result of capital accumulation (or savings). Intertemporal production possibilities curve frontier shows all possible patterns of consumption in different periods. The lower current consumption, the greater the investment in capital goods and thus higher future production and thus also consumption. Future Future consumption consumption Current consumption Current consumption

14 Two cases of intertemporal trade Case a) Case b) C 2 C 2 intertemporal intertemporal budget constraint Q 2 budget constraint 2 U T 2 Q 2 export import U T 1 Q 1 C 1 Q 1 1 C 1

15 The benefits of intertemporal trade Thanks to access to international markets, it is possible to transfer future income to present consumption. This happens when r <r A. Otherwise, the current income would be exported into the future. Intertemporal trade is beneficial if U T > U A In the period 1: y 1 - c 1 = NX 1 <0 external debt market (negative current account balance and a positive capital account balance) In the period 2: NX 2 = y 2 - c 2 - (1 + r) NX 1 > 0 repayment of liabilities (current account surplus and a negative capital account balance) Benefits arising from the so-called smoothing consumption path.

16 Question 4. According to you, which of the following countries has the intertemporal production possibility frontier skewed towards present consumption of goods and which has the production possibility frontier skewed towards future consumption of goods? A country which receives a huge inflow of immigrants (e.g. Argentina or Canada in the XIXth century); A country which is one of the world s leaders in terms of technology, yet its advantage is decreasing systematically (e.g. Great Britain in the XIXth century or the United Stated today); A country that possess huge oil deposits that can be easily extracted (e.g. Saudi Arabia); A country that possess huge oil deposit, yet they can be extracted only after great investment (e.g. Norway that has oil deposits under the North Sea); A country such as South Korea that is quickly approaching the most developed countries.

17 Question 5. In a country A that is a small closed economy, intertemporal production possibility frontier is symmetric against the 45 degrees line (maximum level of present and future production is equal). We also know that this economy is in its long term equilibrium and that present production equals exactly a half of the maximum level of present production. Show this situation on the diagram plotting present and future production and consumption, isovalue line that assures production equilibrium and the indifference curve that assures consumption equilibrium; What will happen to the present and future production and consumption if we open the economy and the world s interest rate is higher than the autarky s interest rate. Show changes on the diagram. What can we say abort the current account balance in present and future?

18 Question 6. Country A has its intertemporal production possibility frontier skewed towards present consumption while country B has its intertemporal production possibility frontier skewed towards future consumption. Both economies are open economies and their indifference curves are symmetric against the 45 degrees line. Show the situation on the diagrams (each country separately). Plot present and future production and consumption, isovalue line that assures production equilibrium and the indifference curve that assures consumption equilibrium. What can we say about the current account balance in both countries in present and future? What will happen if the future discount factor falls? Show changes on the diagrams. What will happen if the world s interest rate falls (r*)? Show changes on the diagrams.

19 Question 7. Suppose now that individual endowments are y 1 and y 2. Assume that endowments are such that the current account is balanced and the utility function is: U(c 1, c 2 ) = log(c 1 ) + β log(c 2 ) with 1 > β > 0 If the capital stock cannot adjust immediately, a wave of immigration reduces income per capita temporarily by reducing the capital stock per head. You can model this as a fall in y 1 with y 2 unchanged. In Israel, during the early nineties they saw a big wave of immigration mainly from former Communist countries. What would you expect to happen to Israel s current account?

20 Question 8. Assume a small country has an initial stock of foreign assets B 0 <0. The latter implies that the stock of assets at the end of the first period is given by B 1 B 0 = rb 0 + Y 1 C 1. Assume it is β (1 + r) = 1. The first period endowment is y 1 and there is no endowment in period 2. The utility function is: U(c 1, c 2 ) = log(c 1 ) + β log(c 2 ) with 1 > β > 0 Write down the country s intertemporal budget constraint and the optimality condition for consumption. What would be the effect of a debt reduction (an increase in B 0 ) on the present value of the country s income. What is its effect on consumption and the current account in the first period?

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