1 Two Period Production Economy

Transcription

1 University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout # 3 1 Two Period Production Economy We shall now extend our two-period exchange economy model to incorporate production decisons as well. As before, consider a closed economy inhabited by a representative agent who lives two periods. You could think of this economy instead as being inhabited by a continuum of identical agents of measure one. Let the agent s preferences be given by V = u (c 1 ) + u (c 2 ) where the utility function u is continuous and strictly concave with u 0 > 0; u 00 < 0. c 1 denotes consumption now and c 2 is consumption tomorrow. < 1 denotes the discount factor which captures how much the agent values today relative to tomorrow. Suppose the agent receives the endowment stream y 1 now and y 2 tomorrow. We shall continue to assume that agents operate in competitive markets for goods. Let us use p 1 and p 2 to denote the (utility based) market prices of these goods. But now, there also exists a technology to convert the non-storable good into capital which can be used to produce more non-storable goods tomorrow. In particular, there exists a technology which can convert k units of investment today into y units of the good tomorrow with y = f (k) We shall assume throughout that the production function f is increasing and strictly concave in k, i.e., f 0 > 0 and f 00 < 0. Firm s rent capital from households to produce output in period 2. There goal is pro t 1

2 maximzation. The pro ts of a typical rm are given by = p 2 f (k) qk where q is the rental price of capital. Equivalently, the rm could buying the capital directly in which case q would be the price of capital. Of course, in this event we would have to add on the end-value of capital (the left over capital after production and depreciation) to the value of the rm. Firms are owned by the households. Hence, the representative agent s budget constraint is p 1 c 1 + p 2 c 2 p 1 (y 1 x) + p 2 y 2 + p 2 (1 ) x + qx + where x denotes the total amount of investment in capital made by the household. Here is the rate of depreciation of capital. Note that this formulation implies that the capital after current production and depreciation can be converted one-for-one into current output. Clearly, it must be the case that p 1 = q + p 2 (1 ) since the cost of creating capital and the net returns that it fetches must be the same given competitive markets. Before proceeding we will rst again de ne the competitive equilibrium for this economy as a set of allocations fc 1 ; c 2 ; x; kg and prices fp 1 ; p 2 ; qg such that: (a) the allocations are consistent with agents maximizing their utility given their budget constraints taking these prices as given; (b) rms maximize pro ts given prices and technology; and (c) at these prices all markets clear. The market clearing condition in this economy follow the same principle as in the pureexchange economy we analyzed earlier except they get modi ed as c 1 + k = y 1 c 2 = y 2 + f (k) + (1 ) k x = k The third condition essentially says that the supply of capital (x) must equal its demand (k). The rst two conditions are just analogs of the conditions in the pure exchange case. 2

3 The representative agent s utility maximization problem is to maximize: L = u (c 1 ) + u (c 2 ) + [p 1 (y 1 x) + p 2 [y 2 + (1 ) x] + qx + p 1 c 1 p 2 c 2 ] where is the Lagrange mulitplier on the agent s budget constraint. Maximizing this with respect to c 1 and c 2 taking p 1 and p 2 as given gives the standard necessary conditions for utility maximization u 0 (c 1 ) = p 1 u 0 (c 2 ) = p 2 p 1 = q + p 2 (1 ) where we have ignored the binding budget constraint condition which always arises under our preferences. As before, combining the rst two conditions gives u 0 (c 2 ) u 0 (c 1 ) = p 2 p 1 : Note that the third condition is one that we anticipated earlier: the net return to capital must equal its production cost. Pro t maximization by rms implies that at an optimum we must have p 2 f 0 (k) = q; that is, the marginal bene t from renting capital must equal the rental cost of capital for a pro t maximizing rm. Noting the fact that q = p 1 p 2 (1 ) from above, we can rewrite this condition as p 1 p 2 = f 0 (k) + 1 In order to solve for the equilibrium of this economy we need to do two things. First, we need to combine the household and rm optimality conditions to get u 0 (c 1 ) u 0 (c 2 ) = [f 0 (k) + 1 ] This essentially says that both households and rms must be optimizing at the given set of prices. Hence, the marginal rate of substitution for households between c 1 and c 2 must equal the marginal rate of transformation of the two goods by the rm. 3

4 Second, we now need to impose the market clearing conditions. Substituting these into the above gives u 0 (y 1 k) u 0 (y 2 + f (k) + (1 ) k) = [f 0 (k) + 1 ] This is e ectively one equation in one unknown and provides the general solution for k in terms of the exogenous variables y 1 ; y 2 and the parameters of the model. One we have the solution for k then from the market clearing conditions we also have the solutions for c 1 and c 2. The solution for the equilibrium price ratio then is determined by substituting these solutions into the household s rst order condition u 0 (c 2 ) u 0 (c 1 ) = p 2 p 1 : This expression provides a solution for the equilibrium price ratio p 2 p 1 in terms of the primitives of the economy. At this price ratio, the allocations c 1 and c 2 are consistent with both optimality and market clearing as is the allocation for k. It is also useful to describe the Pareto optimum in this case. The planner will maximize the representative agent s lifetime utiltiy subject to the feasibility constraints c 1 + k y 1 ; c 2 y 2 + (1 ) k + f (k) : This optimum is often easier to solve than the competitive equilibrium. Since the welfare theorems apply in economies such as ours without distortions, externalities and information frictions, we often set up the planning problem instead of the decentralized version. The planning problem is easier to solve because it doesn t have any prices. 2 Lucas Critique We now turn to a variant of the production economy analyzed above to illustrate one of the most fundamental points made by Robert Lucas Jr. back in 1976 which revolutionized macroeconomics and ushered in modern micro-founded macroeconomics as the prevailing paradigm for its study. 4

5 Consider our two-period representative agent economy but with the following twist. Suppose there is no capital but now the representative agent has a time endowment of one unit which can be allocated to either work or leisure. Let n denote work e ort. Thus, the agent s welfare is given by V = ln c 1 + ln c 2 + ln (1 n 1 ) + ln (1 n 2 ) The agent receives wage income w per unit of time worked. She takes these wages and the interest rate r as given when choosing her consumption and labor supply. The government imposes a wage tax at the at rate 1 and 2. Hence, her budget constraint is c 1 + s = (1 1 ) w 1 n 1 c 2 = (1 2 ) w 2 n 2 + () s The rst order conditions for the agent s optimal consumption and labor supply decisions are c 2 = () c 1 c 1 = w 1 (1 1 n 1 1 ) c 2 = w 2 (1 1 n 2 1 ) Combining the last two conditions with the Euler equation relation (the rst condition) then gives 1 n 2 = (1 1) w 1 () 1 n 1 (1 2 ) w 2 Now, the agent s lifetime budget constraint (which can be derived by combining the two period budget constraints) gives c 1 + c 2 = (1 1) w 1 n 1 + (1 2) w 2 n 2 Using the Euler equation the lifetime constraint reduces to c 1 (1 + ) = (1 1 ) w 1 n 1 + (1 2) w 2 n 2 5

6 which solves for optimal consumption in period 1. The Euler equation can then be used to derive c 2 directly. This expression reveals the previously noted feature of optimal consumption decisions: agent s base their consumption decisions on the present discounted value of lifetime income rather than current income. Optimal labor supply in period 1 can then be derived from the relation 1 n 1 = c 1 = (1 1 ) w 1 : The optimal n 2 follows from the relationship 1 n 2 1 n 1 = (1 1) w 1 (1 2 ) w 2 (). Thus, all relevant decision variables have been determined as functions of w 1 ; w 2 ; 1 ; 2 ; r and the parameters of the model. Now suppose the government attempts to determine the e ect of a change in the period 1 wage tax on employment. The straightforward approach would be to use the optimal solutions for n 1 and n 2 and determine the comparative static e But this will lead to erroneous conclusions for two reasons. First, what the government wants to determine is the e ect on employment, not just the e ect on labor supply. But to gure out the e ect on employment we also need to know what the labor demand schedule looks. In particular, the optimal labor supply decisions take as given the wages w 1 and w 2. But these will change when labor supply changes. By how much and in which direction? That depends on the slope of the labor demand function. To determine that we need a general equilibrium model. This partial equilibrium structure will not reveal the correct answer. The analog in actual policymaking is that running crosssectional regressions of labor hours on wages and taxes will not reveal the answer. One needs a full-blown general equilibrium model for this question. Second, the agent s labor supply decision depends not just on 1 but also 2. Hence, attempting to determine the e ect of a change in 1 on n 1 without also specifying what will the policy do 2 will lead to wrong conclusions. We need to specify the full path of the policy change proposed in order to be able to evaluate its e ect on employment. Again, the partial equilibrium approach will not work nor will the cross-sectional regression approach. 6

7 3 Ricardian Equivalence Yet another powerful yet controversial economic question relates to the issue of government budget de cits and their e ects on the macroeconomy. This issue has been particularly topical over the last year in light of the large scal stimulus that has been injected into most of the industrial economies recently. A benchmark result to judge outcomes against is provided by a result known as Ricardian Equivalence. It is named after David Ricardo even though it owes its modern incarnation to Robert Barro. Ricardian Equivalence holds that for a given path of government spending changes in the time path of taxes have no e ect on the economy, i.e., tax policy is ine ective. The argument is easy to illustrate using just the intertemporal budget constraints of the government and private agents in the context of our two period exchange economy model. Let T denote taxes and G denote government spending. The key notion here is that the government needs to balance its budget intertemporally even though there may be de cits or surpluses in any given time period. The government s intertemporal budget constraint is G 1 + G 2 = T 1 + T 2 Suppose that T denotes lump-sum taxes and let G 1 and G 2 be given constants. The private agent s corresponding intertemporal constraint is c 1 + c 2 = y 1 + y 2 T 1 T 2 where the right hand side just gives the present discounted value of the agent s lifetime income net of taxes. Let the agents lifetime utility be given by V = ln c 1 + ln c 2 Since taxes are lump sum, the agent s optimal decisions, which are based on marginal assessments of changing choice variables, are una ected by them. Speci cally, we shall continue to have c 2 c 1 = () = R 7

8 The market clearing conditions for goods in periods 1 and 2 dictate that we must have c 1 = y 1 G 1 c 2 = y 2 G 2 Hence, the equilibrium interest rate for this economy will be which clearly is independent of T 1 and T 2. yields R = y 2 G 2 (y 1 G 1 ) Combining the government s intertemporal constraint with the private agent s constraint c 1 + c 2 = y 1 + y 2 G 1 G 2 : This expression makes clear that for a given path of government spending, private consumption is completely una ected by changes in T 1 and T 2 (recall that r is also independent of taxes as we showed above). Hence, varying taxes will have no e ect on the aggregate economy. Intuitively, for a given path of government spending, a cut in T 1 would imply that T 2 has to rise to balance the budget intertemporally, i.e., dt 1 = dt 2. So how would private agent s 1+r react to the tax cut in period 1? Anticipating the tax increase tomorrow private agent s will save the additional disposable income today in the form of government bonds which pay the going interest rate r. The government, which nances its tax cut by borrowing, will hence be able to sell its de cit nancing bonds. When the next period rolls around the bonds would have to be redeemed with additional interest payment r. the government would do so by raising T 2. Private agents would pay up and the government would turn around and hand these revenues right back to them by redeeming their bonds!! This extreme result of tax ine ectiveness is clearly sensitive to some key assumptions that were made above. Just two examples of these are: (a) if taxes were distortionary rather than lump-sum, then private behavior would be a ected by changes in taxes; (b) if there were heterogeneity amongst agents so that the tax relief went to some groups while the tax hike falls on others then the equivalence result would disappear again. 8