1. Explain why the interest rate equals the marginal product of capital at an optimum in the firm's problem. Also, explain why savings in period 0 is taken as given in the dynamic general equilibrium model with production.

using sources:

As was the case in the static general equilibrium model, the firm takes the opposite side of every market. That is, it supplies output and demands both labour and capital in each period. The key object in the firm's problem is the production function. Our assumptions concerning the production function were discussed in detail in the Module 4 notes. For our purposes here, the key assumptions are (i) the production function exhibits constant returns to scale, (ii) that the marginal products of both labour and capital are positive and diminishing, and (iii) the production function is strictly concave. As was the case in Module 4, since we will be analyzing the firm's problem using calculus, we must also assume that the production function is twice continuously differentiable.

Denote the firm's production function by F(Kt,Ntd) and let zt be the level of total factor productivity in period t. This means that

Yt = ztF(Kt,Ntd), represents the firm's profits in period t.

As stated in the introduction to this version of the model, the households own the capital.

As a result, the firm simply enters into the capital and labour markets in each period and chooses how much of each factor to employ given the prices. Since there is nothing connecting the periods for the firm, these decisions are made independently. Hence the firm facesa sequence of static optimization problems. This means the representative firm's profit maximization problem is simply:

Definition 1. A competitive equilibrium is: (1) an allocation (i.e. quantities): and (2) prices, {ux, roll_o, such that: (i) the household chooses {Co, Ci, N, Ni, lo, , Si, So} to marimize its lifetime utility given its constraints, taking all prices as given, (ii) the firm chooses {Yo, Y1, NJ, NY, Ko, K, } to marimize profits given its production tech- nology, while taking all prices as given, and (ini) the goods, capital, and labour markets all clear in each period: N = N, t = 0,1, Yo = Ot + Sil t = 0,1, and Se = K t= 0, 1.All that remains is to clear the markets. From the equilibrium definition, we can see that the market clearing conditions are: for t = 0, 1, (2.22) Y = Ci+ Si+1 for t = 0, 1, (2.23) S = K. for t = 0, 1. (2.24) That's quite a lot of equations. It is convenient to substitute some of the variables out of the system so as to reduce its size. A useful reduction yields the following system of equilibrium equations: u2 (Co, lo) = zoFz(Ko, h - lo) U1 (Co, lo) (2.25) us (C1,41) = 2F2(Ki, h - 4) us (C1,4) (2.26) w1(Co, lo) = =BFI(Ki, h - h)(Ci,h) (2.27) Co + K1 = zOF(Kosh - 40) (2.28) C1 = =F(Ki, h - l) (2.29)