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- Suppose the return and cost of entrepreneurship curves are described by the following equations (with numbers measured in the thousands): R= 600 – 100N C= 200 + 150N, where R= returns to entrepreneurship, C= cost of entrepreneurship, and N= number of entrepreneurs.Consider a Diamond model with no technology growth (g = 0), a Cobb-Douglas production function Y = K (AL)(1-a), and a logarithmic utility function (0 = 1). Suppose the government taxes each young individual an amount T and uses the proceeds to pay benefits to old individuals; thus each old person receives (1 + n)T. Find the Euler equation.a) A planner in a small, open economy has a utility function U = x1X2 and production takes place according to the production functions Y1 = 10L, – 0.5L Y2 = 10L2 – L The world prices of the goods are p1 10 and p2 5. The total amount of labour %| %3D available in the economy is 5 units. (i) Formulate the Lagrange equation. (ii) Write down the Karush-Khun-Tucker conditions. (iii) Determine a possible solution.
- A closed economy which is shown below, endogenous variables are Y, I, C, i and exogenous variables are Go and Mo and parameters are a,b,t,d,e, k. Y= C+l+G C= a+ b(1-t)Y |= d-ei G= Go Md= Ms Ma= ky-li Ms=Mo Calculate the equlibrium level of I by using inverse matrix rule.1. Consider the deterministic Brock-Mirman model studied in class. The discount factor is β ∈ (0, 1), the per-period utility function is u(c) = log(c) and the production function is f(k) = kα + (1 − δ)k, where δ = 1 and α ∈ (0, 1). Assume β = 0.99 and α = 0.3. (a) Compute the model's steady-state k∗. (b) Use the F and G that we derived in class by solving the Bellman equation by guess and verify to plot the optimal decision rules for k and c in the range [0.005, ¯k], where ¯k is the upper bound on the state space. Can you figure out such ¯k. If not, set ¯k to twice the steady-state capital stock. (c) Let the initial capital stock be k0 = 0.5k∗. Compute the time series for kt, t = 1, 2, ..., 200. Plot the time series. Now do the same for k0 = 1.5k∗. Do the sequences converge to k∗? 2. Now solve the model in question (1), with the same parameter values, by…Consider the path of output per person over time in the Romer model shown below, where the y-axis is the natural log of output per person (which is equivalent to using a ratio scale on the vertical axis). population Output per Person ideas efficiency parameter 3.5 3 2.5 0.5 O share of labor engaged in production saving rate O growth rate of the population 0 0 2 ——————- 4 Time At time t = 3, the change in the trajectory of this economy can be explained by a change in one of the parameters of the Romer model, namely an increase in 6 8 10 Pa
- In a Heckscher-Ohlin model, if capital-owners are relatively better off, then it is consistent with the labour- capital amount (L/K) demanded becoming for the capital-intensive good. Paragraph v B I U ...Consider the one-sector Schumpeterian model in discrete time analyzed in the previous exercise, except that now (.) denotes the probability of innovation, and each innovation improves the quality of a machine q to λq, where λ > 1. Suppose that when a new innovation arrives a fraction ϕ of workers employed in the final good production are unable to adapt to this new technology and need to remain unemployed for one time period to “retool.” (a) Define the equilibrium and steady-state (BGP) allocations. [Hint: also specify the number of unemployed workers in equilibrium.] (b) Define the appropriate generalization of the steady state for this economy, and determine the number of unemployed workers in this equilibrium. (c) Show that the economy experiences bursts of unemployment followed by periods of full employment. (d) Show that a decline in ρ increases the average growth rate and the average unemployment rate in the economy.Assume a Heckscher-Ohlin model. Home and Foreign produce food (F) and machinery (M) using labor (L) and capital (K) as the unique inputs in both sectors. Both countries face the same technologies. In both countries the input requirements are fixed at akf = 2, aLF = 1, aKM = 2, and aLM = 3 (i.e., 3 units of labor are required to produce one unit of machinery etc.). Home is endowed with K = 800 units of capital-hours and L= 600 units of labor-hours. Foreign is endowed with K* = 600 units of capital-hours and L* = 400 units of labor-hours. The output of food is denoted by QF. The output of machinery is denoted by QM- a) Determine Home's capital constraint. Home's capital constraint is b) Which of the two goods is capital-intensive in the production? is capital-intensive. c) Which output mix will be realized in Home if it uses all its resources? The optimal output mix is units of food and units of machinery. d) Which good will be exported by Home? Home will export
- Consider now the two-period model in general equilibrium, so that prices, investment, and labor supply are endogenous, i.e. the production economy. Analyze and carefully explain graphically and in words the general equilibrium effects of a decrease in TFP for a benchmark economy with no frictions.Based on two period model: population growth results in a decrease in the relative amount allocated to the second period and a lower present value of the marginal user cost. True or False?Exercises 1 (Solutions on p. 217) 1.1 Consider the growth equation i = tu, with x(0) = 0, x(1) = 1, and with the cost function J = u? dr. Show that u = 3t is a successful control, with x* = r and J* = 3 the corresponding trajectory and cost. If u = u* + v is another successful control, show that vt dt = 0, and, by finding the cost for this control, show that u* is the optimal control.