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Suppose we have a constant returns to scale production function with capital, labor, and land, X, which is in fixed supply. The production function is: Y = K"[A(t)L(t)18xx The labor force grows at rate n and labor-augmenting technological progress grows at a fixed rate p. a. What is the sum of the exponents (a + B + A)?. b. Dividing both sides of the production function by L, show that you get the following result: y = k"A x where y, k, and x are measured per worker. c. Show that this implies the following equation (take logs and differentiate):. By = ag + Bp - An where "g" denotes the growth-rate. d. Finally, assume that we are in a steady state where y and k grow at the same rate g*. Find the value for g*. What conditions are necessary for there to be growth in per-capita output in the steady-state? Explain your results intuitivelyConsider an individual making choices over two goods,x and y with initial prices px= 5 and py= 2, with incomel=100: A) If the individual's preferences can be represented by the utility function u= 4x+ 2y; find the income, substitution and total effects of a decrease in the price of x to px= 3. B) If the individual's preferences can be represented by the utility function u= min(4x,2y); find the income, substitution and total effects of a decrease in the price of x to px= 3. C) If the individual's preferences can be represented by the utility function us4xy, find the income, substitution and total effects of a decrease in the price of x Px= 41. Consider an agent with preferences represented by the utility function: u (x,y) =xy 4 (e) Give an example of another utility function that represents the same preferences that u (x,y) = xy represents. Explain why your example matches the preferences repre- sented by u (x,y) = xyz. Why can two functions represent the same preferences? (5 points) (F Are these preferences rational? Continuous Strongly or weakly monotone? Strictly convex. Define cach property and then explainsprove your answer for each property (14 points)4. Consider an agent with preference relations defined over the non-negative real numbers, i.e., (x, y) ER, , represented by the following utility function: u(x, y) = a log(x) + (1 -a) log(y), if x>0 and > >0 a E (0, 1) fixed -oo, otherwise Argue each of the following items: (a) The preference relations represented by u are strictly convex (when (r.y) >> (0.0)). (b) The preference relations represented by w/ are strongly monotone (when (x. v) )> (0.0)). (c) it represents the same preference relations as utility function v(r.very " ae (0, 1) fixedQuestion : Consider the simple Solow Model without technological change. The production possibilities in an economy can be captured with a constant returns to scale production function, Y=KOL1- where Y is the total output in the economy, K is the capital stock and L is the total labor (workers) in the economy. The total labor force grows at an annual rate n. Capital depreciates at an annual rate d (so that a fraction 1-d of capital is left after every year). Consumers in the economy save a constant fraction s of their income. 1. Show that in this model, capital per worker grows according to the equation, k = sy - (n + )k . where k is the capital per worker, and y is the output per worker. 2. Explain, using the Solow diagram, why the capital per worker would converge to a stead state value in this model. 3. Suppose the savings rate increases. Explain, using the Solow diagram, what would happen to the steady state level of income per worker, and the growth rate of income per