A Malthusian Economy with Productivity Growth. (40 points) Let us consider a Malthusian economy populated at time t with N, identical adults/workers living for one period. The size of the adult population in the next period is given by the following law of motion equation: where 71, >: 0 denotes the number of kid(s) per adult born at time /. Each adult allocates & unit(s) of time to work at every period and earns at time + y, unit(s) of good that can be allocated between consumption c and child rearing. Therefore, the budget constraint of each adult can be written as: where p denotes the cost or rearing one child. At every time &, the level of utility of each adult denoted by Up is given by the following logarithmic utility function: U. = Inc + plan; where y > 0 is a preference parameter, The aggregate level of income/output at time f denoted by *, can be described by the following aggregate Cobb-Douglas production function: Y =2X"L- where we(@ 1) is a production parameter, Z, > 0 stands for the aggregate productivity level at time 4 X denotes a fixed aggregate stock of land used in agriculture and &, > 0 stands for the aggregate labour input defined as the aggregate amount of time allocated to work: Let = eNt Let us assume that productivity grows at a constant rate a =(-1,1): Let x: stand for the land per adult at time / defined as: x, = X/N, and let y, = Y:/N, represent the output per adult at time t a. Under the assumption that each adult maximizes his utility function subject to his budget constraint with p and y, given, write-down the utility maximizing problem of the representative adult at time fand derive his utility maximizing demand for kid. (10 points) b. Using your answer to a., show that the growth rate of the aggregate population of adults from time / to (+I can be written as a function of the income per adult at time & (5 points) C. In a two-dimensional diagram, plot the relationship found in b. and identify the level of income per adult compatible with no growth in the adult population. (5 points) d. Show that the output per adult at time & can be written as a function of the productivity level at time / and the land per adult at time & How does the aggregate population of adult at time &affect the output per adult at time t? (5 points) e. Using the production function per adult found in d., derive a relationship between the growth rate of the output per adult from time / to f+1 and the growth rate of the aggregate population of adult from time / to *+/ given a. (5 points) f. Derive both the steady-state population growth rate and the steady-state output per adult compatible with no income per adult growth? (10 points)