Explain the following questions
Question 28 - Question 31: Use the income-expenditure model to answer the questions. Suppose a small open economy with fixed prices can be described by the following equations: Consumption: C = 600 + 0.7YD Government spending: G = 300 Planned investment: IPlanned = 500 - 400i Taxes: T = 60 + 0.06Y Exports: X =600 - 10EFC/DC Transfers: TR = 280 -0.04Y Imports: IM = 214 + 0.03Y + 25EFC/DC NOTE: YD = disposable income. Unless otherwise stated, interest rate, i, is held constant at 0.1 (i.e., 10%) and exchange rate, EFC/DC, is held constant at 4 (FC per DC). For each question, keep your answer to 2 decimal places if necessary and be sure to show your work. Question 28 Find the equilibrium level of output. Output: Question 29 (Continued from Question 28) Find the country's trade balance. Does the economy have a trade surplus or trade deficit?Problem 3. Let X, Y be two linear spaces. A linear mapping or linear map or linear transformation T: Xr Y takes any vector x E X as input and produces a vector y E Y as its output in a linear way: If y = TX, v = To then ax + do) = a'Tx + 3TH so the output is try + 3!: namely T preserves linear combination in its output. As simple consequences {i} oT+ DS is a linear map or linear transform if T.S are linear maps (ii) If T is linear map than T0 = 0. where D on the left is the zero 1lrector in X, and D on the right is the zero vector in Y. If X= R", Y = R\Problem 5. Let X, Y be two linear spaces with bases X =[x1.x2. x} and 'h'={y1.y3. m, ym}. so dim X=n, dim Y=m. Let T: X-r Y be a linear map. We say that the m-by-n matrix A is the representation of T over the two bases if TX = VA. This is the defintion for matrix A, and it can be used directly to gure out A for given ERA". Alternatively. to figure out A, we rewrite this matrix equation column by column: ij ='v'aj where aJ- is the j-th column of A. For different applications, we might find one of the two methods more convenient to use than the other: {a} Let X = Y = IF\" . Determine the matrix A to represent T= % over the standard basis {1. t. t2. t"' 1}. (bl Determine the matrix A to represent T of Problem 3(a) over the standard basis {a 1. e2] for X = R2 and {911 e2. e3}for Y= R3. [c] Let X= Y= B\". Let the linear map T: X r Y be a nby-n matrix. Let a basis of R" be contained in a n-by-n matrix 5 so it is invertible and its columns are the basis. Represent T by matrix A over the basis specified by 5. Note: strictly follow definition and you can use 5' 1 in the formula for A. Warning: show your steps leading to A. not just the nal result plus its verification. The table below shows Hanne's demand for mini cupcakes. a) Complete the table below. [4 marks] Quantity MU TU 56 2 100 30 151 167 6 8 8 170 b) At what quantity is total utility maximized? [1 marks] c) What is the value of marginal utility at this quantity? (1 mark] d) What do your answers to (b) and (c) tell us? [2 marks]
